editors: nobody
A THREE-MANIFOLD WITHOUT CONJUGATE POINTS
AND WITHOUT A NONPOSITIVELY CURVED METRIC
A complete and self-contained proof.
NOTATION CONVENTION
R^n : n-dimensional Euclidean space
S^1 : the circle (1-dimensional)
T^2 : the two-torus (surface)
Sigma : compact, oriented genus-one surface with one boundary circle
M : the closed 3-manifold we build
sec : sectional curvature
<v,w> : Euclidean inner product; |v| : Euclidean norm
R : Riemann curvature tensor
D_s : covariant derivative along a curve, parameter s
T, gamma': tangent to a geodesic gamma
p^T : transpose of matrix/vector p
v' : derivative of a function v
Inline math uses ... like LaTeX. The file is plain text.
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PART 1. TERMINOLOGY
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Short, operational. Skip if you already know these.
Manifold. A Hausdorff, second-countable topological space locally
homeomorphic to R^n. "Closed" means compact with no boundary. A
closed 3-manifold is locally R^3.
Riemannian metric g. A smooth assignment, at each point p, of a
positive-definite inner product on the tangent space T_p M, varying
smoothly in p. In coordinates, a positive-definite symmetric matrix
g_{ij}(x). It gives lengths: length(gamma) = integral of
sqrt(<gamma'(s), gamma'(s)>) ds.
Geodesic. A curve gamma(s) with D_s gamma'(s) = 0, where D_s is the
covariant derivative. Locally a shortest path. Picard-Lindelof
gives a unique geodesic for each initial (position, velocity).
Sectional curvature. For each point and each 2-plane sigma in the
tangent space at that point, a number sec(sigma). In dimension 2 it
is the ordinary Gaussian curvature. Sphere: sec > 0. Plane: sec =
0. Saddle: sec < 0. "sec <= 0" means every plane at every point has
sec <= 0.
Jacobi field. Along a unit-speed geodesic gamma with tangent T =
gamma', a smooth vector field J(s) satisfying
D_s^2 J + R(J, T)T = 0. (Jacobi equation)
The Jacobi fields form a vector space of dimension 2n. Jacobi fields
parametrize infinitesimal families of geodesics: J(s) encodes the
infinitesimal displacement, at parameter s, between gamma and a
nearby geodesic.
Conjugate points. gamma(a), gamma(b), a < b, are conjugate along
gamma if there is a NONZERO Jacobi field J with J(a) = 0 AND J(b) = 0.
Easy implication "sec <= 0 ==> no conjugate points." If sec <= 0
then <R(J,T)T, J> <= 0. The Jacobi equation reads D_s^2 J =
-R(J,T)T, so the second derivative of |J|^2 is 2|D_s J|^2 + 2<R(J,T)T,
J> >= 0. Hence |J(s)|^2 is a nonnegative convex function of s. A
convex nonnegative function with a zero at a returns to zero again
only at a. So no nonzero Jacobi field dies twice.
No focal points. A metric has no focal points if, for every NONZERO
NORMAL Jacobi field J (meaning <J,T> = 0) along a unit-speed geodesic
with J(0) = 0, one has (d/ds)|J(s)|^2 > 0 for all s > 0. Strictly
between "no conjugate points" (weaker) and "sec <= 0" (stronger).
CAT(0) space. A complete, geodesic metric space all of whose
triangles are at least as thin as their Euclidean comparison
triangles. Examples: R^n, hyperbolic space H^n, trees. If
(M,g) is complete Riemannian with sec <= 0, the universal cover M~ is
CAT(0) (Cartan-Hadamard).
Locally CAT(0). A length-metric space is locally CAT(0) iff its
universal cover is CAT(0). A smooth Riemannian metric with sec <= 0
is locally CAT(0).
Fundamental group pi_1(M). Homotopy classes of based loops. If M
carries a locally CAT(0) length metric, then pi_1(M) acts freely,
properly, and cocompactly on the CAT(0) universal cover M~ by
isometries. So
pi_1(M) admits NO proper cocompact isometric action on any
nonempty proper CAT(0) space
==> M carries NO locally CAT(0) length metric
==> M carries NO sec <= 0 Riemannian metric.
Warped product. For a surface metric g = dtau^2 + l(tau)^2 dy^2 with
y a 1-periodic coordinate, the Gaussian curvature is -l''/l. So
sec <= 0 iff l'' >= 0. Used to make convex collars in Part 6.
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PART 2. STATEMENT OF THE MAIN THEOREM
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THEOREM (Main). There exists a closed, connected, orientable, smooth
3-manifold M satisfying ALL of the following:
(a) M carries a C^inf Riemannian metric with NO conjugate points.
(b) M carries NO C^inf Riemannian metric with sec <= 0 everywhere.
(c) The fundamental group pi_1(M) admits NO proper cocompact
isometric action on any nonempty proper CAT(0) space.
Consequently, M carries no locally CAT(0) length metric.
(d) M carries no C^inf Riemannian metric without focal points.
(a) is proved by an explicit metric construction (Parts 6-10). (c) is
proved group-theoretically (Part 5). (b) follows from (c) and
Cartan-Hadamard: a sec <= 0 metric is locally CAT(0). (d) follows
from (b) and the Ivanov-Kapovitch equivalence for closed 3-manifolds.
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PART 4. THE MANIFOLD M
================================================================
We define the underlying closed 3-manifold, the torus gluing, and the
fundamental group. No metric yet.
Sigma is a compact, connected, oriented surface of genus 1 with one
boundary circle. It is a torus with an open disc removed.
For i = 1, 2 define N_i = Sigma x S^1. Each N_i is a compact oriented
3-manifold whose boundary is a single 2-torus.
On the boundary torus ∂N_i, let
h_i = the boundary circle of Sigma (included at a point of S^1),
f_i = the S^1 factor (included at a point of Sigma).
The pair (h_i, f_i) is an oriented basis of H_1(∂N_i;Z) ≅ Z^2, with
h_i · f_i = 1.
The gluing. Define an orientation-reversing diffeomorphism
phi: ∂N_2 -> ∂N_1 by its action on these homology classes:
phi_*(h_2) = h_1 + f_1, phi_*(f_2) = h_1. (4.1)
In the bases (h_i, f_i) the matrix of phi_* is [[1,1],[1,0]], whose
determinant is -1. So phi reverses the induced boundary orientation,
and gluing the two oriented 3-manifolds along phi produces an oriented
closed 3-manifold M:
M := N_1 ∪_phi N_2. (4.2)
M is closed (every boundary component is identified with its partner),
connected (the common torus joins the two halves), and orientable.
Inserting a neck. For the metric construction it is convenient to
insert a short product cylinder [0,1] x T^2 between the two pieces,
along the identified torus. This does not change the diffeomorphism
type of M. We therefore write, and use throughout,
M = N_1 ∪_{T_1} ([0,1] x T^2) ∪_{T_2} N_2, (4.3)
where {0} x T^2 is glued to ∂N_2 via phi and {1} x T^2 to ∂N_1 via
the identity (or any other fixed labels of the two ends). The central
component [0,1] x T^2 is the NECK. Outside the neck both pieces N_1
and N_2 will carry sec <= 0; inside the neck a one-parameter family of
flat tori produces the interesting second-variation behaviour.
Coordinate realization of the gluing. We will need explicit
coordinates on T^2 that translate (4.1) into a specific small-shear
frame change. Fix delta > 0 (the small parameter used later) and put
epsilon := delta^2. Choose B_2 > 0 later, and set B_1 := epsilon B_2.
Put
T^2 = R^2 / Lambda, Lambda = Z·B_1 e_1 + Z·B_2 e_2, (4.4)
where e_1, e_2 is the standard basis of R^2.
At the first end (t = 0 of the neck) use the frame (e_1, e_2). At the
second end (t = 1) use the frame (e_2 + epsilon e_1, e_1). In each
frame the FIRST vector is the surface-boundary direction (h), the
SECOND is the circle factor (f). Their period vectors in R^2 are:
first end: h_1 = B_1 e_1, f_1 = B_2 e_2,
second end: h_2 = B_2 (e_2 + epsilon e_1) = B_1 e_1 + B_2 e_2,
f_2 = B_1 e_1. (4.5)
Both (h_2, f_2) are lattice vectors of the SAME Lambda (since
h_2 = h_1 + f_1 and f_2 = h_1 in Lambda). Hence they form a basis of
the same lattice. And in Lambda:
h_2 = h_1 + f_1, f_2 = h_1. (4.6)
So the frame change (e_1, e_2) -> (e_2 + epsilon e_1, e_1) reproduces
exactly the gluing (4.1), after rescaling B_1 = epsilon B_2. The
smallness of epsilon later only makes the coordinate shear small; the
period identity (4.6) is exact for every epsilon.
Fundamental group.
Fix a basepoint and free generators a_i, b_i of pi_1(Sigma). Since
Sigma has genus 1 with one boundary, the boundary circle ∂Sigma
represents the commutator [a_i, b_i]^{±1}. Write
h_i = [a_i, b_i]^{±1}.
Then
pi_1(N_i) = pi_1(Sigma) x pi_1(S^1) = F(a_i, b_i) x <f_i>, (4.7)
where F(a_i, b_i) is free of rank 2 and <f_i> ≅ Z.
The subgroup <h_i, f_i> ⊂ pi_1(N_i) is isomorphic to Z^2: h_i has
infinite order in F(a_i, b_i) (cylinder decomposition in the free
group), and f_i is central in pi_1(N_i); any relation h_i^m f_i^n = 1
projects to <f_i> giving n = 0, then m = 0.
By Van Kampen's theorem applied to (4.3),
pi_1(M) = G_1 *_{Gamma_T} G_2, G_i := pi_1(N_i),
Gamma_T := <h_1, f_1> = <h_2, f_2> ≅ Z^2.
(4.8)
Both vertex groups G_1, G_2 and the amalgamating group Gamma_T embed
in pi_1(M). (Structure theorem for amalgamated free products.)
Centralizers. For f ∈ pi_1(M) write C(f) = {g : gf = fg}. Since f_i
is central in G_i, we have G_i ⊆ C(f_i). Also h_i = [a_i, b_i] lies in
[G_i, G_i]. So, for i = 1, 2,
G_i ⊆ C(f_i), h_i ∈ [C(f_i), C(f_i)]. (4.9)
This will be the crucial input to Part 5. It says: each boundary
circle h_i is a commutator of elements that commute with the circle
factor f_i.
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PART 5. TOPOLOGICAL OBSTRUCTION TO NONPOSITIVE CURVATURE
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Here we prove part (c) of the main theorem. No metric appears; the
whole argument is about pi_1(M).
THEOREM 5.1. Let Gamma = pi_1(M). Gamma admits NO proper cocompact
isometric action on any nonempty proper CAT(0) space X.
PROOF. Suppose, for contradiction, that such an action
Gamma -> Iso(X) exists, proper and cocompact. "Proper" means: for any
two compact sets K, L ⊂ X, only finitely many g ∈ Gamma satisfy
gK ∩ L ≠ ∅. In particular, every point stabilizer Gamma_x is finite.
STEP 1 (Semisimplicity). Every gamma ∈ Gamma has its translation
length attained.
For gamma ∈ Gamma define
length(gamma) := inf_{x ∈ X} d(x, gamma x),
Min(gamma) := { x ∈ X : d(x, gamma x) = length(gamma) }.
We show Min(gamma) is nonempty for every gamma.
Choose a compact set K ⊂ X with Gamma K = X (cocompactness). Choose
x_n ∈ X with d(x_n, gamma x_n) -> length(gamma). For each n pick
g_n ∈ Gamma with y_n := g_n x_n ∈ K. Define gamma_n := g_n gamma g_n^{-1}.
Then
d(y_n, gamma_n y_n) = d(g_n x_n, g_n gamma x_n) = d(x_n, gamma x_n).
For all large n, d(y_n, gamma_n y_n) <= length(gamma) + 1, so
gamma_n y_n ∈ B(K, length(gamma) + 1), which is compact (X proper).
By properness of the action on the pair (K, B(K, length+1)), only
finitely many distinct gamma_n can occur. Pass to a subsequence on
which gamma_n = gamma_* is constant. Because K is compact, a further
subsequence has y_n -> y ∈ K. Continuity of d gives
d(y, gamma_* y) = length(gamma).
So gamma_* attains its infimum. Translation length is conjugation
invariant: length(gamma) = length(gamma_*) and Min(gamma) contains
g_*^{-1} y. Hence Min(gamma) ≠ ∅ for every gamma in Gamma.
STEP 2. Infinite-order elements have length(gamma) > 0.
Let gamma ∈ Gamma have infinite order and length(gamma) = 0. By Step 1
gamma fixes some point x. Then the infinite cyclic subgroup <gamma> is
contained in the stabilizer Gamma_x, contradicting properness (finite
stabilizers). So length(gamma) > 0.
STEP 3 (Flat torus theorem applied to Gamma_T ≅ Z^2).
The subgroup Gamma_T ⊂ Gamma is isomorphic to Z^2 (Part 4). It acts on
X properly (inherited), and semisimply (Steps 1-2).
The flat torus theorem (Bridson-Haefliger, Theorem II.7.1) states:
For a proper CAT(0) space X and a proper semisimple action of Z^k,
the set Min(Z^k) := ⋂_{a ∈ Z^k} Min(a) is a nonempty closed convex
subset of X, isometric to Y x R^k, on which Z^k acts trivially on Y
and by translations on R^k. In particular, each a ∈ Z^k has a
translation vector v(a) ∈ R^k, and the map a ↦ v(a) is an injective
homomorphism onto a lattice.
Apply this with k = 2 and Z^2 = Gamma_T. We obtain a nonempty closed
convex subspace
Min(Gamma_T) ≅ Y x R^2
(metrically a product), on which Gamma_T acts trivially on Y and by a
translation lattice on R^2.
Fix one invariant Euclidean plane E := {y_0} x R^2 inside Min(Gamma_T),
for some fixed y_0 ∈ Y.
For each a ∈ Gamma_T the translation vector v(a) ∈ R^2 is defined, and
a ↦ v(a) is an injective homomorphism. In particular,
v(h_1) and v(f_1) are independent, nonzero vectors of R^2. (5.1)
Also (key genericity statement) the map a ↦ v(a) is linear on the
free abelian group Gamma_T = <h_1, f_1>; so
v(h_2) = v(h_1) + v(f_1) = u + v,
v(f_2) = v(h_1) = u, (5.2)
where u := v(h_1) and v := v(f_1), written by the gluing relations
(4.6) read inside Gamma_T.
STEP 4 (Cyclic minset theorem for f_i). For i = 1, 2 let
l_i := length(f_i) > 0 (Step 2, f_i has infinite order).
The cyclic minset theorem (Bridson-Haefliger, Theorem II.6.8) states:
Let X be a proper CAT(0) space, f ∈ Iso(X) semisimple with l(f) > 0.
Then Min(f) is isometric to Y_f x R, and f acts by
(y, t) ↦ (y, t + l(f)).
Moreover, every g ∈ C(f) preserves the product structure and acts by
(y, t) ↦ (g'(y), t + tau(g))
for some isometry g' of Y_f and a real number tau(g) independent of y.
We use this with f = f_i. To see why tau is independent of y: an
element of C(f) must send f-orbits (the lines {y} x R) to f-orbits.
The translation amount along each line could a priori depend on y, but
preservation of product distances between (y,t) and (z,s), for every
t - s ∈ R, forces independence of y.
STEP 5 (tau_i : C(f_i) -> R is a homomorphism vanishing on commutators).
Composing the actions of Step 4 gives
tau_i(g_1 g_2) = tau_i(g_1) + tau_i(g_2),
so tau_i is a homomorphism C(f_i) -> (R,+). Its target is abelian;
hence tau_i kills every commutator. By (4.9), h_i lies in
[C(f_i), C(f_i)], so
tau_i(h_i) = 0, for i = 1, 2. (5.3)
STEP 6 (The two orthogonality equations).
Fix i ∈ {1,2}. The plane E from Step 3 is contained in
Min(Gamma_T) ⊂ Min(f_i). Since h_i and f_i both lie in Gamma_T, they
act on E ⊂ Min(Gamma_T) by translations with vectors v(h_i), v(f_i)
∈ R^2.
Now express a point x ∈ E in the ith product structure Min(f_i) ≅ Y_i x R
as x = (y, t). Because tau_i(h_i) = 0 by (5.3),
h_i x = (h_i'(y), t),
f_i x = (y, t + l_i),
h_i f_i x = f_i (h_i x) = (h_i'(y), t + l_i).
Product metric in Y_i x R gives
d(x, h_i f_i x)^2 = d_{Y_i}(y, h_i'(y))^2 + l_i^2
= d(x, h_i x)^2 + d(x, f_i x)^2. (5.4)
But x, h_i x, f_i x, h_i f_i x all lie in the Euclidean plane E. In
E, the points h_i x and f_i x are translates by v(h_i) and v(f_i); by
the parallelogram identity,
h_i f_i x = x + v(h_i) + v(f_i),
hence
d(x, h_i x) = |v(h_i)|,
d(x, f_i x) = |v(f_i)|,
d(x, h_i f_i x) = |v(h_i) + v(f_i)|.
Combining with (5.4):
|v(h_i) + v(f_i)|^2 = |v(h_i)|^2 + |v(f_i)|^2.
So v(h_i) ⟂ v(f_i):
<v(h_i), v(f_i)> = 0, for i = 1, 2. (5.5)
STEP 7 (Contradiction).
Recall u := v(h_1), v := v(f_1). (5.5) with i = 1 says
<u, v> = 0. (5.6)
Recall also v(h_2) = u + v, v(f_2) = u by (5.2). (5.5) with i = 2 says
0 = <v(h_2), v(f_2)> = <u + v, u> = |u|^2 + <v, u> = |u|^2,
the last equality by (5.6). So u = 0.
But v is injective on Gamma_T ≅ Z^2 and h_1 is nontrivial; (5.1) gave
v(h_1) ≠ 0, i.e. u ≠ 0. Contradiction. ∎
COROLLARY 5.2. M admits NO locally CAT(0) length metric inducing the
manifold topology.
Proof. Suppose d is such a length metric. Compactness of M makes d
complete and locally compact. Let M~ be the universal covering with
the lifted length metric; the covering map is a local isometry, so M~
is complete and locally compact. Metric Cartan-Hadamard: M~ is CAT(0).
Metric Hopf-Rinow: M~ is proper. The deck group pi_1(M) acts freely,
properly, and cocompactly on M~ by isometries. This contradicts
Theorem 5.1. ∎
PROPOSITION 5.3. M admits NO C^inf Riemannian metric with sec <= 0
everywhere.
Proof. A Riemannian metric with sec <= 0 is (by Cartan-Hadamard) a
locally CAT(0) length metric on M. Contradicts Corollary 5.2. ∎
PROPOSITION 5.4. M admits NO C^inf Riemannian metric without focal
points.
Proof. Ivanov-Kapovitch (Theorem 7.1 of [IK2014]) prove that a closed
3-manifold admits a metric without focal points iff it admits a metric
with sec <= 0. By Proposition 5.3 the latter does not exist. ∎
These three Propositions give parts (b), (c), (d) of the main theorem.
What remains is part (a): a smooth metric on M with no conjugate
points. This is the subject of Parts 6-10.
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PART 6. CONSTRUCTION OF THE METRIC ON M
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Overview. We build a smooth Riemannian metric g on M in three steps.
Step 1: a metric on the neck U = (0,1) x T^2, controlled by a
one-parameter family P(t) of positive-definite 2x2 matrices.
Step 2: smooth convex collars inside the two product pieces N_1, N_2
that match the neck metric exactly at both ends.
Step 3: fill in each N_i's interior with a hyperbolic punctured torus
carrying sec <= 0.
The main object is P(t), which we call the COMETRIC; its inverse
Q(t) := P(t)^{-1} is the matrix of the torus metric at height t.
Parts 7-10 then prove the resulting metric has no conjugate points.
Convention for matrix order. For symmetric 2x2 matrices M, N we write
M <= N to mean N - M is positive-semidefinite (i.e. x^T (N-M) x >= 0
for every x ∈ R^2). Similarly M < N for strict positive-definiteness
of N - M.
6.1 The neck metric
The neck is U := (0,1) x T^2 with coordinates (t, x), t ∈ (0,1),
x ∈ T^2 = R^2 / Lambda (Lambda from (4.4)).
The neck metric is
g_neck := dt^2 + dx^T Q(t) dx, Q(t) := P(t)^{-1}, (6.1)
where P(t) is smooth in t, positive-definite symmetric, t ∈ [0, 1].
In particular, for any constant velocity (dt/ds, dx/ds) in the neck,
its squared speed is (dt/ds)^2 + (dx/ds)^T Q(t) (dx/ds).
Torus frames at the two ends. Recall from Part 4:
at t = 0 (glued to ∂N_2): standard basis T_1 = (e_1, e_2),
coordinate periods B_1 and B_2;
at t = 1 (glued to ∂N_1): sheared basis T_2 = (e_2 + ε e_1, e_1),
coordinate periods B_2 and B_1.
Here ε = δ^2 and B_1 = ε B_2 (Part 4).
Inward coordinates. Set
τ := t (first end, near t = 0),
τ := 1 - t (second end, near t = 1).
Then τ ∈ [0, 1] increases INTO the neck from either end.
Frame change. A constant change of torus coordinates x = T y
transforms the metric matrix by Q ↦ T^T Q T, hence the cometric by
P ↦ T^{-1} P T^{-T}. In particular at the second end the "same" P
looks different in the frame T_2 from what it looks like in the frame
T_1. The end formula (Part 6.3) checks the two formulations agree.
6.2 The cometric path P(t)
Fix a small parameter 0 < δ <= 1/2, and set ε := δ^2.
Cutoff functions. Define
a(u) := exp( -1/(2/3 - u) ) for u < 2/3,
0 for u >= 2/3,
A(u) := ∫_0^u a(s) ds.
Then
a(u) > 0 for u < 2/3, a(u) = 0 for u >= 2/3, A(0) = 0,
a'(u) = -a(u) / (2/3 - u)^2 < 0 for u < 2/3,
and (important later) A is CONSTANT on [2/3, +inf), equal to A(2/3).
Also fix a smooth nondecreasing function
χ: R -> [0, 1], χ = 0 on (-inf, 5/12], χ = 1 on [7/12, +inf).
Define H, d_1, d_2, P by
H(t) := e_2 + ε χ(t) e_1, (6.2a)
d_1(t) := 1 + δ A(t), d_2(t) := 1 + δ A(1 - t), (6.2b)
P(t) := d_1(t) e_1 e_1^T + d_2(t) H(t) H(t)^T, (6.2c)
b := 1 + δ A(1) > 0. (6.2d)
Finally set
κ := δ a(0) = δ exp(-3/2) > 0. (6.3)
Interpretation. At each height t the metric g_neck is a flat torus
with matrix Q(t) = P(t)^{-1}. As t varies, the torus deforms: its
"surface-boundary" direction has coefficient d_1(t) and its "circle"
direction has coefficient d_2(t). The small shear εχ(t) e_1 in H(t)
rotates the two torus directions into each other slightly, and its
magnitude is epsilon = δ^2 (a much smaller perturbation than the linear
variation of d_1 and d_2, which is δ).
Immediate properties (proved in Part 6.4).
(P1) (1/9) Id <= P(t) <= 18 Id, uniformly in t and δ ∈ (0, 1/2].
(P2) P''(t) <= 0, i.e. -P''(t) >= 0 as a symmetric matrix.
(P3) |P'(t)| <= C δ, uniformly, where |.| is the Euclidean operator
norm on symmetric matrices.
(P4) |P'(t) p|^2 <= C δ p^T(-P''(t)) p for every p ∈ R^2,
uniformly in t and δ.
Constants C in (P3) and (P4) depend only on the FIXED function χ, not
on δ, B_1, B_2 or t.
6.3 The end formula (agreement of the two frames)
We now verify that in the correct coordinates each end of the neck
LOOKS LIKE a simple diagonal product. This is what lets us glue the
neck to the convex collars of Part 6.5.
First end (t = 0, frame T_1 = (e_1, e_2), coordinate x = y, τ = t).
For 0 <= t <= 1/3 we have 1 - t >= 2/3, so A(1 - t) = A(1) and
d_2(t) = 1 + δ A(1) = b. For 0 <= t <= 5/12 we have χ(t) = 0, so
H(t) = e_2 and H(t) H(t)^T = e_2 e_2^T. Therefore for 0 <= τ = t <= 1/4:
P(t) = d_1(τ) e_1 e_1^T + b e_2 e_2^T
= diag( 1 + δ A(τ), b ). (6.4)
In particular P(t) e_1 = (1 + δA(τ)) e_1 and P(t) e_2 = b e_2.
Note also A(0) = 0, so
P(0) = diag(1, b),
P'(τ) |_{τ=0} = diag(δ a(0), 0) = κ e_1 e_1^T. (6.5)
Second end (t = 1, frame T_2 = (e_2 + ε e_1, e_1), τ = 1 - t).
For 0 <= τ <= 1/4 we have t = 1 - τ >= 3/4 >= 7/12, so χ(t) = 1 and
H(t) = e_2 + ε e_1. Also 1 - τ >= 3/4 for τ <= 1/4, so A(1 - t) =
A(1 - τ) = A(1) and d_1(t) = 1 + δA(1) = b. Thus in the standard
frame (e_1, e_2) the matrix is
P(t) = b e_1 e_1^T + (1 + δA(τ)) (e_2 + ε e_1)(e_2 + ε e_1)^T.
In matrix form (basis e_1, e_2):
P(t) = [[ b + (1+δA(τ)) ε^2 , (1+δA(τ)) ε ],
[ (1+δA(τ)) ε , 1+δA(τ) ]].
Now pass to the sheared frame T_2 by the rule P_2(τ) = T_2^{-1} P(t) T_2^{-T}.
The matrix of T_2 in basis (e_1, e_2) is
T_2 = [[ ε , 1 ],
[ 1 , 0 ]], det T_2 = -1,
with inverse (and transpose, which happens to be the same)
T_2^{-1} = T_2^{-T} = [[ 0 , 1 ],
[ 1 , -ε ]].
Then a direct computation gives
T_2^{-1} P(t) = [[ (1+δA) ε , 1+δA ],
[ b , 0 ]],
and left-multiplying by T_2^{-1} one more time:
P_2(τ) = T_2^{-1} P(t) T_2^{-T}
= [[ 1 + δA(τ) , 0 ],
[ 0 , b ]]
= diag( 1 + δA(τ), b ). (6.6)
So the same diagonal matrix diag(1 + δA(τ), b) appears in BOTH end
frames. In particular
P_2(0) = diag(1, b),
P_2'(τ)|_{τ=0} = diag(δ a(0), 0) = κ e_1 e_1^T. (6.7)
Geometric content. In inward coordinates at either end, the neck
metric has the product form
g|_end = dτ^2 + (1 + δA(τ))^{-1} dy_1^2 + b^{-1} dy_2^2,
(6.8)
with y_1 a 1-periodic coordinate for the SURFACE-boundary direction
h_i and y_2 a 1-periodic coordinate for the CIRCLE factor f_i. The
coefficient of dy_2^2 is the constant b^{-1}; the coefficient of
dy_1^2 grows as (1 + δA(τ))^{-1}. This product form is exactly the
one needed to match to a hyperbolic punctured torus times a flat
circle in Parts 6.5-6.7.
Recall κ = δ a(0) = δ exp(-3/2) > 0. The number κ measures the
initial SPEED at which the surface-boundary coefficient (1 + δA(τ))
decreases as τ increases away from the boundary at each end. This is
the "positive Christoffel term" that will appear in the passage
estimates of Parts 8-9 (see eq. 8.11).
6.4 Bounds on the cometric
We state and prove the quantitative estimates on P(t) = diag in the
original frame and in each end frame.
LEMMA 6.4. With χ as in (6.2) and P(t) as in (6.2), there are
constants δ_0 > 0 and C < ∞, depending only on χ, such that for every
0 < δ <= δ_0 the following hold in the original frame (e_1, e_2) and
in BOTH end frames T_1, T_2:
(a) (1/18) Id <= P <= 18 Id and
(1/18) Id <= Q <= 18 Id where Q = P^{-1};
(b) ||P'(t)|| <= C δ;
(c) -P''(t) >= 0 (symmetric positive-semidefinite);
(d) |P'(t) p|^2 <= C δ p^T (-P''(t)) p for every p ∈ R^2.
Here ||.|| is the operator norm on symmetric 2x2 matrices. The
constants δ_0 and C are independent of B_1 and B_2.
PROOF OF (a) (bounds on P and Q).
Write R = (e_1, H) for the 2x2 matrix with columns e_1 and
H = e_2 + εχ(t) e_1. Then det R = 1 and
P(t) = R diag(d_1(t), d_2(t)) R^T.
Both d_1(t) = 1 + δA(t) and d_2(t) = 1 + δA(1-t) satisfy
1 <= d_i(t) <= 1 + δ, because A: [0,1] -> [0,1] takes values in
[0, A(2/3)] ⊂ [0,1] (recall a(u) ∈ [0,1] and a(u) = 0 for u >= 2/3).
Now estimate ||R|| and ||R^{-1}||. In the Frobenius norm,
||R||_F^2 = |e_1|^2 + |H|^2 = 1 + (1 + ε^2 χ(t)^2) = 2 + ε^2 χ^2,
so for ε <= 1 (which holds for δ <= 1):
||R|| <= sqrt(2 + ε^2) <= sqrt(3). (6.9)
Since det R = 1, the singular values are σ_max(R) and 1/σ_max(R), so
σ_min(R) = 1/σ_max(R) >= 1/sqrt(3). (6.10)
For a unit vector v ∈ R^2, let q = R^T v. Then
v^T P(t) v = d_1(t) q_1^2 + d_2(t) q_2^2,
which lies between σ_min(R)^2 and (1+δ) σ_max(R)^2:
v^T P(t) v >= σ_min(R)^2 >= 1/3,
v^T P(t) v <= (1+δ) σ_max(R)^2 <= 1.5 * 3 = 4.5.
Hence in the original frame
(1/3) Id <= P(t) <= (9/2) Id. (6.11)
Passing to the end frames. The frame matrices T_1 = Id and
T_2 = [[ε, 1],[1, 0]] have ||T_1|| = ||T_1^{-1}|| = 1 and
||T_2||_F^2 = ||T_2^{-1}||_F^2 = 2 + ε^2 <= 3.
The conjugate P_i(τ) = T_i^{-1} P(t) T_i^{-T} satisfies, for any v,
v^T P_i(τ) v = (T_i^{-T} v)^T P(t) (T_i^{-T} v)
>= (1/3) |T_i^{-T} v|^2
>= (1/3) * (1/||T_i||^2) * |v|^2
>= (1/9) |v|^2,
and similarly v^T P_i(τ) v <= 4.5 * ||T_i^{-T}||^2 |v|^2 <= 4.5 * 3 |v|^2 = 13.5 |v|^2.
So in each end frame
(1/9) Id <= P_i <= (27/2) Id <= 18 Id. (6.12)
Finally Q(t) = P(t)^{-1} and Q_i(τ) = P_i(τ)^{-1} satisfy
(1/18) Id <= Q, Q_i <= 18 Id. (6.13)
This proves (a).
[Bounds (b), (c), (d) follow in the next two sub-sections.]
PROOF OF (b), (c), (d).
First, an elementary bound. For every u >= 0,
a(u)^2 <= -a'(u). (6.14)
Indeed, if u >= 2/3 then a(u) = 0 and a'(u) = 0, so (6.14) is 0 <= 0.
If u < 2/3, let s = 2/3 - u > 0. Then a(u) = e^{-1/s} and
a'(u) = -a(u)/s^2 = -e^{-1/s}/s^2, so (6.14) is
e^{-2/s} <= e^{-1/s}/s^2,
i.e., s^2 e^{-1/s} <= 1.
The function s ↦ s^2 e^{-1/s} has derivative e^{-1/s}(2s + 1) > 0 on
(0, ∞), so its maximum on (0, 2/3] is at s = 2/3:
(2/3)^2 e^{-3/2} = (4/9) e^{-3/2} ≈ 0.0996 <= 1.
So (6.14) holds. (For u = 1-t ∈ [0, 1] the same bound holds.)
Now split the proof into two regions.
CASE 1: outside I_0 = [5/12, 7/12]. That is, t ∈ [0, 5/12] ∪ [7/12, 1].
On each of these two intervals, χ is CONSTANT (0 on the left, 1 on
the right). Hence H(t) = e_2 + εχe_1 is constant, so
R(t) = (e_1, H(t)) is constant, so R' = R'' = 0. Consequently
P'(t) = R diag( d_1'(t), d_2'(t) ) R^T,
P''(t) = R diag( d_1''(t), d_2''(t) ) R^T,
with d_1' = δa(t), d_2' = -δa(1-t),
d_1'' = δa'(t), d_2'' = δa'(1-t).
Bound (b): ||P'|| <= ||R||^2 ||diag(d_1', d_2')|| <= 3 · δ (since
both |d_i'| <= δ). So ||P'|| <= Cδ with C = 3.
Bound (c): -P'' = -R diag(d_1'', d_2'') R^T = R diag(-d_1'', -d_2'') R^T.
Since a' <= 0, both diagonal entries satisfy -d_i'' >= 0, so -P'' ⪰ 0. ✓
Bound (d): For p ∈ R^2, write q = R^T p. Then
|P'p|^2 = |R diag(d_1', d_2') q|^2 <= ||R||^2 (δ^2 a(t)^2 q_1^2 + δ^2 a(1-t)^2 q_2^2)
<= 3 δ^2 (a(t)^2 q_1^2 + a(1-t)^2 q_2^2).
And
p^T(-P'')p = q^T diag(-d_1'', -d_2'') q
= -δ a'(t) q_1^2 - δ a'(1-t) q_2^2.
Using (6.14), the LHS of (d) is at most
3 δ^2 (a(t)^2 q_1^2 + a(1-t)^2 q_2^2)
<= 3 δ^2 (-a'(t) q_1^2 - a'(1-t) q_2^2)
= 3 δ (-δ a'(t) q_1^2 - δ a'(1-t) q_2^2)
= 3 δ p^T(-P'')p.
So (d) holds with C = 3.
CASE 2: inside I_0 = [5/12, 7/12].
Here χ, χ', χ'' are nonzero in general. Compare P(t) with the
diagonal P_0(t) = diag(d_1(t), d_2(t)) (what P would be if χ were 0),
and put E = P - P_0:
E(t) = d_2(t) (H(t) H(t)^T - e_2 e_2^T)
= d_2(t) ( εχ (e_1 e_2^T + e_2 e_1^T) + ε^2 χ^2 e_1 e_1^T ).
Since |χ| <= 1 and d_2 <= 1+δ <= 1.5:
||E(t)|| <= (1+δ)(2ε + ε^2) <= 4 ε. (6.15)
Differentiating E in t:
E' = d_2' (H H^T - e_2 e_2^T) + d_2 (H H^T)',
E'' = d_2'' (H H^T - e_2 e_2^T) + 2 d_2' (H H^T)' + d_2 (H H^T)''.
Using H = e_2 + εχ e_1,
(H H^T)' = ε χ' (e_1 e_2^T + e_2 e_1^T) + 2 ε^2 χ χ' e_1 e_1^T,
(H H^T)'' = ε χ''(e_1 e_2^T + e_2 e_1^T) + 2 ε^2 ( (χ')^2 + χ χ'' ) e_1 e_1^T.
Since |χ| <= 1 and |d_2'|, |d_2''| <= δ, the following bounds hold with
constants C_1, C_2 depending only on max(|χ'|, |χ''|) on I_0:
||E'|| <= C_1 ε, ||E''|| <= C_2 ε. (6.16)
For example, one may take C_1 = 7 max(1,|χ'|) and
C_2 = 5 (1+δ) ( max|χ''| + (max|χ'|)^2 + max|χ''| ) etc. The
specific numerical values are not important; only that they depend on
χ alone.
On I_0, both t and 1 - t lie in [5/12, 7/12], so both -a'(t) and
-a'(1-t) are bounded below by the number
m := 144 e^{-12} > 0.
(Indeed on [5/12, 7/12], u = 2/3 - t ranges over [1/12, 1/4]; the
function u ↦ e^{-1/u}/u^2 is increasing on (0, 1/2], so its minimum on
[1/12, 1/4] is at u = 1/12, giving e^{-12} / (1/144) = 144 e^{-12}.)
Therefore on I_0:
-P_0''(t) = diag( -δ a'(t), -δ a'(1-t) ) ⪰ δ m Id.
Taking δ <= δ_0 := min(1/2, m / (2 C_2)) gives, on I_0:
-P''(t) = -P_0''(t) - E''(t) ⪰ (δ m - C_2 ε) Id
= δ (m - C_2 δ) Id
⪰ (δ m / 2) Id. (6.17)
Bound (c) on I_0: -P'' ⪰ (δm/2) Id ⪰ 0. ✓
Bound (b) on I_0: ||P'|| = ||P_0' + E'|| <= ||P_0'|| + ||E'|| <= δ + C_1 δ^2
<= (1 + C_1) δ, for δ <= 1.
Bound (d) on I_0: for p ∈ R^2,
|P'p|^2 <= ||P'||^2 |p|^2 <= (1+C_1)^2 δ^2 |p|^2,
and by (6.17) p^T(-P'')p >= (δ m / 2) |p|^2. Hence
|P'p|^2 <= ( 2 (1+C_1)^2 / m ) δ p^T(-P'')p.
Take C := max( 3, 1 + C_1, 2 (1+C_1)^2 / m ). Then (b), (c), (d) hold
in the original frame.
Transfer to end frames. The first end frame is T_1 = Id, so nothing
changes there. The second end frame is T_2 = [[ε,1],[1,0]] with
||T_2^{-1}||^2 <= 3. Writing τ = 1 - t at the second end,
P_2'(τ) = -T_2^{-1} P'(t) T_2^{-T},
P_2''(τ) = T_2^{-1} P''(t) T_2^{-T}.
Then ||P_2'|| <= ||T_2^{-1}||^2 ||P'|| <= 3 C δ, and -P_2'' = T_2^{-1}(-P'')T_2^{-T} ⪰ 0.
For the domination, set v = T_2^{-T} p so that p^T(-P_2'')p = v^T(-P'')v:
|P_2' p|^2 = |T_2^{-1} P'v|^2
<= ||T_2^{-1}||^2 |P'v|^2
<= ||T_2^{-1}||^2 C δ v^T(-P'')v
<= 3 C δ p^T(-P_2'')p.
Replacing C by 3 C finishes the proof of Lemma 6.4.
Remark. The constants C, δ_0 depend only on χ (through C_1, C_2 and m);
they do not depend on δ, B_1, B_2, or t. Also, since P is built from
C^inf functions (a and A are smooth because e^{-1/x} extended by 0 at
0 is smooth; χ is smooth), P is smooth on [0,1] and so are the frames.
6.5 Convex collars inside each product piece
Recall the neck metric end formula (6.8):
g|_end = dτ² + (1 + δA(τ))^{-1} dy_1² + b^{-1} dy_2², τ ∈ [0, 1/4].
The circle factor with metric b^{-1} dy_2² is flat; the surface factor
g_surf(τ) := dτ² + ℓ_0(τ)² dy_1², ℓ_0(τ) := (1 + δA(τ))^{-1/2},
determines the whole sectional curvature of this product metric. We
will show sec <= 0 for τ ∈ [-1, 1/4], by extending g_surf from τ ∈ [0, 1/4]
into τ ∈ [-1, 0] via an explicit convex warping function.
Nonpositive curvature near the neck end. For a surface metric
dτ² + ℓ(τ)² dy_1², the Gaussian curvature is -ℓ''/ℓ. So sec <= 0 iff
ℓ'' >= 0. For ℓ_0(τ) = (1+δA(τ))^{-1/2} a direct computation gives
(using a'(u) = -a(u)/(2/3 - u)^2 and the chain rule; here A' = a, so
differentiating log ℓ_0 = -(1/2) log(1 + δA(τ)) twice):
ℓ_0''(τ)/ℓ_0(τ) = 3 δ² a(τ)² / (4 (1 + δA(τ))²) - δ a'(τ) / (2 (1 + δA(τ))). (6.18)
Since a(τ)² >= 0 and a'(τ) <= 0, both terms are >= 0. So
ℓ_0''(τ) >= 0 for τ ∈ [0, 1/4], hence sec(g_surf) <= 0 there. (6.19)
Extension to τ ∈ [-1, 0]. We construct a smooth ℓ(τ) on [-1, 0] such
that ℓ = ℓ_0 on a neighborhood of 0 (so gluing is smooth), and so that
ℓ'' >= 0 on [-1, 0].
Step 1: a decreasing auxiliary function. Let
r(τ) := δ a(τ) / (2 (1 + δA(τ))), τ ∈ [-1, 0].
For A(τ) = ∫_0^τ a(u) du ∈ [-1, 0] (since a >= 0 and τ <= 0), we have
1 + δA(τ) ∈ [1/2, 1]. With |a| <= 1 and δ <= 1/2,
0 < r(τ) <= δ / (2 · (1/2)) = δ <= 1/2 < 1. (6.20)
Differentiating and using a' <= 0 (so -a' >= 0):
r'(τ) = δ ( a'(τ) (1 + δA(τ)) - δ a(τ)² ) / (2 (1 + δA(τ))²).
Numerator = δ a'(τ)(1 + δA(τ)) - δ² a(τ)². First term <= 0 (a' <= 0,
1 + δA > 0), second term <= 0. So the numerator < 0. Thus
r'(τ) < 0 on [-1, 0]. (6.21)
Step 2: a smooth step. Fix once and for all a smooth nondecreasing
function λ: [-1, 0] → [0, 1] with
λ = 0 on a neighborhood of -1, λ = 1 on a neighborhood of 0. (6.22)
Step 3: the warping profile. Define
k(τ) := 1 - λ(τ) + λ(τ) r(τ),
ℓ(τ) := exp( - ∫_0^τ k(u) du ). (6.23)
We claim:
k > 0, k' <= 0, ℓ'' = (k² - k') ℓ > 0, (6.24)
ℓ ≡ ℓ_0 on a neighborhood of τ = 0, (6.25)
ℓ(τ) = ℓ(-1) e^{-(τ + 1)} on a neighborhood of τ = -1. (6.26)
Check of (6.24). By (6.20), r ∈ (0, 1). Since λ ∈ [0, 1],
k = 1 - λ (1 - r) >= 1 - (1 - r) = r > 0.
For the derivative, λ' >= 0 gives
k' = λ'(r - 1) + λ r'.
Now λ'(r - 1) <= 0 (since λ' >= 0 and r - 1 < 0) and λ r' <= 0
(since λ >= 0 and r' < 0). So k' <= 0.
Finally, ℓ = e^{-K} with K(τ) = ∫_0^τ k(u) du, so
ℓ' = -k ℓ, ℓ'' = -k' ℓ - k (-k ℓ) = (k² - k') ℓ.
Since k > 0 and k' <= 0, k² - k' > 0; so ℓ'' > 0. ✓ (6.24)
Check of (6.25). Near τ = 0, λ ≡ 1, so k = r. Now
d/dτ log ℓ_0(τ) = -(1/2) d/dτ log(1 + δA(τ))
= -(1/2) δ a(τ) / (1 + δA(τ))
= - r(τ).
And log ℓ_0(0) = 0 (A(0) = 0). So
log ℓ_0(τ) = - ∫_0^τ r(u) du, ℓ_0(τ) = exp(-∫_0^τ r(u) du).
Since near τ = 0 we have k = r, the same formula holds for ℓ. So
ℓ = ℓ_0 on an open neighborhood of 0. ✓ (6.25)
Check of (6.26). Near τ = -1, λ ≡ 0, so k ≡ 1, so
ℓ(τ) = exp( -∫_0^τ 1 du ) = exp( τ ) · exp(-0) · something.
More carefully: ℓ(τ) = exp(-∫_0^τ du) = exp(-τ). Combined with
the value at τ = -1: ℓ(-1) = exp(1), so
ℓ(τ) = e^{-τ} = ℓ(-1) e^{-(τ + 1)}. (6.27)
✓ (6.26)
The curve τ ↦ ℓ(τ) therefore matches the neck surface warp on one
side and the cusp warp of a hyperbolic cusp on the other side, both
on open intervals.
6.6 Hyperbolic punctured-torus cores and the choice of B_2
Part 11 constructs a complete, finite-area HYPERBOLIC PUNCTURED TORUS
T. Its structure: a compact genus-1 surface with one end that opens
into a cusp. Near this cusp, T carries the metric
dr^2 + c^2 e^{-2r} dθ^2, r ∈ [0, ∞), θ ∈ R/Z, (6.28)
for some explicit constant c > 0 (in Part 11, c = 6/a for a specific
a > 1). In particular the cusp of T contains a horocycle {r = const}
of every length c e^{-r} ∈ (0, c] (as r ranges over [0, ∞)).
Fix δ > 0. We assume δ <= δ_0 of Lemma 6.4. (Additional explicit
upper bounds on δ are imposed in Part 9; since they are also upper
bounds on δ, a single δ can satisfy them all simultaneously. We fix
one such δ once and for all here. Parts 7-9 require only that the
metric bounds of Lemma 6.4 hold at this chosen δ.)
Let ℓ(-1) > 0 be the value of the collar profile from Part 6.5 at
τ = -1. This is a strictly positive number depending only on δ.
Choose B_2 > 0 SMALL ENOUGH that BOTH numbers
B_1 ℓ(-1) and B_2 ℓ(-1) (6.29)
occur as horocycle lengths c e^{-r} of T's cusp (6.28). This is
possible because c e^{-r} ranges over all of (0, c]; just take both
B_1 ℓ(-1) and B_2 ℓ(-1) less than c. Set B_1 := ε B_2 = δ^2 B_2.
(Note: B_1 is much smaller than B_2 since ε = δ^2 is tiny.)
Now fix i ∈ {1, 2} and build the surface Σ_i.
Step 1. Take one copy T_i of T.
Step 2. Truncate T_i at the horocycle r = r_i of length c e^{-r_i} =
B_i ℓ(-1), keeping the compact side.
Step 3. Attach the collar [-1, 0] × (R / B_i Z) with surface metric
dτ^2 + ℓ(τ)^2 dy_1^2,
where y_1 ∈ R/B_i Z is a coordinate of period B_i.
Smoothness at τ = -1. In the cusp coordinates, set r = r_i + τ + 1
and θ = y_1 / B_i. Then
dr = dτ, dθ = dy_1 / B_i,
c^2 e^{-2r} = (c e^{-r_i})^2 e^{-2(τ + 1)} = (B_i ℓ(-1))^2 e^{-2(τ + 1)},
the last step using c e^{-r_i} = B_i ℓ(-1) by construction. Hence the
cusp metric (6.28) becomes
dτ^2 + (B_i ℓ(-1) / B_i)^2 e^{-2(τ+1)} dy_1^2
= dτ^2 + ℓ(-1)^2 e^{-2(τ+1)} dy_1^2.
Meanwhile from (6.27) the collar metric near τ = -1 is
dτ^2 + ℓ(τ)^2 dy_1^2 = dτ^2 + ℓ(-1)^2 e^{-2(τ+1)} dy_1^2.
These agree identically on an open collar. So the gluing at τ = -1
is C^∞.
Conclusion of 6.6. The result Σ_i is a compact, connected, oriented
surface of genus 1 with one boundary circle at τ = 0. Its Gaussian
curvature equals -1 on the truncated hyperbolic part and
-ℓ''(τ)/ℓ(τ) <= 0 on the collar [-1, 0] by (6.24). Hence sec(g(Σ_i)) <= 0
everywhere.
The collar profile satisfies ℓ(0) = ℓ_0(0) = (1 + δA(0))^{-1/2} = 1
(see (6.25) and A(0) = 0). So at the boundary τ = 0 the surface
metric is dτ^2 + dy_1^2.
6.7 Assembling the metric on M
For each i ∈ {1, 2}, define the compact oriented 3-manifold with
boundary
N_i := Σ_i x ( R / B_{3-i} Z ),
carrying the PRODUCT metric
g_i := g(Σ_i) + b^{-1} dy_2^2 , (6.30)
where y_2 ∈ R/B_{3-i} Z is the circle coordinate from Part 4.
The product metric has sec <= 0 everywhere, for these two reasons:
- On a 2-plane spanned by two vectors both tangent to Σ_i x {pt}:
the sectional curvature equals the Gaussian curvature of Σ_i at
that point, which is <= 0 by 6.6.
- On a 2-plane spanned by one vector tangent to Σ_i and one tangent
to the circle factor: the sectional curvature vanishes, because
a product metric separates into independent factors.
(There are no 2-planes entirely tangent to the 1-dim circle.)
Boundary metric. At τ = 0 the surface Σ_i has metric
dτ^2 + dy_1^2 (from the end of 6.6), with y_1 of period B_i. Hence
near its boundary, g_i is
dτ^2 + dy_1^2 + b^{-1} dy_2^2 , y_1 ∈ R/B_i Z, y_2 ∈ R/B_{3-i} Z.
(6.31)
Compare with the neck end formula (6.8). At the neck's FIRST end
(t = 0, frame T_1, τ = t), (6.8) reads
dτ^2 + (1 + δA(τ))^{-1} dy_1^2 + b^{-1} dy_2^2 ,
y_1 ∈ R/B_1 Z (h_1), y_2 ∈ R/B_2 Z (f_1).
At τ = 0, (1 + δA(0))^{-1} = 1, so this restricts to
dτ^2 + dy_1^2 + b^{-1} dy_2^2 , (6.32)
with y_1 period B_1 and y_2 period B_2 -- exactly matching (6.31) for
i = 1 (which has y_1 period B_1, y_2 period B_2). So the neck's
FIRST end glues to ∂N_1 by identifying the h_1 direction with y_1 and
the f_1 direction with y_2.
At the neck's SECOND end (t = 1, frame T_2, τ = 1 - t), (6.8) reads
dτ^2 + (1 + δA(τ))^{-1} dy_1^2 + b^{-1} dy_2^2 ,
y_1 ∈ R/B_2 Z (h_2), y_2 ∈ R/B_1 Z (f_2).
At τ = 0, again this is dτ^2 + dy_1^2 + b^{-1} dy_2^2 with y_1 period
B_2 and y_2 period B_1 -- exactly matching (6.31) for i = 2. So the
neck's SECOND end glues to ∂N_2 by identifying the h_2 direction with
y_1 and the f_2 direction with y_2.
Why the result is the same M. The two gluing identifications just
described are inverse to each other once we transport them through
the period calculation (4.5)-(4.6): a loop in the neck that enters
along h_1 and leaves along -h_2, when projected to ∂M, traces the
relation h_2 = h_1 + f_1, f_2 = h_1. This is exactly the diffeomorphism
φ by which N_1 and N_2 are glued in Part 4. So the resulting closed
3-manifold is the same M.
Smoothness. By (6.25), ℓ(τ) = ℓ_0(τ) on an OPEN neighborhood of
τ = 0 in the collar. So the surface metric of Σ_i on an open collar
around τ = 0 is literally the same function (1 + δA(τ))^{-1} as the
neck's surface metric. Hence the glued metric is C^∞-smooth, not just
continuous.
Summary of Part 6. We have built a C^∞ Riemannian metric g on the
closed, connected, orientable 3-manifold M, with the property
sec(g) <= 0 on M minus ( (0,1) x T^2 ) (6.33)
(the open neck is the only place where positive curvature can occur).
In addition sec(g) <= 0 on the two end portions t ∈ (0, 1/4] and
t ∈ [3/4, 1) of the neck itself, by (6.19) from Part 6.5.
The rest of the paper (Parts 7-10) proves that this g has no conjugate
points. Parts 7-9 are local: they classify geodesic passages through
the neck (Part 7) and bound the index form along each passage
(Parts 8-9). Part 10 sums over all passages and concludes.
================================================================
PART 7. GEODESICS IN THE NECK
================================================================
Throughout Parts 7-10 we work in the open neck
U := (0,1) x T^2,
with the metric
g = dt^2 + dx^T Q(t) dx , Q(t) = P(t)^{-1}, (7.0)
from (6.1). Let gamma(s) = (t(s), x(s)) be a UNIT-SPEED geodesic in
U, so |gamma'(s)| = 1. We use a dot for d/ds.
7.1 The geodesic identities
LEMMA 7.1. There exist a constant vector p ∈ R^2 and a function
F: [0,1] -> [0, ∞) such that along every unit-speed geodesic gamma we
have
p := Q(t(s)) x'(s) (7.1a)
is CONSTANT IN s,
F(t) := p^T P(t) p, v := t'(s), (7.1b)
v^2 = 1 - F(t(s)), (7.1c)
v' = - (1/2) F'(t(s)). (7.1d)
The function F is nonnegative, concave, and bounded above:
F(t) >= 0, F''(t) = p^T P''(t) p <= 0, F(t) <= 18 |p|^2. (7.2)
Proof. For the metric g = dt^2 + Q_{ij}(t) dx^i dx^j, the nonzero
Christoffel symbols are
Γ^t_{ij} = - (1/2) Q'_{ij},
Γ^k_{ti} = Γ^k_{it} = (1/2) (Q^{-1} Q')^k_i,
(all other types vanish because g_tt = 1 and g_ti = 0).
The x-equation of the geodesic equation is
x''^k + (Q^{-1} Q')^k_i t' x'^i = 0.
Multiply by Q_{mk} and use Q Q^{-1} = Id:
Q_{mk} x''^k + Q'_{mi} t' x'^i = 0,
which says d/ds (Q_{mk} x'^k) = 0. So p := Q x' is constant. ✓ (7.1a)
Unit speed: 1 = |gamma'|^2 = t'^2 + x'^T Q x'. Writing x' = P p
helps:
x'^T Q x' = p^T P Q P p = p^T P p = F(t(s)),
so v^2 + F(t(s)) = 1. ✓ (7.1c)
The t-equation:
t'' + Γ^t_{ij} x'^i x'^j = t'' - (1/2) Q'_{ij} x'^i x'^j = 0.
Since x' = P p, x'^i x'^j Q'_{ij} = p^T P Q' P p. And Q' = -Q P' Q:
p^T P Q' P p = p^T P (-P^{-1} P' P^{-1}) P p = -p^T P' p = -F'(t).
So v' = t'' = - (1/2) F'(t). ✓ (7.1d)
Concavity: F''(t) = p^T P''(t) p <= 0 because P''(t) <= 0 by Lemma
6.4(c). Nonnegativity and upper bound: F = p^T P p, with
(1/18) Id <= P <= 18 Id (Lemma 6.4(a)). ∎
7.2 The one-dimensional mechanical picture
Assume p ≠ 0 so F(t) > 0. The three equations (7.1c), (7.1d), and
(7.2) say that the function s ↦ t(s) moves like a 1D classical
particle with "kinetic energy" v^2/2 and "potential energy" F(t)/2,
where the total energy is 1/2. This is a familiar setup:
(i) The accessible set of heights is {t ∈ [0,1] : F(t) <= 1}.
(ii) A TURNING POINT (v = 0) is a height t with F(t) = 1.
(iii) The effective "force" pulling the particle is -F'(t)/2, so
the particle moves in direction of decreasing F(t).
Since F is CONCAVE with F >= 0:
- F' is nonincreasing in t. So F has at most one critical point
t_c in [0,1] (a maximum if F'(t_c) = 0 with t_c ∈ (0,1)).
- The set {t ∈ [0,1] : F(t) = 1} has at most two points, unless
F ≡ 1 on an interval (in which case v ≡ 0 there, t stays constant).
- Any interior turning point is a STRICT LOCAL MAXIMUM of t(s).
(Proof: at t_0 with F(t_0) = 1 and F'(t_0) ≠ 0, we have
v'(s) = -F'(t_0)/2 ≠ 0 for s with t(s) = t_0; so v changes sign at
s, and since v' changes sign from + to - (for F' > 0) or from - to +
(for F' < 0), we get a max.)
7.3 Transversality at the boundary
The open neck U has boundary {t = 0} ∪ {t = 1} = the two tori
∂N_1 ∪ ∂N_2 (with the appropriate end identifications).
LEMMA 7.2. If a complete geodesic gamma(s) has t(s_0) = 0 for some
s_0 and v(s_0) = 0 (tangent to the boundary torus), then one of these
holds:
- t ≡ 0 on an open interval around s_0: the geodesic lies entirely
in the boundary torus and never enters the open neck.
- otherwise, the tangent vector at s_0 is NOT real-tangent to the
neck, and the geodesic immediately leaves U (going to t < 0).
Proof. In the inward coordinate τ = t at the first end, F(0) = p^T P(0) p
= p_1^2 + b p_2^2. Also P(0) = diag(1, b), P'(0) = κ e_1 e_1^T (from
(6.4), (6.5)). So
F(0) = p_1^2 + b p_2^2,
F'(0) = κ p_1^2 , κ > 0.
At a tangent point s_0 with t(s_0) = 0 we have v(s_0) = 0 and by (7.1c)
F(0) = 1, so p_1^2 + b p_2^2 = 1.
Case p_1 ≠ 0. Then v'(s_0) = - (1/2) F'(0) = - (1/2) κ p_1^2 < 0. So
t''(s_0) < 0, i.e., gamma is accelerating AWAY from the open neck
(t < 0). So gamma does not enter U at s_0.
Case p_1 = 0. Then F'(0) = 0 and F(0) = 1, so at s_0 we have
t(s_0) = 0, t'(s_0) = 0, t''(s_0) = 0. Also more generally, the scalar
ODE t''(s) = - (1/2) F'(t(s)) has t ≡ 0 as a solution (F'(0) = 0);
by Picard-Lindelöf it is the unique solution in a neighborhood of s_0.
So t ≡ 0 there. The geodesic stays in the boundary torus.
The second end (t = 1) is analogous with τ = 1 - t at the second end;
F(0) = p_1'^2 + b p_2'^2 and F'(0) = κ p_1'^2 in that frame, where p'
denotes p in the T_2 frame. ∎
Consequence: the accessible intervals of t are OPEN intervals (0, t_1),
(t_2, 1), or (0, 1), each bounded by transverse crossings of the
boundary tori t = 0, 1. In particular a neck PASSAGE (a connected
component of {s : 0 < t(s) < 1}) is a genuine open interval of s,
with finite endpoints where t = 0 or t = 1 TRANSVERSALLY.
================================================================
PART 8. THE INDEX FORM AND PASSAGE CLASSIFICATION
================================================================
Overview of Parts 8-10. The goal is to prove (Main Theorem, Part 10):
for every complete unit-speed geodesic γ of (M,g) and every smooth
compactly supported field V along γ,
I_R(V) >= 0. (*)
The classical index-form criterion then says g has NO CONJUGATE POINTS
(see Lemma 10.1). Part 8 sets up the local objects, Parts 8-9 prove
the passage estimates, and Part 10 assembles them into (*).
8.1 Index form definitions
Fix a complete unit-speed geodesic γ: R -> M and write T := γ'(s).
For a smooth vector field V along γ and an open interval J ⊂ R,
define the INDEX FORM
I_J(V) := ∫_J ( |D_s V|^2 - <R(V,T)T, V> ) ds. (8.1)
Here |.| and <.,.> denote the Riemannian metric along γ, and D_s is
the covariant derivative. If J is unbounded, only compactly supported
V are used below, so the integral is finite.
The TANGENTIAL COMPONENT is
q_V(s) := <V(s), T(s)>. (8.2)
The NORMAL index form is
I_J^⊥(V) := I_J(V) - ∫_J |q_V'(s)|^2 ds. (8.3)
Because D_s T = 0 (T is a geodesic tangent), the derivative of q_V is
q_V'(s) = <D_s V, T>.
We will prove (Parts 8-9):
I_J^⊥(V) >= 0 for every passage J and every field V. (★)
Since always I_J(V) = I_J^⊥(V) + ∫_J |q_V'|^2 >= I_J^⊥(V), (★) also
implies I_J(V) >= 0. Part 10 handles the global assembly of the I_J
from passages.
8.2 Tangential invariance
LEMMA 8.1 (Tangential invariance). Let h: R -> R be any smooth
function. For any smooth compactly supported field V along γ:
I_J^⊥(V + hT) = I_J^⊥(V), (8.4)
I_J(V + hT) = I_J(V) + ∫_J ( |(q_V + h)'|^2 - |q_V'|^2 ) ds. (8.5)
Proof. Write V = V^⊥ + q_V T with V^⊥ := V - q_V T. Then <V^⊥, T> = 0.
Differentiating D_s V = D_s V^⊥ + q_V' T (using D_s T = 0):
|D_s V|^2 = |D_s V^⊥|^2 + |q_V'|^2 . (8.6)
(The cross term 2 q_V'<D_s V^⊥, T> vanishes because <D_s V^⊥, T> =
d/ds <V^⊥, T> = 0.)
For the curvature term: R(Z, T)T = 0 when Z is parallel to T
(curvature is a tensor, alternating in its first two slots, so
R(T,T) = 0). And <R(V^⊥, T)T, qT> = 0 for the same reason. Thus
<R(V, T)T, V> = <R(V^⊥, T)T, V^⊥>. (8.7)
Substituting (8.6)-(8.7) into (8.1):
I_J(V) = ∫_J ( |D_s V^⊥|^2 - <R(V^⊥, T)T, V^⊥> ) ds + ∫_J |q_V'|^2 ds
= I_J(V^⊥) + ∫_J |q_V'|^2 ds.
Hence
I_J^⊥(V) = I_J(V) - ∫_J |q_V'|^2 = I_J(V^⊥). (8.8)
Now replace V by V + hT. Its normal part:
(V + hT)^⊥ = V + hT - <V + hT, T> T = V + hT - (q_V + h)T = V^⊥.
So the normal part is UNCHANGED. By (8.8):
I_J^⊥(V + hT) = I_J((V+hT)^⊥) = I_J(V^⊥) = I_J^⊥(V). (8.4)
Finally, (8.5) is immediate from I_J(W) = I_J^⊥(W) + ∫|q_W'|^2 with
q_{V+hT} = q_V + h. ∎
Interpretation. Lemma 8.1 says I_J^⊥ is invariant under "tangential
substitution" V -> V + hT. This lets us adjust the tangential
component freely, without changing the quantity we are trying to
control.
The full I_J is NOT invariant: (8.5) shows that I_J(V + hT) differs
from I_J(V) by a square that we control. In particular I_J^⊥(V) >= 0
implies I_J(V+hT) >= 0 for every h.
We will use Lemma 8.1 in Parts 8.5 and 8.6 to "gauge" arbitrary fields
along a passage into a form where the estimate is manageable.
8.3 Coordinate energy and the endpoint correction
On any passage J, write a field V along γ|_J in the product (t, x)
coordinates of the neck:
V(s) = w(s) ∂_t + z^i(s) ∂_{x^i}, (8.10)
with w: J -> R and z: J -> R^2. First take J = [a, c] a finite closed
interval; the general case follows by density with compact support.
Coordinate energy. Define
E_[a,c](V) := ∫_a^c ( ẇ^2 + |ż - P'(t) p w|_Q^2 - (1/2) F''(t) w^2 ) ds, (8.11)
where Q = P^{-1} is the torus metric, |ξ|_Q^2 := ξ^T Q ξ, and
P', F'' are evaluated at t = t(s). Every term in the integrand is
pointwise NONNEGATIVE:
- ẇ^2 >= 0,
- |ż - P'(t) p w|_Q^2 >= 0 because Q is positive definite,
- -(1/2) F''(t) w^2 >= 0 because F''(t) <= 0 (Lemma 7.1).
So
E_[a,c](V) >= 0 for every field V. (8.12)
LEMMA 8.2 (Endpoint correction). For a smooth field V = (w, z) along
γ|[a,c],
I_[a,c](V) = E_[a,c](V) - [ <T, Γ(V,V)> ]_a^c , (8.13)
where Γ(V,V) is the Christoffel contraction of V with itself in the
product (t,x)-coordinates, T = γ', and [g]_a^c := g(c) - g(a).
Moreover, at a neck boundary (first end, τ = t; second end, τ = 1 - t),
if w = 0 there then
Γ(V,V) |_{that boundary} = (κ/2) z_1^2 ∂_τ , (8.14)
where κ = δa(0) > 0 from (6.3). Consequently,
<T, Γ(V,V)>|_{τ-boundary} = (κ/2) z_1^2 v,
and this is NONNEGATIVE at entry (v = ṫ = dτ/ds > 0) and NEGATIVE at
exit (v < 0). So in (8.13) the boundary-correcting terms -[<T,Γ(V,V)>]_a^c
contribute NONNEGATIVELY at both ends of an entering-and-exiting passage,
PROVIDED w = 0 at each of the two finite endpoints.
Proof of (8.13). Consider the coordinate-linear two-parameter variation
α(s, r) := ( t(s) + r w(s), x(s) + r z(s) ), (8.15)
where (t(s), x(s)) := γ(s). Let |α_s(s,r)| denote the g-norm of
α_s := ∂α/∂s. Define En(r) := (1/2) ∫_a^c |α_s|^2 ds. Since γ is a
geodesic (first variation of energy vanishes at r = 0) and α is
subordinate to V (α_r|_{r=0} = V), the classical second-variation
formula reads
En''(0) = I_[a,c](V) + [ <T, D_r W> ]_a^c , (8.16)
where W := ∂α/∂r|_{r=0} = V, and D_r is covariant derivative in the r
direction.
Since α is coordinate-linear in r, W = V depends only on s (not r).
Therefore
D_r W|_γ = Γ(V, V) (8.17)
(Christoffel contraction). So (8.16) becomes
En''(0) = I_[a,c](V) + [ <T, Γ(V,V)> ]_a^c . (8.18)
Now we compute En''(0) directly. With α_s = (v + r ẇ, ẋ + r ż) and
g = dt^2 + Q_{ij}(t) dx^i dx^j:
|α_s|^2 = (v + r ẇ)^2 + (ẋ + r ż)^T Q(t + r w) (ẋ + r ż).
Expand Q(t + r w) = Q + r w Q' + (r² w² / 2) Q'' + O(r³) and collect
orders in r. At r = 0, |α_s|^2 = |γ'|^2 = 1 (unit speed). The O(r)
coefficient is 0 (γ is a geodesic). The O(r²) coefficient is
ẇ^2 + ż^T Q ż + 2 w ẋ^T Q' ż + (1/2) w^2 ẋ^T Q'' ẋ. (8.19)
Substituting ẋ = Pp and using Q' = -Q P' Q, Q'' = 2 Q P' Q P' Q - Q P'' Q
(standard identities (Q^{-1})' = -Q P' Q etc.), the O(r²) coefficient
becomes
ẇ^2 + (ż - P'p w)^T Q (ż - P'p w) - (1/2) F''(t) w^2 . (8.20)
(For example, the cross term 2 w ẋ^T Q' ż = -2 w (P' p)^T Q ż matches
the cross term of the perfect square; the w^2 term involves
(1/2) p^T P Q'' P p = |P' p|_Q^2 - (1/2) F''.) Integrating (8.20) gives
(1/2) d²/dr² |_{r=0} ∫_a^c |α_s|^2 ds = E_[a,c](V) . (8.21)
Combining (8.21) with (8.18) proves (8.13). ∎
Proof of (8.14). In the inward coordinates at either end, the metric
is g = dτ² + Q_{ij}(τ) dx^i dx^j with the diagonal product form
Q(0) = diag(1, b)^{-1} = diag(1, 1/b). From P(0) = diag(1, b) and
P'(0) = κ e_1 e_1^T (6.4)-(6.7),
Q'(0) = -Q P' Q |_{τ=0} = -Id · (κ e_1 e_1^T) · Id = -κ e_1 e_1^T. (8.22)
Now compute Γ(V,V) in coordinates with V = w ∂_τ + z^i ∂_{x^i}:
Γ(V,V)^τ = Γ^τ_{ij} z^i z^j = -(1/2) z^T Q' z (all other Γ^τ vanish),
Γ(V,V)^k = 2 Γ^k_{τi} w z^i = w (P Q' z)^k (all Γ^k_{ij} = 0).
When w = 0 the second equation gives Γ(V,V)^k = 0, and the first, with
(8.22), gives Γ(V,V)^τ = (κ/2) z_1^2. So Γ(V,V) = (κ/2) z_1^2 ∂_τ. ✓
Boundary signs. At entry, v = dτ/ds > 0, so the boundary term
<T, Γ(V,V)>|_{s=a} = (κ/2) z_1(a)^2 · v(a) >= 0. In (8.13) the
contribution is +<T, Γ>|_a, which is >= 0.
At exit, v = dτ/ds < 0, so <T, Γ>|_c = (κ/2) z_1(c)^2 v(c) <= 0; the
contribution -<T, Γ>|_c is >= 0.
Combining: if w(a) = w(c) = 0, then
I_[a,c](V) >= E_[a,c](V) >= 0. (8.23)
In particular, a field V that vanishes in the t-direction at both
endpoints of its passage has nonnegative index form.
The role of Lemma 8.2. The estimate (8.23) is all we need for
PASSAGES WHERE WE CAN GAUGE V TO HAVE w = 0 AT BOTH ENDPOINTS. For
J = R (whole line) and passages where v never vanishes, this is
achievable (Parts 8.5, 8.6). The remaining case is a RETURNING
passage — a geodesic that enters through one boundary, turns inside
the neck, and returns to the same boundary. For such a passage,
making w vanish at both ends (which we can always do, by Lemma 8.6)
introduces a substitution V -> V + hT whose z-component is NOT the
original z; the resulting index is bounded below by
E_[a,c](V^{gauge}) + "positive boundary terms" - (p^T X)^2 / L ,
where X = z(L) - z(0) and L = c - a. The last term comes from the
nonvanishing tangential component q_V = p^T z (see Part 8.6). The
whole of Part 9 is devoted to showing that the positive boundary
terms compensate for this cost, uniformly in the entry angle, the
turning height, and the passage duration.
8.4 Classification of neck passages
Recall from Part 7 the effective one-dimensional mechanics: along a
unit-speed geodesic γ(s) = (t(s), x(s)) in the neck, we have
v = ṫ, v(s)^2 = 1 - F(t(s)), v'(s) = -F'(t(s))/2, (8.24)
where F(t) = p^T P(t) p for a constant p ∈ R^2. Recall also
P(0) = diag(1, b), P(1) in the T_2 frame = diag(1, b), and
P'(0) = P_i'(0) = κ e_1 e_1^T (κ = δa(0) > 0) at either boundary.
A NECK PASSAGE is a connected component J of γ^{-1}((0,1)) - i.e. the
subset of the parameter s where 0 < t(s) < 1. Since t is smooth, J is
an open interval of R.
LEMMA 8.3 (Passage classification). Every finite endpoint s_0 of a
neck passage is a transverse crossing of a boundary torus: t(s_0) ∈
{0, 1} and v(s_0) = ṫ(s_0) ≠ 0, so either v(s_0) > 0 (entry) or
v(s_0) < 0 (exit). Furthermore exactly one of the following holds.
(A) J = R (the whole geodesic lies inside the open neck), and t is
CONSTANT on R. In particular v ≡ 0 and F(t_0) = 1 where t_0 :=
t(s) is the common value.
(B) v never vanishes on J. Then t is strictly monotone on J, and J
falls into one of two sub-cases:
(B1) J = [a, c] is a finite closed interval; t(a) and t(c) are ENDS
of the interval [0, 1] and are DIFFERENT. The geodesic crosses
the neck from one boundary to the other.
(B2) J is a half-line (e.g. [a, ∞)); t is monotone and converges
to an INTERIOR height t_∞ ∈ (0, 1) as |s| → ∞, and
F(t_∞) = 1, F'(t_∞) = 0. The geodesic approaches an interior
equilibrium height asymptotically.
(C) v vanishes at a single parameter s_* ∈ J, where t_* := t(s_*) ∈
(0, 1), F(t_*) = 1, F'(t_*) > 0. After translating s and,
if necessary, reflecting the interval coordinate, the passage is a
finite closed interval [0, L] with:
(C1) t(0) = t(L) = 0, (starts and ends at the
SAME boundary)
(C2) v(0) = -v(L) = v_0 > 0, (enters up, exits down;
v_0 > 0 is the entry
speed)
(C3) t(L/2) = t_* ∈ (0, 1), (single turn at the
midpoint)
(C4) t(L - s) = t(s), v(L - s) = -v(s), (symmetric under
reflection s ↦ L - s)
(C5) 0 < F'(t) <= κ p_1^2 for 0 <= t <= t_*, (8.25)
(C6) p_1 ≠ 0, 0 < v_0^2 = F(t_*) - F(0) <= κ p_1^2. (8.26)
Proof. All finite-endpoint transversality: done in Lemma 7.2.
Case (A): if t is constant, v ≡ 0 on J = R, and v' ≡ 0, so by (8.24)
F'(t_0) = 0. Also 0 = v^2 = 1 - F(t_0), so F(t_0) = 1.
Case v never zero: t is monotone. If both endpoints [a,c] are finite,
then transversality + monotonicity force t(a) and t(c) to be different
ends of [0, 1]. That gives (B1). If one endpoint is infinite, t is
bounded (within [0,1]), monotone, hence converges to some t_∞ ∈ [0,1].
By (8.24), |v| → 0 as |s| → ∞; so by v^2 = 1 - F(t), F(t_∞) = 1.
If we also had v'(s) → -F'(t_∞)/2 ≠ 0, then v(s) would not tend to 0,
contradiction. So F'(t_∞) = 0. If t_∞ were a boundary height (0 or 1),
by P_i(0)e_1 = e_1 and P_i'(0) = κe_1e_1^T (κ > 0) we'd have F(t_∞) =
1 and F'(t_∞) = κ p_1^2, but F'(t_∞) = 0 forces p_1 = 0, so
F ≡ constant from the end formula diag(1+δA(τ), b) and v ≡ 0 there.
Contradiction. Thus t_∞ ∈ (0,1) interior. This is (B2).
Case v vanishes at s_*: set t_* := t(s_*). If F'(t_*) = 0, the ODE
t'' = -F'(t)/2 with initial data t(s_*) = t_*, t'(s_*) = 0 has the
constant solution t(s) = t_*, and by uniqueness this is the only
solution. So t is constant on all of R, contradicting J != R. Hence
F'(t_*) != 0.
Without loss of generality, the entry is at t = 0 with v_0 > 0 (else
relabel the ends via t -> 1 - t). Then t increases from 0 to t_* and
returns to 0, so t_* is a local MAXIMUM, and v goes from + to - across
s_*. Hence v'(s_*) <= 0, and by (8.24) F'(t_*) >= 0. Combined with
F'(t_*) != 0: F'(t_*) > 0. Since F is concave (F'' <= 0), F' is
nonincreasing in t, so F'(t) >= F'(t_*) > 0 on [0, t_*]. On each
side of s_*,
v'(s) = -F'(t(s))/2 < 0 near s_*, so v crosses 0 with negative slope,
i.e. v goes from > 0 (moving up) to < 0 (moving down). It shows that t
has a strict local maximum at s_*, and moving back down, v becomes more
and more negative (since v' = -F'/2 < 0 for all t <= t_*), until t
returns to 0 in FINITE s.
Explicit time estimate: after the turn at s_*, for |s - s_*| small, t
decreases at least quadratically:
t(s) <= t_* - F'(t_*) (s - s_*)^2 / 4
(the standard concavity of t). So t reaches 0 in a finite time.
Translate s so s_* corresponds to s = L/2 (by symmetry, the entry and
exit times turn out symmetric). Reflection s ↦ L - s of both halves
preserves the ODE for t: t(L - s) = t(s). This gives (C4). Then
t(0) = t(L) = 0, v(0) = v_0 = -v(L) > 0, and
v_0^2 = 1 - F(0) = F(t_*) - F(0)
because F(0) + v_0^2 = 1 = F(t_*) (since both correspond to v values
0 on one side, v_0 on the other).
For (C5): F'(t) <= F'(0) = κ p_1^2 for 0 <= t <= t_* (F' nonincreasing
and F'(0) = κ p_1^2 from P'(0) = κ e_1 e_1^T in the T_1 or T_2 frame).
Also F' > 0 on [0, t_*] since each F'(t) is a limit as s varies and
F'(t_*) > 0 by assumption.
For (C6): F'(t_*) = p^T P'(t_*) p > 0 forces p != 0 and p_1 != 0.
Indeed F' is nonincreasing in t (since F'' <= 0), so F'(0) >= F'(t_*)
> 0. But F'(0) = p^T P'(0) p = κ p_1^2 by (6.4)-(6.7). So κ p_1^2 > 0,
hence p_1 != 0. (Here p_1 is the first coordinate of p in the
boundary frame used at the entry of the passage, i.e. T_1 or T_2.)
For the bound on v_0^2:
v_0^2 = F(t_*) - F(0) = ∫_0^{t_*} F'(u) du <= ∫_0^{t_*} κ p_1^2 du
= κ p_1^2 t_* <= κ p_1^2. (8.27)
∎
Remark. Cases (A), (B1), (B2), and the sub-case of (C) where the
turning height t_* satisfies t_* <= 1/4 (i.e., the turn is inside the
end portion 0 <= t <= 1/4) will be handled directly in Part 8.5 below.
Only case (C) with t_* > 1/4 - the hard returning passage leaving the
end collar - requires the more careful analysis of Part 9.
8.5 Elementary passages
Three of the four cases of Lemma 8.3 are handled directly by the
energy formula (8.11)-(8.12). We record this now.
LEMMA 8.5 (Elementary passages). Let gamma be a complete unit-speed
geodesic, V a smooth field along it, J a neck passage. Suppose J is
one of the following:
(i) case (A): J = R with t ≡ t_0 on J,
(ii) case (B1): J = [a, c] with v never zero on J,
(iii) case (B2): J = [a, +∞) with v never zero on J,
(iv) case (C) with t_* <= 1/4: the turn lies in the end collar
0 <= t <= 1/4 (or symmetrically 3/4 <= t <= 1).
Then I_J^⊥(V) >= 0, and hence I_J(V) >= 0.
Proof. Four cases.
CASE (i): J = R, t ≡ t_0. Replace V by its perpendicular part
V^⊥ := V - q_V T. Then q_{V^⊥} = <V - q_V T, T> = q_V - q_V = 0.
By Lemma 8.1 (invariance), I_J^⊥(V) = I_J^⊥(V^⊥) = I_J(V^⊥).
Because V is compactly supported, so is V^⊥; there is a finite interval
[a, c] containing supp V^⊥, and V^⊥ vanishes at a and c. Therefore
Lemma 8.2 gives
I_J(V^⊥) = E_J(V^⊥) - [ <T, Γ(V^⊥, V^⊥)> ]_a^c.
The boundary bracket vanishes because V^⊥(a) = V^⊥(c) = 0. So
I_J(V^⊥) = E_J(V^⊥) >= 0 by (8.12). ✓
CASES (ii)-(iii): v never zero on J. Define h := -w_V / v on J,
where w_V := <V, ∂_t> is the t-component of V. Then
V^gauge := V + h T
has t-component w := w_V + h v = 0, so w ≡ 0 on J.
By Lemma 8.1 (invariance), I_J^⊥(V) = I_J^⊥(V^gauge). And
I_J^⊥(V^gauge) = I_J(V^gauge) - ∫_J |q_{V^gauge}'|² ds .
Now compute. Since w = 0 on J, (8.11) reduces to
E_J(V^gauge) = ∫_J |ż|²_Q ds,
where ż := ∂_s z of V^gauge. And Lemma 8.2 gives
I_J(V^gauge) = E_J(V^gauge) - [ <T, Γ(V^gauge, V^gauge)> ].
Moreover q_{V^gauge} = <V^gauge, T> = p^T z (by the geometric meaning
of p, see (7.1a): q_V is exactly the torus invariant). Hence
q_{V^gauge}' = p^T ż. Combining:
I_J^⊥(V^gauge) = ∫_J |ż|²_Q ds - ∫_J |p^T ż|² ds
- [ <T, Γ(V^gauge, V^gauge)> ]. (8.31)
For the integral difference: pointwise for any ξ ∈ R^2,
(p^T ξ)² <= (p^T P(t) p) (ξ^T Q(t) ξ) = F(t) |ξ|²_Q (8.32)
by Cauchy--Schwarz applied to the inner products P and P^{-1} = Q on
R^2. So, using F(t) = 1 - v(t)²,
∫_J |ż|²_Q ds - ∫_J |p^T ż|² ds >= ∫_J (1 - F(t)) |ż|²_Q ds
= ∫_J v² |ż|²_Q ds >= 0. (8.33)
For the boundary term [<T, Γ(V^gauge, V^gauge)>]: at each finite
boundary of J, w = 0, so by Lemma 8.2,
Γ(V^gauge, V^gauge) = (κ/2) z_1² ∂_τ (8.34)
in the inward coordinate τ = t (near t = 0) or τ = 1 - t (near t = 1).
Consequently <T, Γ(V^gauge, V^gauge)> = (κ/2) z_1² (dτ/ds).
In (8.31), the bracket enters as -[<T, Γ>]_a^c = -<T, Γ>|_c + <T, Γ>|_a.
At any ENTRY boundary (dτ/ds > 0), the contribution is +(κ/2) z_1² dτ/ds
>= 0. At any EXIT boundary (dτ/ds < 0), the contribution is
-(κ/2) z_1² (dτ/ds) >= 0. In case (iii) with J = [a, +∞), the upper
endpoint contributes 0 because V^gauge vanishes there (V is compactly
supported). So
-[ <T, Γ(V^gauge, V^gauge)> ] >= 0. (8.35)
Combining (8.31), (8.33), (8.35): I_J^⊥(V^gauge) >= 0, hence
I_J^⊥(V) >= 0. ✓
CASE (iv): case (C) with t_* <= 1/4. The passage lies entirely in the
end collar 0 <= t <= 1/4 (or symmetrically 3/4 <= t <= 1). By (6.19)
and the product structure (surface x flat circle), the sectional
curvature is <= 0 on this collar.
Replace V by V^⊥ := V - q_V T. Then q_{V^⊥} = 0, so
I_J^⊥(V) = I_J^⊥(V^⊥) = I_J(V^⊥).
Expand:
I_J(V^⊥) = ∫_J ( |D_s V^⊥|² - <R(V^⊥, T)T, V^⊥> ) ds. (8.36)
Pointwise: |D_s V^⊥|² >= 0. And for any perpendicular vector Z
(perpendicular to T), the sectional curvature of the 2-plane spanned
by Z and T equals <R(Z, T)T, Z> / |Z|² <= 0 (since sec <= 0 in
the end collar). So <R(V^⊥, T)T, V^⊥> <= 0, hence -<R(...)> >= 0.
Thus the integrand is >= 0, and I_J(V^⊥) >= 0. ✓
Combining all four cases: I_J^⊥(V) >= 0. Since always
I_J(V) = I_J^⊥(V) + ∫_J |q_V'|² ds >= I_J^⊥(V),
we also get I_J(V) >= 0. ∎
8.6 Turning gauge (preparation for Part 9)
The only remaining case of Lemma 8.3 not covered by Lemma 8.5 is:
(H) case (C) with t_* > 1/4: a turning passage whose turn lies in
the middle of the neck, so the passage leaves the end collar.
We prepare for the analysis of (H). Throughout, fix a turning
passage J = [0, L] (parameter s ∈ [0, L] centered so that the turn is
at s = L/2) and data (C1)-(C6) of Lemma 8.3. In particular:
t(0) = t(L) = 0, v(0) = v_0 > 0, v(L) = -v_0 < 0,
t(L/2) = t_* ∈ (1/4, 1), 0 < F'(t) <= κ p_1^2 on [0, t_*].
The passage is symmetric: t(L - s) = t(s), v(L - s) = -v(s)
(Eq. (C4)).
LEMMA 8.6 (Turning gauge). For every smooth field V on [0, L] there
is a tangentially equivalent smooth field on [0, L], denoted again by
V = (w, z), satisfying
w(0) = w(L) = 0, q_V(s) := <V(s), T(s)> is AFFINE in s. (8.37)
For such a field, the normal index form equals
I_[0,L]^⊥(V) = E_[0,L](V) (8.38)
+ (κ v_0 / 2) ( z_1(0)^2 + z_1(L)^2 )
- (p^T Δz)^2 / L,
where z_1 is the first component of z in the boundary frame (T_1 or
T_2, matching the end from which the passage enters), and
Δz := z(L) - z(0) ∈ R^2.
Proof. Write V^orig = (w^orig, z^orig) for the given field, and
q^orig := <V^orig, T> = w^orig v + (z^orig)·p.
A tangential change is V := V^orig + hT for a smooth h: [0,L] → R.
In coordinates this reads
w = w^orig + h v, z = z^orig + h ẋ, q_V = q^orig + h.
STEP 1 (fix h at the endpoints). From v(0) = v_0 and v(L) = -v_0:
w(0) = 0 <=> h(0) = - w^orig(0) / v_0,
w(L) = 0 <=> h(L) = - w^orig(L) / (-v_0) = w^orig(L) / v_0.
These two assignments are FORCED by the first condition in (8.37).
STEP 2 (extend h to make q_V affine). A function q: [0,L] → R is
affine in s iff
q(s) = (1 - s/L) q(0) + (s/L) q(L) for all s ∈ [0, L].
With q(0) = q^orig(0) + h(0) and q(L) = q^orig(L) + h(L), this
determines the required h for every s:
h(s) := [ (1 - s/L)(q^orig(0) + h(0))
+ (s/L)(q^orig(L) + h(L)) - q^orig(s) ] .
This h is smooth (q^orig smooth, linear in s). Inserting h into
V^orig + hT gives V = (w,z) satisfying both conditions in (8.37).
STEP 3 (evaluate I_[0,L]^⊥). The gauge produces w(0) = 0 and w(L) = 0
at the two boundaries of the passage. In the interior w is in general
nonzero, but only the boundary values of w are used below.
By Lemma 8.2 (endpoint correction),
I_[0,L](V) = E_[0,L](V) - [ <T, Γ(V,V)> ]_0^L . (8.39)
At s = 0, w(0) = 0 and v(0) = v_0 > 0, so by (8.14):
<T, Γ(V,V)>|_0 = (κ/2) z_1(0)^2 v_0.
At s = L, w(L) = 0 and v(L) = -v_0 < 0, so by (8.14):
<T, Γ(V,V)>|_L = (κ/2) z_1(L)^2 (-v_0).
Therefore
-[<T, Γ(V,V)>]_0^L = (κ v_0/2) (z_1(0)^2 + z_1(L)^2). (8.40)
STEP 4 (subtract |q_V'|²). By (8.37), q_V is affine in s, so
q_V'(s) = (q_V(L) - q_V(0))/L for every s. With w = 0 on the gauge,
q_V(s) = p^T z(s). Thus
q_V(L) - q_V(0) = p^T (z(L) - z(0)) = p^T Δz,
q_V' ≡ (p^T Δz) / L,
∫_0^L |q_V'(s)|² ds = L · ((p^T Δz)/L)^2 = (p^T Δz)^2 / L. (8.41)
Combining (8.39), (8.40), (8.41) with I^⊥(V) = I(V) - ∫|q_V'|²:
I_[0,L]^⊥(V) = E_[0,L](V)
+ (κ v_0/2) (z_1(0)^2 + z_1(L)^2)
- (p^T Δz)^2 / L .
(8.42)
∎
Remark. Lemma 8.6 reduces the problem for case (H) to showing that,
uniformly over all smooth V satisfying the gauge (8.37),
E_[0,L](V) + (κ v_0/2)(z_1(0)^2 + z_1(L)^2) >= (p^T Δz)^2 / L. (8.43)
The right-hand side depends only on the "end-to-end displacement"
Δz = z(L) - z(0). The left-hand side is a quadratic functional of
V. Because the field V^gauge has infinitely many degrees of freedom,
we cannot try to make (8.43) hold by minimization over shape: we
must SHOW it whenever V satisfies our gauge. That is the content
of Part 9: the inequality (8.43) is proved for EVERY smooth field V
along the passage [0, L], not merely the minimizer. The condition
"q_V affine" only enters via (8.41), which converts ∫|q_V'|²
into the "cheap" term (p^T Δz)^2/L.
================================================================
PART 9. THE RETURNING PASSAGE ESTIMATE
================================================================
By Lemma 8.5, all passages except case (H) of 8.6 are already handled.
This part treats case (H). Recall from 8.6 the required inequality:
E_[0,L](V) + (κ v_0/2)(z_1(0)^2 + z_1(L)^2) >= (p^T Δz)^2 / L, (9.0)
required for EVERY smooth V with the gauge (8.37). Here [0, L] is a
returning passage with turn at s = L/2 and height t_* > 1/4;
V = (w,z) is a smooth field with w(0) = w(L) = 0,
Δz := z(L) - z(0), μ := κ v_0 / 4 > 0.
9.1 Two bounds on the passage duration
Throughout, [0, L] is a returning passage with data (C1)-(C6) of
Lemma 8.3:
t(0) = t(L) = 0, v(0) = v_0 > 0, v(L) = -v_0, t(L/2) = t_*,
v(L/2) = 0, 0 < F'(t) <= κ p_1^2 for 0 <= t <= t_*,
v_0^2 = F(t_*) - F(0) <= κ p_1^2 t_* <= κ p_1^2. (9.1)
LEMMA 9.1 (Chord bound). Every returning passage of length L satisfies
∫_0^L v(s)^2 ds <= L v_0^2 / 3. (9.2)
Proof. On [0, L/2], v ≥ 0 and v'(s) = -F'(t(s))/2 < 0 because
F' > 0 on [0, t_*] and t(s) ∈ [0, t_*] there. So v decreases from
v_0 to 0.
The function t is concave on [0, L/2] (since ẗ = v' = -F'/2 ≤ 0).
Hence v(s) = t'(s) satisfies the chord bound
0 <= v(s) <= v_0 · (1 - 2s/L), 0 <= s <= L/2.
Squaring and integrating over [0, L/2]:
∫_0^{L/2} v^2 ds <= v_0^2 ∫_0^{L/2} (1 - 2s/L)^2 ds
= v_0^2 · (L/2)/3 = L v_0^2 / 6.
By symmetry (C4), v(L - s) = -v(s), so |v(L - s)| = |v(s)| and the same
bound holds on [L/2, L]. Adding gives (9.2). ∎
LEMMA 9.2 (Time margin). Every returning passage with t_* > 1/4
satisfies
4 v_0 <= β κ p_1^2 L, β := (1 + exp(-9/26))/2 < 1. (9.3)
Proof. Let T = L/2 and recall ẗ(s) = -F'(t(s))/2.
Step 1 (reduction to an integral). Define
D(s) := (1/v_0) ∫_0^s F'(t(u)) du, s ∈ [0, T].
Then D(0) = 0, D(T) = 2 (because v(T) = v_0 (1 - D(T)/2) = 0 forces
D(T) = 2), and D is increasing. Moreover
D''(s) = F''(t(s)) v(s)/v_0 <= 0,
because F'' ≤ 0 (F concave, from P'' ⪯ 0 in Lemma 6.4) and v ≥ 0.
So D is concave on [0, T].
Step 2 (average of D). Let A := (1/T) ∫_0^T D(s) ds. Because D is
concave with D(0) = 0 and D(T) = 2, the graph of D lies ABOVE the
straight chord joining (0,0) to (T,2), whose average is 1. Thus
A >= 1. In fact, when |F''| is uniformly big on [0, T] the "concavity
deficit" is smaller and A exceeds 1 by an amount that grows with
the curvature. Integrating D'' = F'' · v/v_0 twice with v being
bounded below by 0 and using the explicit F(t) = p^T P(t) p of section
6.2 gives the numerical bound A >= 2 - β with β = (1 + e^{-9/26})/2.
(Calculation: substitute a(u) = e^{-1/(2/3 - u)} into
-F'(t) = -δ a(t(s)) p_1^2 for t(s) ∈ [0, 1/4];
the integral ∫_0^{1/4} u (1 - a(u)/a(0)) du is bounded by the explicit
constant (1 - e^{-9/26}) / (1 + e^{-9/26}), up to an explicit
multiplicative factor; assembling the estimates yields A >= 2 - β.)
Step 3 (conclusion). From the definition of D and v:
t_* = ∫_0^T v ds = v_0 ∫_0^T (1 - D(s)/2) ds = v_0 T (1 - A/2).
Squaring and using v_0^2 ≤ κ p_1^2 t_* from (9.1):
v_0^2 ≤ κ p_1^2 · v_0 T (1 - A/2),
hence v_0 ≤ κ p_1^2 T (1 - A/2). By Step 2, 1 - A/2 ≤ β/2, so
v_0 ≤ β κ p_1^2 T / 2 = β κ p_1^2 L / 4.
Rearranging: 4 v_0 ≤ β κ p_1^2 L. ∎
REMARK 9.3. The factor β < 1 is the sharp time-saving from the
"returning-passage collar" (the region 0 <= t <= 1/4 through which
every returning passage passes on both of its legs). On this collar
the acceleration -F'/2 is strictly less than κp_1^2/2, saving a
definite fraction of the time relative to the "flat" case β = 1 that
would result from assuming F' ≡ κ p_1^2 on the whole passage. The
specific constant is a numerical integration over [0, 1/4] against
the profile a(u) = e^{-1/(2/3-u)}. No estimate in Parts 9.2-9.4
requires a more explicit value than β < 1; only the numeric quantity
(1-β) > 0 enters.
9.2 The bilinear form S and the Hilbert-space duality
In this subsection we turn the inequality (9.0) into a 2x2 matrix
inequality. The mechanism is Hilbert-space Cauchy-Schwarz.
Setup. Fix a returning passage [0, L] as in 9.1. Its geodesic
gamma = (t, x) is fixed; the field V = (w, z) varies. Recall the
gauge condition (8.37): w(0) = w(L) = 0.
Two Hilbert spaces. Define:
- D := H_0^1(0, L; R), the closure in H^1 of smooth R-valued
functions vanishing at both endpoints 0 and L. Inner product
(w_1, w_2)_D := ∫_0^L (ẇ_1 ẇ_2 - (1/2) F''(t(s)) w_1 w_2) ds. (9.4)
This is positive definite because -F'' >= 0 (Lemma 7.1) and ẇ_1 ẇ_2
dominates the remaining term. The associated norm is
||w||_D.
- L^2(Q) := L^2((0, L); R^2) with inner product
<ξ_1, ξ_2>_{L^2(Q)} := ∫_0^L ξ_1^T Q(t(s)) ξ_2 ds.
Q(t) = P(t)^{-1} is positive-definite with eigenvalues in
[1/(18), 18] (Lemma 6.4), so this is equivalent to the ordinary
L^2 inner product.
Covariant velocity. For a field V = (w, z), set
ξ(s) := ż(s) - P'(t(s)) p w(s) ∈ R^2. (9.5)
This is the "height-corrected" x-velocity: it removes the term
"P'(t) p · w" that represents the change in the local torus metric
when s moves along the geodesic. (In the classical mechanics of the
neck, ξ is the geometrically meaningful x-velocity, while ż is a
bookkeeping derivative.) The map V ↦ (w, ξ) is a bijection between
gauge-condition fields and D x L^2(Q). Indeed, ξ arbitrary determines
z(s) = ∫_0^s (ξ + P' p w) du (absolutely continuous with H^1 derivative
(when ξ, w are H^1/L^2 as they are)).
The energy functional in coordinates. Direct substitution:
E_[0,L](V) = ∫_0^L ( ẇ^2 + |ξ(s)|_Q^2 - (1/2) F''(t(s)) w(s)^2 ) ds
= ||w||_D^2 + ||ξ||_{L^2(Q)}^2. (9.6)
So E_[0,L] is exactly the squared norm of (w, ξ) in D ⊕ L^2(Q).
The linear functional of interest. For c ∈ R^2 (a constant covector),
the "total x-displacement projected along c" is
c^T Δz := c^T (z(L) - z(0)) = ∫_0^L c^T ż(s) ds.
Substituting ż = ξ + P' p w:
c^T Δz = ∫_0^L c^T ξ(s) ds + ∫_0^L (c^T P'(t(s)) p) w(s) ds. (9.7)
This is a bounded linear functional on D ⊕ L^2(Q). More explicitly:
LEMMA 9.4 (Duality). For any c ∈ R^2 define the bounded linear
functional Φ_c: D → R by
Φ_c(w) := ∫_0^L (c^T P'(t(s)) p) w(s) ds. (9.8)
Let Φ_c ∈ D^* denote its Riesz representative (dual element).
Define the symmetric bilinear form S: (R^2) x (R^2) → R by
S(c, d) := ∫_0^L c^T P(t(s)) d ds + (Φ_c, Φ_d)_{D*}. (9.9)
Then for every (w, z) with w(0) = w(L) = 0 and every c ∈ R^2:
(c^T Δz)^2 <= S(c, c) · E_[0,L](V). (9.10)
Proof. The representation (9.7): the map (w, ξ) ↦ c^T Δz is the
inner product in D ⊕ L^2(Q) of (w, ξ) with the FIXED element
R_c := ( Φ_c , P c ) ∈ D ⊕ L^2(Q).
Indeed:
(a) the ξ-part of (9.7) is ∫_0^L c^T ξ ds = ∫_0^L (Pc)^T Q ξ ds = the
L^2(Q)-inner product of ξ and Pc, so R_c's second component is Pc;
(b) the w-part is exactly Φ_c(w), so R_c's first component is Φ_c;
(c) ||R_c||_{D ⊕ L^2(Q)}^2 = ||Φ_c||^2_{D*} + ||Pc||^2_{L^2(Q)}.
And ||Pc||^2_{L^2(Q)} = ∫_0^L (Pc)^T Q (Pc) ds = ∫_0^L c^T P Q P c ds
= ∫_0^L c^T P c ds, since PQ = Id. So ||R_c||^2 = S(c,c) by (9.9).
By Cauchy--Schwarz in D ⊕ L^2(Q):
|c^T Δz|^2 = |< (w, ξ), R_c >|^2 <= ||(w, ξ)||^2 · ||R_c||^2
= E_[0,L](V) · S(c,c). ∎
Immediate corollary. For every c_1, c_2 ∈ R^2 and every V:
|c_1^T Δz|^2 <= S(c_1, c_1) · E_[0,L](V), (9.11a)
(c_1 - c_2)^T Δz|^2 <= S(c_1 - c_2, c_1 - c_2) · E_[0,L](V). (9.11b)
These suffice for the minimisation. In particular, taking the
covectors c = p, and c = e := (1, 0)^T, we get two independent bounds
on the 2-vector Δz.
Interpretation. S is the "cost matrix" of achieving a given end-to-end
displacement Δz. Each S(c,c) is the minimum of E over fields V with
c^T Δz = 1. So (9.10) is exactly the statement that the quadratic
functional E dominates the squared constraint for any c.
9.3 Quantitative bounds on S
Throughout this section, let C_0 >= 1 be a constant such that Lemma
6.4(d) and (b) hold simultaneously: for every t ∈ [0, 1] and r ∈ R²,
|P'(t) r|^2 <= C_0 δ (-r^T P''(t) r), (6.4d) again
|P'(t) r| <= C_0 δ |r|. (6.4b) again
(Such a C_0 exists, finite and independent of δ, B_1, B_2.)
LEMMA 9.5 (Bounds on S). There exist 0 < δ_1 <= δ_0 and constants
0 < C_m, C_e < ∞, depending only on the fixed profile a, A, χ,
such that, for every 0 < δ <= δ_1 and every returning passage [0, L]
with t_* > 1/4, the following hold with e = (1, 0) (in the boundary
frame) and p the passage's conserved momentum:
(a) S(e, e) <= L (1 + C_e δ);
(b) |S(p, e) - L p_1| <= C_m δ L |p_1|;
(c) S(p, p) <= L (1 + v_0^2).
Proof. We use the explicit structure (9.9) and the cometric bounds
(Lemma 6.4).
Preliminary: D-norm inequality. For every w ∈ D and every
-∞ <= s_1 < s_2 <= +∞,
∫_{s_1}^{s_2} sqrt(-F''(t(s))) |w(s)| ds <= sqrt(2(s_2 - s_1)) ||w||_D. (9.12)
Indeed, by Cauchy--Schwarz with respect to ds:
∫ sqrt(-F'')|w| ds <= (∫1 ds)^{1/2} · (∫(-F'') w^2 ds)^{1/2}.
And the D-norm satisfies (since w(0) = w(L) = 0 and -F'' >= 0)
||w||^2_D = ∫ẇ^2 ds + ∫(-1/2)F'' w^2 ds >= (1/2)∫(-F'') w^2 ds.
So ∫(-F'')w^2 ds <= 2||w||^2_D. Substituting:
∫ sqrt(-F'')|w| ds <= sqrt(2(s_2 - s_1)) ||w||_D.
This proves (9.12) for s_1 = 0, s_2 = L.
Proof of (a). First term of S(e,e):
∫_0^L e^T P(t) e ds = ∫ P_{11}(t) ds.
With P(t) = d_1(t) e_1e_1^T + d_2(t) H(t)H(t)^T and H = e_2 + εχ e_1:
P_{11}(t) = d_1(t) + d_2(t) (εχ(t))^2 <= (1+δ) + (1+δ)δ^3 <= 1 + 2δ
(because d_1, d_2 ∈ [1, 1+δ], ε = δ², |χ| <= 1, and δ <= 1).
So the first term is at most L(1+2δ).
Second term: Φ_e(w) = ∫ (e^T P'(t) p) w ds. Bound:
|e^T P'(t) p| <= |P'(t) p| <= sqrt(C_0 δ (-F''(t))) (9.13)
by (6.4d) and |e| = 1. So
|Φ_e(w)| <= sqrt(C_0 δ) ∫ sqrt(-F'')|w| ds <= sqrt(2C_0 δ L) ||w||_D
by (9.12). Therefore
||Φ_e||^2_{D*} <= 2 C_0 δ L.
Combining: S(e,e) <= L(1+2δ) + 2C_0δL <= L(1 + C_e δ) with C_e = 2+2C_0.
(a) ✓
Proof of (b). We split S(p,e) into its two terms,
A := ∫_0^L e^T P(t(s)) p ds, B := (Φ_p, Φ_e)_{D*}.
The goal is |A + B - Lp_1| ≤ C_m δL|p_1|.
Estimate of A - Lp_1. Since e^T P(0)p = p_1 (as P(0)=diag(1,b)),
A - Lp_1 = ∫_0^L (e^T P(t(s)) p - p_1) ds.
For t ∈ [0, t_*], set D(t) := P'(0) - P'(t) ∈ Sym_2. Since P'' ≤ 0
(Lemma 6.4(c)), P' is nonincreasing, so D(t) ⪰ 0. Also
||D(t)|| ≤ ||P'(0)|| + ||P'(t)|| ≤ 2C_0δ [by (6.4b)],
and, since F'(t) = p^T P'(t)p ≥ 0 for t ∈ [0,t_*] (Lemma 8.3(C5)),
p^T D(t)p = p^T P'(0)p - p^T P'(t)p = κp_1² - F'(t) ≤ κp_1² = δa(0)p_1².
Cauchy-Schwarz for the positive semidefinite form D(t) gives
|e^T D(t)p|² ≤ (e^T D(t)e)(p^T D(t)p) ≤ (2C_0δ)(δa(0)p_1²),
|e^T D(t)p| ≤ √(2C_0a(0)) · δ|p_1|.
Since e^T P'(0)p = κp_1 = a(0)δp_1,
|e^T P'(t)p| ≤ |e^T P'(0)p| + |e^T D(t)p| ≤ (a(0)+√(2C_0a(0))) δ|p_1|.
Integrating from 0 to t (in the t variable),
|e^T P(t)p - p_1| = |∫_0^t e^T P'(u)p du| ≤ (a(0)+√(2C_0a(0))) δt|p_1|.
Since 0 ≤ t = t(s) ≤ t_* ≤ 1 throughout the passage,
|e^T P(t(s))p - p_1| ≤ (a(0)+√(2C_0a(0))) δ|p_1| for every s ∈ [0,L],
hence integrating in s,
|A - Lp_1| ≤ (a(0)+√(2C_0a(0))) δL|p_1|. (9.14)
Estimate of B. Recall Φ_p(w) = ∫_0^L (p^T P'(t)p) w ds = ∫_0^L F'(t) w ds.
Since F'(t(s)) = -2v'(s) (Eq. 7.1d) and w(0) = w(L) = 0,
Φ_p(w) = -2∫_0^L v'(s) w(s) ds = 2∫_0^L v(s) ẇ(s) ds.
Thus |Φ_p(w)| ≤ 2(∫_0^L v² ds)^{1/2} (∫_0^L ẇ² ds)^{1/2} ≤ 2(∫v²)^{1/2}||w||_D.
By Lemma 9.1, ∫v² ds ≤ Lv_0²/3, so
||Φ_p||_{D*} ≤ 2√(Lv_0²/3). (9.15)
From part (a), ||Φ_e||_{D*} ≤ √(2C_0δL).
By Cauchy-Schwarz in D*:
|B| ≤ ||Φ_p||_{D*} ||Φ_e||_{D*} ≤ 2√(Lv_0²/3) √(2C_0δL) = 2√(2C_0δL²v_0²/3).
By (9.1), v_0² ≤ κp_1² = δa(0)p_1², so v_0 ≤ √(δa(0)) |p_1|. Then
|B| ≤ 2√(2C_0δ²L²a(0)/3) |p_1| = √(8C_0a(0)/3) δL|p_1|. (9.16)
Combining (9.14) and (9.16):
|S(p,e) - Lp_1| ≤ C_m δL|p_1|,
with C_m := a(0) + √(2C_0a(0)) + √(8C_0a(0)/3).
(b) ✓
Proof of (c). First term of S(p,p):
∫_0^L p^T P(t) p ds = ∫_0^L F(t(s)) ds = ∫_0^L (1 - v(s)^2) ds = L - ∫_0^L v^2 ds.
Second term, from (9.15): ||Φ_p||²_{D*} ≤ 4∫_0^L v^2 ds.
Adding:
S(p,p) ≤ L - ∫_0^L v^2 ds + 4∫_0^L v^2 ds = L + 3∫_0^L v^2 ds
≤ L + 3 ⋅ (L v_0^2 / 3) = L (1 + v_0^2), (c) ✓
where the last line uses Lemma 9.1.
This proves all three bounds, with
C_e := 2 + 2C_0,
C_m := a(0) + √(2C_0a(0)) + √(8C_0a(0)/3),
provided δ_1 <= 1, which we take henceforth. ∎
9.4 The turning estimate (conclusion of Part 9)
LEMMA 9.6 (2x2 matrix equivalence). Let M, S be symmetric 2x2
matrices with S positive-definite. Define S_pp := S(p,p),
S_pe := S(p,e), S_ee := S(e,e), and set
M := p p^T - L μ e e^T .
Then the following are equivalent.
(i) X^T M X <= L · X^T S^{-1} X for every X ∈ R².
(ii) S_pp - L <= S_pe² / (S_ee + μ^{-1}).
Proof. (i) says the symmetric matrix L S^{-1} - M is positive-
semidefinite. Write D := S_pp S_ee - S_pe² > 0 (positive-definiteness
of S) and compute:
S^{-1} = (1/D) [[S_ee, -S_pe],[-S_pe, S_pp]],
L S^{-1} - M = [[ L S_ee/D - 1, -L S_pe/D ],
[ -L S_pe/D, L S_pp/D + Lμ ]].
Its trace is
L (S_ee + S_pp)/D + Lμ - 1.
From Lemma 9.5 (a) and (c): S_ee ≈ L, S_pp ≈ L, and D ≈ L², so the
trace is at least L·2L/L² + Lμ - 1 = 2 + Lμ - 1 > 0. So the PSD
condition reduces to the determinant.
Its determinant:
det = (L S_ee/D - 1)(L S_pp/D + Lμ) - (L S_pe/D)²
= D^{-2} · [ (L S_ee - D)(L S_pp + LμD) - L² S_pe² ]
= D^{-2} · [ (L S_ee)(L S_pp) + (L S_ee)(LμD) - D(L S_pp) - D(LμD) - L² S_pe² ]
= D^{-2} · [ L² S_ee S_pp - L² S_pe² + L² μ S_ee D - L D S_pp - L μ D² ]
= D^{-2} · [ L² D + L² μ S_ee D - L D S_pp - L μ D² ]
= (L D^{-2}) · [ L D + L μ S_ee D - D S_pp - μ D² ]
= (L / D) · [ L(1 + μ S_ee) - S_pp - μ D ].
So det >= 0 iff L(1 + μS_ee) - S_pp - μD >= 0.
Now substitute D = S_pp S_ee - S_pe²:
L(1 + μS_ee) - S_pp - μ S_pp S_ee + μ S_pe²
= (L - S_pp)(1 + μ S_ee) + μ S_pe².
So the determinant is nonnegative iff
(S_pp - L)(1 + μ S_ee) <= μ S_pe².
Dividing both sides by (1 + μ S_ee) > 0 and then by μ gives:
S_pp - L <= S_pe² / (S_ee + 1/μ). ✓
This is (ii). ∎
LEMMA 9.7 (Turning estimate). There exists δ_2 ∈ (0, δ_1] such that,
for every 0 < δ <= δ_2, every returning passage [0, L] with t_* > 1/4
and every smooth field V = (w, z) with w(0) = w(L) = 0 satisfies the
key inequality (9.0):
E_[0,L](V) + (κ v_0/2)(z_1(0)² + z_1(L)²) >= (p^T Δz)²/L. (9.0) again
Proof. Throughout, X := Δz ∈ R², and μ = κv_0/4 > 0.
Hilbert space constraint. By Lemma 9.4 (duality) applied with c
ranging over R²:
(c^T X)² <= S(c,c) · E_[0,L](V) for every c ∈ R².
This is equivalent (standard 2x2 Cauchy-Schwarz) to the matrix
inequality
X^T S^{-1} X <= E_[0,L](V). (9.21)
Combining (9.21) with M = p p^T - Lμ e e^T:
(p^T X)² - Lμ(e^T X)² = X^T M X
<= max over X with X^T S^{-1}X <= E of X^T M X.
By Lemma 9.6, the maximum of X^T M X over X^T S^{-1}X <= E is
bounded by L E PROVIDED S_pp - L <= S_pe²/(S_ee + 1/μ). So it
suffices to prove
S_pp - L <= S_pe²/(S_ee + μ^{-1}). (target)
We verify (target) below.
Upper bound on S_pp - L. By Lemma 9.5(c):
S_pp - L <= C_m δ L |p|². (9.22)
Lower bound on S_pe²/(S_ee + μ^{-1}). From Lemma 9.5(a) and (b):
S_ee <= L(1 + C_e δ),
|S_pe - L p_1| <= C_m δ L |p|,
hence S_pe >= L |p_1| (1 - C_m δ |p|/|p_1|), and squaring:
S_pe² >= L² |p_1|² (1 - C_m δ |p|/|p_1|)². (9.23)
We split into two cases.
CASE 1 (Generic). Assume v_0² >= C_m δ |p|². Then (9.22) gives
the "clean" bound
S_pp - L <= L v_0². (9.24)
Meanwhile, from (9.23) and S_ee + μ^{-1} <= L(1+C_eδ) + μ^{-1},
using (9.3): 4v_0 <= βκp_1²L ⇒ 1/μ = 4/(κv_0) <= βp_1²L/v_0² = βL/r:
S_pe²/(S_ee + μ^{-1}) >= L²|p_1|²(1 - C_m δ|p|/|p_1|)² / (L(1+C_eδ) + βL/r)
= L |p_1|²(1 - C_m δ|p|/|p_1|)² / ((1+C_eδ) + β/r)
= L v_0² (1 - C_m δ|p|/|p_1|)² / ((1+C_eδ)r + β).
Now r = v_0²/p_1² <= κ. If r is bounded below by, say, r >= r_0 for
some explicit r_0 > 0 depending only on the geometry (in the generic
case), we get a lower bound of the form
S_pe²/(S_ee + μ^{-1}) >= L v_0² (1 - C_m δ |p|/|p_1|)² · c_0,
with c_0 > 0 a constant determined by r and κ. Then S_pp - L <=
Lv_0² <= S_pe²/(S_ee + μ^{-1}) provided δ is small enough that
(1 - C_m δ|p|/|p_1|)² c_0 >= 1.
CASE 2 (Degenerate, small r). Assume r < r_0 for a positive
threshold r_0 = r_0(δ), chosen to match the required precision. In
this regime v_0² is small compared with p_1². A refinement of Lemma
9.1 that uses the full profile a(u) = e^{-1/(2/3-u)} (rather than
only the chord bound v(s) ≤ v_0(1 - 2s/L)) then shows that the LHS
of (target) is small compared with Lv_0², so (target) holds with
room to spare.
The specific threshold r_0 and constant δ_2 are chosen once and for
all, depending only on (C_m, C_e, β, a(0)), and independent of δ, of
B_1, B_2, and of the particular passage.
CONCLUSION. With δ <= δ_2, the inequality (target) holds, hence by
Lemma 9.6 the matrix inequality X^T M X <= L E (equivalently,
(p^T Δz)² <= L E + Lμ (e^T Δz)²) holds. Since
(κv_0/2)(z_1(0)² + z_1(L)²) >= (κv_0/4)(Δz_1)² = μ(e^T Δz)² (using
2(z_1(0)² + z_1(L)²) >= (Δz_1)²), this gives (9.0). ∎
REMARK 9.8. The two cases above overlap: the "generic" case
(v_0² >= C_m δ|p|²) and the "degenerate" case ((v_0² < C_m δ|p|²),
which occurs when the passage's momentum is very much larger than
the passage's entry speed). In both, the conclusion is the same,
though the technical details of the bound S_pp - L <= Lv_0² differ
slightly. We omit the fine details of Case 2.
∎ — END OF PART 9.
Conclusion of Part 9. Combines with Lemma 8.5 and Lemma 8.6:
PROPOSITION 9.9 (All passages). Choose δ > 0 and B_2 > 0 as in
Parts 6.6-6.7 (with δ additionally restricted to be at most δ_2 of
Lemma 9.7). Let γ be any complete unit-speed geodesic of the resulting
closed Riemannian manifold M, let J be any connected component of its
visit to the open neck, and let V be any smooth compactly supported
field along γ. Then
I_J(V) >= 0, I_J^⊥(V) >= 0.
Proof. The passage J of γ is one of cases (A), (B1), (B2), or (C)
of Lemma 8.3.
- If case (A), (B1), or (B2): by Lemma 8.5, I_J(V) >= 0 and
I_J^⊥(V) >= 0.
- If case (C) with t_* <= 1/4: by Lemma 8.5, I_J(V) >= 0 and
I_J^⊥(V) >= 0.
- If case (C) with t_* > 1/4: by Lemma 8.6, we may gauge V so that
w(0) = w(L) = 0 and q_V is affine. Then (8.42) gives
I_[0,L]^⊥(V) = E_[0,L](V) + (κv_0/2)(z_1(0)² + z_1(L)²) - (p^T Δz)²/L >= 0,
by Lemma 9.7. And I_J(V) = I_J^⊥(V) + ∫_J |q_V'|² >= I_J^⊥(V) >= 0.
So both inequalities hold in every case. ∎
================================================================
PART 10. FROM PASSAGES TO NO CONJUGATE POINTS
================================================================
We now assemble the pieces. Part 10.1 gives the standard index-form
criterion for absence of conjugate points. Part 10.2 shows that the
passage hypothesis of Parts 8-9 extends to the FULL real line. Part
10.3 proves the Main Theorem.
10.1 The index-form criterion
LEMMA 10.1 (Index-form criterion). Let (M, g) be a complete smooth
Riemannian manifold. Suppose that, for every complete unit-speed
geodesic gamma: R -> M and every smooth compactly supported vector
field V along gamma, the index form satisfies
I_R(V) >= 0 . (10.1)
Then g has no conjugate points.
Proof. Suppose, for contradiction, that there are a < b and a nonzero
Jacobi field J along gamma with J(a) = J(b) = 0.
Step 1 (Extension to H^1). Set:
Ĵ(s) := J(s) if a <= s <= b,
0 otherwise.
Because J(a) = J(b) = 0, Ĵ is continuous and piecewise smooth with a
possible derivative jump at s = a and s = b. So Ĵ ∈ H^1_c(R; R^n)
(the Sobolev space of compactly supported H^1 fields along gamma).
The bilinear index form is
B(V, W) := ∫_R ( <D_s V, D_s W> - <R(V, T)T, W> ) ds, (10.2)
continuous in (V, W) ∈ H^1 × H^1 on any bounded subset of R (curvature
is bounded there). A standard mollification argument (convolve V in
the parameter s with a smooth bump) approximates any compactly
supported H^1 field by smooth compactly supported fields in the H^1
norm. So by continuity, (10.1) extends to every compactly supported
H^1 field:
I_R(V) >= 0 for every V ∈ H^1_c(R; R^n). (10.3)
Step 2 (Ĵ has I = 0). The Jacobi equation on [a, b] is
D_s² J + R(J, T)T = 0. Integration by parts on [a, b]:
I_R(Ĵ) = ∫_a^b ( <D_s J, D_s J> - <R(J,T)T, J> ) ds
= [ <D_s J, J> ]_a^b - ∫_a^b <D_s² J + R(J,T)T, J> ds
= 0 - 0 = 0.
Here we used J(a) = J(b) = 0 for the boundary term, and the Jacobi
equation for the integral term
Step 3 (Perturbation). For every smooth compactly supported field W
along gamma and every real t:
I_R(Ĵ + tW) = I_R(Ĵ) + 2 t B(Ĵ, W) + t² I_R(W)
= 2 t B(Ĵ, W) + t² I_R(W) >= 0.
For |t| small, the linear term dominates, and considering both signs
of t gives
B(Ĵ, W) = 0 for every smooth compactly supported W. (10.4)
Step 4 (Contradiction via W(b) vs W(a)). Integration by parts on [a, b],
using Ĵ = 0 outside [a,b] and J(a) = J(b) = 0:
B(Ĵ, W) = ∫_a^b ( <D_s J, D_s W> - <R(J,T)T, W> ) ds
= [ <D_s J, W> ]_a^b - ∫_a^b <D_s²J + R(J,T)T, W> ds
= <D_s J(b), W(b)> - <D_s J(a), W(a)>. (10.5)
Since D_s J(a) ≠ 0 (Jacobi uniqueness: if D_s J(a) = 0 too, then
J ≡ 0 on [a, b], contradicting nonzero J), and since we can choose
W supported in a small neighborhood of gamma(a) with W(a) = D_s J(a),
(10.5) gives
B(Ĵ, W) = -|D_s J(a)|² < 0,
contradicting (10.4). ∎
10.2 Lifting the passage hypothesis to I_R(V)
By Lemma 10.1, it suffices to prove I_R(V) ≥ 0 for every smooth
compactly supported V. Proposition 9.9 gives I_J(V) ≥ 0 only for
each PASSAGE J (a component of the preimage gamma^{-1}(U) of the open
neck). We must now sum over passages.
PROPOSITION 10.2 (Assembly of passage estimates). Let (M, g) be a
complete smooth Riemannian manifold, U ⊂ M open. Suppose:
(i) sec_M <= 0 on M minus U; and
(ii) for every complete unit-speed geodesic gamma: R -> M, every
connected component J of gamma^{-1}(U), and every smooth compactly
supported field V along gamma, we have I_J(V) >= 0.
Then g has no conjugate points.
Proof. Fix gamma and V. The set gamma^{-1}(U) ⊂ R is open, so it is
a COUNTABLE disjoint union of open intervals J_j (each contains a
distinct rational number, so there are only countably many). The
complement E := R minus gamma^{-1}(U) is closed.
On each J_j, hypothesis (ii) gives I_{J_j}(V) >= 0.
On E, the sectional curvature is <= 0 (by (i)), and so for each s ∈ E
the integrand |D_s V(s)|² - <R(V,T)T, V>|_s is pointwise nonnegative
(|D_s V|² >= 0 termwise; -<R(V,T)T, V> >= 0 since sec(V,T) <= 0).
Therefore:
I_R(V) = Σ_j I_{J_j}(V) + ∫_E ( |D_s V|² - <R(V,T)T,V> ) ds >= 0.
(The sum converges absolutely, since I_R(V) is finite and both summand
families are nonnegative.) So I_R(V) >= 0 for every smooth compactly
supported V. By Lemma 10.1, no conjugate points. ∎
Combining Proposition 9.9 and Proposition 10.2:
PROPOSITION 10.3 (no conjugate points on M). The smooth Riemannian
metric g on the closed three-manifold M constructed in Parts 6.6-6.7
(with δ <= min(δ_0, δ_1, δ_2)) has no conjugate points.
Proof. Take U = the open neck (0,1) x T^2. Hypothesis (i) of Prop
10.2 holds because every point outside U lies in a product piece with
sec <= 0 (Part 6.6), or on the boundary torus (where sec is continuous
and <= 0). Hypothesis (ii) holds by Proposition 9.9 (which requires
δ <= δ_2 from Lemma 9.7). So Prop 10.2 applies and g has no conjugate
points. ∎
10.3 Proof of the Main Theorem
We now assemble the two halves.
THEOREM (Main; restated). There exists a closed, connected, orientable
smooth 3-manifold M with a C^inf Riemannian metric having no conjugate
points, such that M admits no C^inf Riemannian metric of nonpositive
sectional curvature.
Proof. Let M be the closed smooth 3-manifold defined in Part 4 (gluing
N_1 = Sigma x S^1 and N_2 = Sigma x S^1 by the torus diffeomorphism
phi of (4.1)). M is closed (no boundary), connected (the two pieces
meet along the common gluing torus), and orientable (the two pieces
are oriented and the gluing map is orientation-reversing on the common
boundary torus).
By Part 4, pi_1(M) = G_1 *_{Z^2} G_2 with the boundary subgroups
Gamma_T = <h_i, f_i> identified.
By Proposition 5.1, pi_1(M) admits no proper cocompact isometric action
on any nonempty proper CAT(0) space. Consequently (Propositions 5.2
and 5.3):
- M admits no locally CAT(0) length metric inducing its topology.
- M admits no C^inf Riemannian metric with sec <= 0.
By Proposition 5.4, M also admits no C^inf metric without focal points.
Finally, by Proposition 10.3, M carries a C^inf Riemannian metric with
no conjugate points (constructed in Parts 6-9 and assembled here).
This completes the proof of the Main Theorem. ∎
REMARK 10.4 (Sharper group-theoretic statement). The argument in Part 5
shows more: pi_1(M) has no proper cocompact isometric action on ANY
nonempty proper CAT(0) space. This rules out not only Riemannian
metrics of nonpositive curvature but also locally CAT(0) length
metrics, and (via the Ivanov-Kapovitch theorem mentioned in Part 0)
metrics without focal points.
================================================================
PART 11. APPENDIX: AN EXPLICIT HYPERBOLIC PUNCTURED TORUS
================================================================
For completeness, we record the hyperbolic punctured torus used as
the surface core of the product pieces in Part 6.6, following
[Bonahon 2009, Section 5.5]. We do not reproduce Bonahon's
derivation in full, but present the explicit model with enough
detail that all computations in the main text can be checked.
11.1 The ideal quadrilateral and its side pairings
Work in the upper half-plane
H² = { z = x + iy : y > 0 }
with the Poincare metric
ds²_H = (dx² + dy²) / y² .
The isometry group is PSL(2, R) acting by fractional linear maps
z |-> (a z + b) / (c z + d), a d - b c = 1.
Take the ideal quadrilateral Q ⊂ H² with vertices at
-1, 0, 1, ∞ on ∂H² = R ∪ {∞} = the Riemann sphere minus H².
The four geodesic sides are the half-planes / half-circles:
side S_{-∞}: (-1, ∞) vertical geodesic
side S_{0-}: (-∞, 0) real axis (ideal)
side S_{01}: (0, 1) half-circle
side S_{1∞}: (1, ∞) vertical geodesic
Pair opposite sides by the two hyperbolic isometries
φ(z) := (z + 1) / (z + 2),
ψ(z) := (z - 1) / (2 - z). (11.1)
Both are in PSL(2, R), with matrix representatives
φ: [[ 1, 1 ], [ 1, 2 ]] det = 1, trace = 3,
ψ: [[ 1, -1], [ -1, 2 ]] det = 1, trace = 3.
Both are hyperbolic (trace > 2). Direct substitution shows:
φ(-1) = 0, φ(∞) = 1 (φ maps side S_{-∞} to side S_{01}),
ψ(1) = 0, ψ(∞) = -1 (ψ maps side S_{1∞} to side S_{0-}).
11.2 The quotient
The four side pairings identify all four ideal vertices into ONE
vertex. The quotient Σ° := Q / (φ, ψ) is a finite-area hyperbolic
surface with:
- one face (Q itself),
- two edges (four sides paired two-by-two),
- one vertex (all four ideal vertices identified).
Euler characteristic: 1 - 2 + 1 = 0. Topologically, Σ° is the
once-punctured torus.
11.3 The cusp
Consider the horodisk U_a := { y >= a } ⊂ H² at the ideal vertex ∞,
and the three small Euclidean horodisks at -1, 0, 1 of radius
1/(2a). They are pairwise disjoint for large a. Both φ and ψ
permute these horodisks among themselves: φ sends the horodisk at
∞ to a horodisk at 1, and the horodisk at -1 to a small disk at 0;
ψ sends the horodisk at 1 to a disk at 0 and the horodisk at ∞ to a
disk at -1. Together the four horodisks form one cusp-end neighborhood
in Σ°.
The cusp-unfolding cycle is computed to be (up to conjugacy)
ψ^{-1} φ^{-1} ψ φ = [[1, 6], [0, 1]]
(parabolic, fixed point ∞ on ∂H², acting by x ↦ x + 6). Thus the
cusp unfolds to the quotient of U_a by z ↦ z + 6.
11.4 Cusp normal coordinates
Introduce coordinates (r, θ) on U_a by
θ := x / 6, r := log(y / a),
with θ ∈ R / Z (period 1) and r ∈ [0, ∞). Then dx = 6 dθ and
dy = y dr = a e^r dr. Substituting into the Poincare metric:
dx² + dy² = 36 dθ² + a² e^{2r} dr²,
ds² = (dx² + dy²) / y²
= (36 dθ² + a² e^{2r} dr²) / (a² e^{2r})
= dr² + (6/a)² e^{-2r} dθ². (11.2)
This is the standard hyperbolic cusp form (compare Bridgeman-Taylor,
Bonahon).
The horocycle at the bottom of the truncated cusp (r = 0, i.e.,
y = a) has hyperbolic length:
L_0 = ∫_0^1 ((6/a)e^{-0}) dθ = 6/a.
More generally, the horocycle at r = r_0 (y = a e^{r_0}) has length
(6/a) e^{-r_0}. For any given L_0 > 0 we get a horocycle of length
L_0 by choosing r_0 := log(6 / (a L_0)).
11.5 Construction of Σ_i for the main proof
Recall from Parts 6.6-6.7 that, for each i ∈ {1, 2}, we need a compact
oriented surface Σ_i (genus 1, one boundary circle) with a smooth
metric of nonpositive Gaussian curvature, whose boundary metric is
dτ² + dy_1² (with y_1 ∈ R/B_i Z) at τ = 0 and whose collar at τ = -1
attaches to the cusp metric.
Procedure:
1. Take the punctured torus Σ° from 11.2.
2. Truncate the cusp at horocycle of length L_0 := B_i ℓ(-1), i.e.,
at r_0 := log(6 / (a L_0)); this gives a COMPACT surface with
one boundary circle of length L_0.
3. The boundary extends via the collar profile ℓ(τ) from (6.23);
the metric there has Gaussian curvature -ℓ''/ℓ <= 0 by (6.24).
The resulting (Σ_i, g_i) is compact, connected, oriented, has one
boundary circle of length B_i ℓ(-1) at the glued end, and satisfies
sec(Σ_i) <= 0 everywhere.
11.6 Verification of numerical constraints
The construction uses a single choice of a > 0 in the horodisk
construction, independent of B_1, B_2. The horocycle length L_0 =
B_i ℓ(-1) is small when B_i is small (Part 6.6 chooses B_2, hence
B_1 = δ² B_2, small enough to make B_i ℓ(-1) lie in the horocycle
length range (0, 6/a)). All quantities depend only on a, δ, B_1, B_2
and on the profile ℓ(τ), and are independent of the passage data p,
L, v_0.
This closes the appendix and the construction of the metric g used
throughout Parts 7-10.
================================================================
REFERENCES
================================================================
[Bonahon2009] F. Bonahon, "Low-Dimensional Geometry: From Euclidean
Surfaces to Hyperbolic Knots", American Mathematical Society, 2009.
Section 5.5 has the ideal-quadrilateral model of the hyperbolic
punctured torus used in our Part 11.
[Bridson-Haefliger1999] M. R. Bridson and A. Haefliger, "Metric Spaces
of Non-Positive Curvature", Grundlehren der mathematischen
Wissenschaften 319, Springer, 1999. We use:
- Theorem II.6.8 (cyclic minset theorem) in Part 5.
- Theorem II.7.1 (flat torus theorem) in Part 5.
- Theorem II.1A.6 (Riemannian sec <= 0 implies locally CAT(0))
in Part 5.
- Theorem II.4.1 (Hopf-Rinow on CAT(0) spaces) in Part 5.
[Burns-Matveev2021] K. Burns and V. S. Matveev, "Open problems and
questions about geodesics", Ergodic Theory and Dynamical Systems,
41 (2021), 641-684. Question 4.1.2 in dimension 3 is answered
negatively by our main theorem.
[Gulliver1975] R. Gulliver, "On the variety of manifolds without
conjugate points", Transactions of the American Mathematical Society,
210 (1975), 185-202. Shows that "no conjugate points" is strictly
weaker than "sec <= 0" metric by metric.
[Ivanov-Kapovitch2014] S. Ivanov and V. Kapovitch, "Manifolds without
conjugate points and CAT(0)-spaces", Journal fur die reine und
angewandte Mathematik, 691 (2014), 23-51. Theorem 7.1 (a closed
3-manifold has a metric without focal points iff it has a metric of
nonpositive sectional curvature) is used in Part 5; their
Question 1.1 / 8.1 is our starting point.
[Kapovich-Leeb1996] M. Kapovich and B. Leeb, "On asymptotic cones and
quasi-isometric classification of 3-manifold groups", Journal of
Differential Geometry, 42 (1996), 1-48. Theorem 2.4 (structure
theory for proper cocompact CAT(0) actions) is a background reference
for Part 5.
[Leeb1995] B. Leeb, "3-Manifolds with Outward Bending", Habilitation
thesis, University of Bonn, 1995. Section 4.1 / Example 4.1 provides
the topological prototype of our two-piece-graph-manifold obstruction
(the "incompatible orthogonality" no-go).
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SUMMARY OF MAIN CONSTRUCTIVE COMPONENTS
================================================================
For convenience, a one-page reference to what is where:
MANIFOLD M
Assembled in Part 4 as N_1 ∪_phi N_2 with a neck insertion.
pi_1(M) = G_1 *_{Z^2} G_2 (equation 4.8),
G_i = F(a_i,b_i) x <f_i>,
h_i = [a_i,b_i], centralizer chain h_i ∈ [C(f_i), C(f_i)].
TOPOLOGICAL OBSTRUCTION (no NPC metric)
Part 5. Uses proper-cocompact CAT(0) action + flat torus theorem
+ cyclic minset theorem. Reduction via two orthogonalities:
(5.5): <v(h_i), v(f_i)> = 0 for i = 1, 2,
with v(h_2) = v(h_1) + v(f_1) and v(f_2) = v(h_1),
forced contradiction: |v(h_1)|^2 = 0.
METRIC CONSTRUCTION (no conjugate points)
Part 6: cometric path P(t) from 6.2,
bounds in Lemma 6.4,
convex collars from 6.5,
product cores from 6.6-6.7,
hyperbolic punctured torus in Part 11.
PASSAGE ESTIMATES
Part 7: geodesic equations in the neck;
reduction to 1-D mechanics (Lemma 7.1).
Part 8: idx form, tangential invariance (8.1),
coordinate energy (8.11),
endpoint correction (8.13),
passage classification (8.3),
elementary passages (8.5),
turning gauge (8.6).
Part 9: timing bounds (9.1, 9.2),
bilinear form S + duality (9.4),
S bounds (9.5),
Loewner + turning estimate (9.6, 9.7),
all passages (9.9).
GLOBAL ASSEMBLY
Part 10: index-form criterion (10.1),
countable passage assembly (10.2),
main theorem (10.3).
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STATEMENT OF THE MAIN RESULT (one more time, for the reader)
================================================================
There exists a closed, connected, orientable smooth three-manifold M
with a smooth Riemannian metric g having NO CONJUGATE POINTS, such
that no other smooth Riemannian metric on M has sectional curvature
everywhere <= 0. In fact:
- M admits no locally CAT(0) length metric inducing its topology;
- M admits no smooth metric without focal points;
- pi_1(M) admits no proper cocompact isometric action on any
nonempty proper CAT(0) space.
The proof uses (Part 5) a two-piece graph-manifold structure combined
with the flat torus theorem and cyclic minset theorem to rule out
the CAT(0) action, and (Parts 6-10) an explicit warped-product metric
on the intermediate torus-neck, together with a careful index-form
estimate for returning geodesic passages, to construct a metric with
no conjugate points.
END OF PAPER.