1c4d5ac f577e51
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AN L^2 EINSTEIN GAP FOR NONNEGATIVELY CURVED FOUR-MANIFOLDS
A complete proof, written so a careful high-school student can follow
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CONTENTS
0. What this paper proves (and why it is surprising)
1. The assumption we make (and why we make it)
2. Notation and conventions
3. Normalizing a metric with small Einstein defect
4. Two Sobolev inequalities
5. Smoothing with Ricci flow
6. Einstein and sectional-curvature defects survive smoothing
7. Extracting a compact Einstein limit
8. Proving the gap theorem
9. Remarks: what is proven, what is assumed, what is open
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0. WHAT THIS PAPER PROVES (AND WHY IT IS SURPRISING)
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0.1. Manifolds.
To a high-school reader "space" usually means R^3 or the surface of a
ball. A mathematician's word for a smooth shape that locally looks
like Euclidean space of a fixed dimension n is "manifold". In this
paper the dimension is always n = 4.
A "closed" manifold is compact with no boundary (no edge, no rim).
A "simply connected" manifold is one where every loop can be shrunk
continuously to a point. The surface of a ball is simply connected;
the surface of a donut is not, because the loop around the hole cannot
be shrunk.
Throughout, M denotes a closed, simply connected 4-manifold, plus a
Riemannian metric g that says how to measure length, angle, and volume.
0.2. Curvature.
A bug walking on a curved surface can detect curvature: two short
walks made in a "V" from a point will not stay parallel after transport.
The Riemann curvature tensor Rm(X,Y,X,Y) measures this deviation for
two tangent vectors X, Y. If X and Y are orthonormal, the number
sec(X,Y) = Rm(X,Y,X,Y)
is called the sectional curvature of the 2-plane spanned by X and Y.
It is the curvature the bug feels if it walks on the slice cut by that
plane. In 4D, at each point, this is a rule that assigns a real number
to every 2-plane.
The Riemann tensor has two useful traces. With respect to any
orthonormal frame {e_1, e_2, e_3, e_4},
Ric(X, X) = sum_{i=1}^{4} Rm(e_i, X, X, e_i), (Ricci tensor)
R = sum_{i=1}^{4} Ric(e_i, e_i). (scalar curvature)
So Ric condenses 2-plane data into a symmetric 2-tensor; R condenses
it into a single number at each point.
Through the paper we assume the geometric hypothesis
sec(g) >= 0 everywhere on M. (0.1)
The space bends outward (like a sphere) or is locally flat (like R^4).
It never has saddle-shaped negative curvature.
0.3. The Einstein condition.
A metric is Einstein if
Ric = lambda * g (0.2)
for some constant lambda. Remember g is the metric itself, the object
that defines lengths. Equation (0.2) says: the Ricci curvature in any
unit direction is the same constant lambda, at every point. It is a
strong isotropy.
Einstein alone does not control the sign of the individual sectional
curvatures. So "sec >= 0" is genuinely extra information.
0.4. The Einstein defect.
Whatever g we have, split Ric into its pure-trace part (R/4)g and the
rest:
E_g = Ric - (R/4) g. (0.3)
E_g is the trace-free Ricci tensor. A direct calculation shows
E_g = 0 if and only if Ric = (R/4) g (Einstein).
So E_g is a tensor-valued measure of "distance from being Einstein".
Its mean-square size is
D(g) = Int_M |E_g|^2 dV(g), (0.4)
where |E|^2 is the sum of the squares of the components of E in any
orthonormal frame (the result is frame-independent), and dV is the
Riemannian volume measure. Clearly D(g) >= 0, and D(g) = 0 iff the
metric is Einstein.
0.5. Scale invariance in four dimensions.
If c > 0 and we replace g by c*g (a constant rescaling), then
lengths scale by sqrt(c)
volumes scale by c^2
|E_g|^2 scales by c^{-2}
D(c*g) = c^{-2} * c^2 * D(g) = D(g).
The exponent 2 is exactly the dimension n = 4. So D is scale-invariant
in 4D: it does not depend on how big the manifold is. This is why we
may normalize below without loss of generality.
0.6. The gap theorem, statement.
THEOREM 8.1 (L^2 Einstein gap, informal).
Suppose M is a closed, simply connected 4-manifold with a metric g
satisfying sec(g) >= 0 and
D(g) < epsilon_0, (0.5)
where epsilon_0 > 0 is a universal constant that does not depend on M
or g. Then M is diffeomorphic to one of the standard manifolds
S^4, CP^2, S^2 x S^2. (0.6)
"Diffeomorphic" means "same smooth shape". Two metrics can make the
same smooth manifold look geometrically different; diffeomorphism
ignores the ruler and remembers only the underlying smooth shape. A
diffeomorphism is weaker than an isometry (isometry preserves distances
exactly).
0.7. The word "gap".
Put all metrics on all such M on a scatter plot, with vertical axis
D(g). The exactly-Einstein metrics sit at height 0. Theorem 8.1 says:
the interval
0 < D(g) < epsilon_0
is empty. There is a gap. So near-Einstein metrics "remember" the
Einstein classification: small positive defect does not let you leave
the three-model family.
0.8. Why this is hard.
The defect D(g) is a global L^2 integral, so it gives no pointwise
control. A curvature spike on a region of tiny volume contributes
almost nothing to D, even though the metric there looks nothing like
Einstein. Worse, when we smooth using Ricci flow, the spike can blow
up and destroy the flow before it has spread. Ruling this out is the
technical core of the proof.
0.9. The strategy.
Rescale so the average scalar curvature is 12. Prove uniform volume
and diameter bounds. Derive a uniform Sobolev inequality. Run Ricci
flow for a fixed universal time T. Show the Einstein defect and the
negative-sectional-curvature amount both stay small. Extract a
compact limit by Hamilton compactness; the limit is an exact
shrinking Einstein metric with sec >= 0. Assumption 1.1 then
identifies the limit as one of the three standard manifolds.
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1. THE ASSUMPTION WE MAKE (AND WHY WE MAKE IT)
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We will use exactly one nontrivial classification theorem. It is
stated here as an explicit hypothesis, so the reader can see precisely
where it is used.
ASSUMPTION 1.1 (Zero-plane rigidity; classification of the limit).
Let (N,h) be a connected closed Riemannian 4-manifold satisfying
Ric_h = 3h and sec_h >= 0. (1.1)
Then the universal Riemannian cover of (N,h) is isometric to one of:
(i) the round sphere S^4 with sectional curvature 1;
(ii) CP^2 with the Fubini-Study metric normalized so Ric = 3g;
(iii) S^2(1/sqrt 3) x S^2(1/sqrt 3), the product metric on two
round 2-spheres each of radius 1/sqrt 3.
In particular, if N is simply connected, then N itself is one of
S^4, CP^2, or S^2 x S^2 up to diffeomorphism.
Assumption 1.1 is proven in the companion paper "Zero-Plane Rigidity
for Einstein Four-Manifolds", OpenAI Math Release, October 2026
([ZeroPlane] in the bibliography). That paper proves a nontrivial
classification of exact Einstein 4-manifolds with sec >= 0. Our
contribution is different: we show how such an exact classification
passes to metrics that are only L^2-close to Einstein. Crucially,
Assumption 1.1 is used exactly once, on an exact Einstein metric.
No ansatz about "stability" of Einstein metrics, no approximate
version, no bounds on curvature derivatives near the limit.
The reader who accepts Assumption 1.1 may read the rest of this paper
as a complete proof of Theorem 8.1. The reader who wants an entirely
unconditional theorem must first prove Assumption 1.1 (or an
equivalent classification); that proof is not contained in this
document.
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2. NOTATION AND CONVENTIONS
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2.1. Objects.
Letters M and N always denote smooth, compact, boundaryless
4-manifolds. A Riemannian metric g assigns an inner product to each
tangent space in a smooth way. From g we may compute: the Riemann
tensor Rm(g), the Ricci tensor Ric(g), the scalar curvature R(g), the
pointwise norm |T|(x) of any tensor T, the Riemannian volume form
dV(g), the total volume Vol(g), and the diameter diam(g).
2.2. Curvature sign.
We use the sign convention in which S^4 (unit) has positive sectional
curvature 1. So on the round unit sphere, Rm(X, Y, X, Y) = 1 for any
orthonormal X, Y. Traces follow: Ric(X, X) = sum over an orthonormal
basis of Rm(e_i, X, X, e_i), and R = sum_i Ric(e_i, e_i).
2.3. Algebraic curvature tensors and their norms.
Fix a Euclidean 4D space V. The "algebraic curvature tensors" on V
are tensors with the same symmetries as Rm(g) at a point: symmetric
under interchanging the two pairs of indices, antisymmetric within
each pair, and satisfying the first Bianchi identity. These form a
finite-dimensional real vector space.
On this space,
|B| = sqrt(sum of squares of B's components in any ON frame)
is a norm. Two norms on a finite-dimensional space are equivalent
(modulo two universal constants). In particular, there is a constant
C_0 = C_0(4) > 0 such that for any algebraic curvature tensor B with
sec_B >= 0 on all 2-planes,
|B| <= C_0 * max over orthogonal unit X, Y of |sec_B(X, Y)|
(2.1)
Note also that the section sum identity holds in any ON 4-frame:
R = sum_i Ric(e_i, e_i)
= sum_{i != j} Rm(e_i, e_j, e_j, e_i)
= 2 * sum_{i < j} sec(e_i, e_j). (2.2)
Under sec >= 0, every entry on the right is >= 0, so each sectional
curvature satisfies sec(e_i, e_j) <= R / 2. Combining with (2.1)
gives the algebraic bound
|Rm|(x) <= C_0 * R(x) (2.3)
at every point where sec >= 0, with C_0 a dimension-4 constant.
2.4. Shorthands.
We write
Int_M f dV for the Riemannian integral of a scalar f over M;
||f||_p for the L^p(g) norm (Int_M |f|^p dV)^{1/p};
||T||_{L^2} for the L^2 norm of a tensor T (same formula, |T|
pointwise norm).
For a sequence (M_j, g_j), all norms and operators are with respect to
g_j unless stated otherwise. Constants C, C', c, and so on are
positive and may change from line to line. They are "uniform" when
they depend only on the dimension 4, not on the sequence index j.
2.5. Laplacian sign.
The Riemannian Laplacian is Delta f = tr(nabla^2 f), the trace of the
second derivative. On a closed manifold,
-Int_M g(nabla phi, nabla psi) dV = Int_M (Delta phi) psi dV,
i.e. Delta is negative semidefinite. The PDE Delta f = u with
Int_M f dV = 0 always has a unique smooth solution for u smooth with
Int_M u dV = 0.
2.6. Moving frames.
When we flow a metric by Ricci flow (Section 5), we often use an
orthonormal frame e_1, ..., e_4 at each point and each time. Fix such
a frame at t = 0 and transport it forward by
partial_t e_i = Ric(e_i) (2.4)
(the Ric-lift of the frame, which keeps it orthonormal). In this
frame the metric components are constant in space and time; tensors
are easier to differentiate. Norms |T| computed in an orthonormal
frame do not depend on the particular ON frame chosen.
2.7. Lipschitz and approximation.
A function f is Lipschitz if |f(x) - f(y)| <= L * dist(x, y) for some
constant L and all x, y. Sobolev-type inequalities below hold for
Lipschitz functions; they extend by density to functions with
|nabla f| in L^2. We will often argue with smooth f and extend at the
end without comment.
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3. NORMALIZING A METRIC WITH SMALL EINSTEIN DEFECT
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3.1. Sequence of near-Einstein metrics.
Throughout Sections 3-7 we work with a sequence (M_j, g_j) of closed,
simply connected 4-manifolds satisfying
sec(g_j) >= 0 and D(g_j) = Int_{M_j} |E_j|^2 dV_j -> 0. (3.1)
Here E_j = E_{g_j}. We suppress the index j when discussing a single
member of the sequence.
We will also write
V_j = Vol(M_j, g_j)
R_j = R(g_j) (scalar curvature)
Ric_j = Ric(g_j) (Ricci tensor).
3.2. Statement.
PROPOSITION 3.1 (Normalization and initial bounds).
After a constant rescaling of each g_j (which leaves D unchanged),
there exist universal positive constants v_0, V_0, D_0 such that, for
all sufficiently large j:
(1/V_j) Int_{M_j} R_j dV_j = 12, (3.2a)
||R_j - 12||_{L^2} <= 4 ||E_j||_{L^2}, (3.2b)
||Ric_j - 3 g_j||_{L^2} -> 0, (3.2c)
v_0 <= V_j <= V_0, (3.2d)
diam(M_j, g_j) <= D_0. (3.2e)
All L^2 norms are with respect to g_j.
The rest of Section 3 is the proof of Proposition 3.1, in ten steps.
3.3. Step 1: chi(M_j) is at least 2.
Betti numbers of a closed, simply connected 4-manifold:
b_0 = 1 (connected);
b_1 = 0 (simply connected);
b_3 = b_1 = 0 (Poincare duality);
b_4 = b_0 = 1 (Poincare duality);
b_2 >= 0 (a manifold can have any second Betti number).
The Euler characteristic is the alternating sum:
chi(M_j) = b_0 - b_1 + b_2 - b_3 + b_4 = 2 + b_2 >= 2. (3.3)
In particular chi(M_j) > 0.
3.4. Step 2: a lower bound on the squared curvature integral.
The Chern-Gauss-Bonnet formula in 4 dimensions, for any closed Riemannian
4-manifold M, reads
chi(M) = (1/(8 pi^2)) Int_M ( |W|^2 - |rho|^2 / 2 + R^2 / 24 ) dV,
(3.4)
where W is the Weyl tensor and rho = Ric - R g / 4 is the trace-free
Ricci tensor. Splitting |Rm|^2 orthogonally into Weyl, traceless, and
scalar parts gives
|Rm|^2 = |W|^2 + 2 |rho|^2 + R^2 / 6. (3.5)
In particular |W|^2 <= |Rm|^2 and R^2 / 24 <= (1/4) |Rm|^2 (because
|Rm|^2 >= R^2/6). So the integrand in (3.4) is at most
|Rm|^2 + (1/4) |Rm|^2 = 1.25 |Rm|^2.
Since 1/(8 pi^2) < 1, we conclude
chi(M) <= C Int_M |Rm|^2 dV (3.6)
for a universal constant C > 0. Combining with (3.3):
2 <= chi(M_j) <= C Int_{M_j} |Rm_{g_j}|^2 dV_j. (3.7)
3.5. Step 3: the average scalar curvature is positive.
Because sec(g_j) >= 0, identity (2.2) gives R_j >= 0 pointwise. If
(1/V_j) Int R_j dV_j = 0, then R_j = 0 everywhere, and the algebraic
bound (2.3) forces Rm_{g_j} = 0, contradicting (3.7). Hence the average
scalar curvature is strictly positive.
3.6. Step 4: rescaling to average R = 12.
Let c_j = 12 / ((1/V_j) Int_{M_j} R_j dV_j), a positive constant.
Replace g_j by c_j g_j. The Einstein defect is scale-invariant
(Section 0.5), so D(c_j g_j) = D(g_j). The rescaled scalar has integral
average 12. We drop the c_j below and assume (3.2a).
3.7. Step 5: almost-Schur (3.2b).
We follow the argument of De Lellis and Topping. Let f solve
Delta f = R - 12, with Int_M f dV = 0.
Because Int(R - 12) dV = 0 by construction, such f exists (Section 2.5).
Contracted Bianchi. The divergence of the Ricci tensor is half the
gradient of R: nabla^i Ric_{ij} = (1/2) d_j R. Substituting
Ric = E + (R/4) g verifies that
div E = (1/4) dR. (3.8)
First integration by parts. Using Delta f = R - 12,
Int (R - 12)^2 dV = Int (R - 12) (Delta f) dV
= -Int g(nabla R, nabla f) dV. (3.9)
Second integration by parts. Using (3.8),
Int g(nabla R, nabla f) dV = 4 Int g(div E, nabla f) dV
= -4 Int <E, Hess f> dV. (3.10)
(The second equality is integration by parts for symmetric tensors on
a closed manifold; the boundary term vanishes because M is closed.)
Combining (3.9) and (3.10):
Int (R - 12)^2 dV = 4 Int <E, Hess f> dV. (3.11)
Cauchy-Schwarz:
Int (R - 12)^2 <= 4 ||E||_{L^2} ||Hess f||_{L^2}. (3.12)
Bochner bound. Standard Bochner formula and integration by parts on a
closed manifold give
Int |Hess f|^2 = Int (Delta f)^2 - Int Ric(nabla f, nabla f). (3.13)
Since sec >= 0 implies Ric >= 0, the second term is >= 0, so
||Hess f||_{L^2} <= ||Delta f||_{L^2} = ||R - 12||_{L^2}. (3.14)
Feeding (3.14) into (3.12):
||R - 12||_{L^2}^2 <= 4 ||E||_{L^2} ||R - 12||_{L^2}. (3.15)
Divide both sides by ||R - 12||_{L^2} (if it is zero, (3.2b) is
trivially true). This gives (3.2b):
||R - 12||_{L^2} <= 4 ||E||_{L^2}.
3.8. Step 6: (3.2c).
The trace decomposition is orthogonal: Ric - 3 g = E + (R - 12)/4 g,
with E trace-free and (R - 12) g/4 pure-trace. So
||Ric - 3 g||_{L^2}^2
= ||E||_{L^2}^2 + (1/16) ||(R - 12) g||_{L^2}^2
= ||E||_{L^2}^2 + (1/4) ||R - 12||_{L^2}^2
<= ||E||_{L^2}^2 + (1/4) (4 ||E||_{L^2})^2
= 5 ||E||_{L^2}^2.
Since ||E||_{L^2} = D(g)^{1/2} -> 0, this gives (3.2c).
3.9. Step 7: lower volume bound (lower half of (3.2d)).
From average R = 12,
Int R^2 dV = Int ( (R - 12) + 12 )^2 dV
= Int (R - 12)^2 dV + 144 V
(since Int (R - 12) dV = 0, cross term drops).
Combine (3.7) with (2.3):
2 <= C Int |Rm|^2 dV
<= C C_0^2 Int R^2 dV
<= C C_0^2 (144 V + ||R - 12||_{L^2}^2)
<= C C_0^2 (144 V + 16 ||E||_{L^2}^2).
For all sufficiently large j, ||E||_{L^2}^2 is much smaller than 1, so
2 <= C'' V_j + (small),
i.e., V_j >= v_0 for a universal constant v_0 > 0. This is the lower
half of (3.2d).
3.10. Step 8: diameter bound (3.2e), part 1.
Write D_j = diam(M_j, g_j). Choose points p, q with dist(p, q) = D_j.
Define
A_1 = B(p, D_j/4), A_2 = B(q, D_j/4).
Bishop-Gromov comparison. Since Ric >= 0, the ratio
r |--> Vol B(x, r) / r^4 is nonincreasing in r. Applying this to
A_1 = B(p, D_j/4) and the ball B(p, 2 D_j) (which equals all of M_j,
since 2 D_j > diam M_j):
Vol A_1 / (D_j/4)^4 >= V_j / (2 D_j)^4, hence
Vol A_1 >= V_j / 4096.
Let me spell this out: (D_j/4)^4 / (2 D_j)^4 = D_j^4 / 256 / (16 D_j^4)
= 1/4096. Same bound for Vol A_2. This is the standard 8^{-4} factor.
Segment inequality. The Cheeger-Colding segment inequality (valid on
closed M with Ric >= 0) states: for any nonnegative continuous H,
Int_{A_1 x A_2} Int_{gamma_xy} H ds dV_x dV_y
<= C D_j (Vol A_1 + Vol A_2) Int_M H dV. (3.16)
where gamma_xy is the minimizing geodesic from x to y (unique for
almost every (x, y)).
Mean value argument. Apply (3.16) with H = |Ric - 3 g|. Estimate:
Int H dV <= V_j^{1/2} ||Ric - 3 g||_{L^2}
<= V_0^{1/2} ||Ric - 3 g||_{L^2}
Vol A_1 + Vol A_2 <= 2 V_j,
so the right-hand side of (3.16) is at most
2 C D_j V_j V_0^{1/2} ||Ric - 3 g||_{L^2}. (3.17)
On the left, Vol(A_1 x A_2) >= (V_j/4096)^2 = V_j^2 / 16 777 216. If
delta_j denotes the average of Int_{gamma_xy} H over A_1 x A_2, then
the left-hand side of (3.16) is delta_j * Vol(A_1 x A_2), so
delta_j <= (2 C D_j V_j V_0^{1/2} ||Ric - 3 g||_{L^2})
/ (V_j^2 / 16 777 216)
= C' D_j (V_0^{1/2} / V_j) ||Ric - 3 g||_{L^2}
<= delta_j^{max}, (3.18)
with
delta_j^{max} = C'' ||Ric - 3 g||_{L^2} -> 0, (3.19)
where C'' = C' V_0^{1/2}/v_0 absorbs the volume factors. Hence for
every j there exists a pair (x_j, y_j) in A_1 x A_2 such that
Int_{gamma_j} H ds <= delta_j^{max} D_j, (3.20)
where gamma_j is the minimizing geodesic from x_j to y_j. Note
length(gamma_j) = ell_j >= D_j/2, because d(x_j, p) <= D_j/4,
d(y_j, q) <= D_j/4, d(p, q) = D_j, so d(x_j, y_j) >= D_j/2.
3.11. Step 9: diameter bound (3.2e), part 2 (Myers argument).
Let gamma: [0, ell] -> M_j be the geodesic from (3.20), arclength
parametrized. For any collection of piecewise smooth vector fields
V_1, ..., V_k along gamma vanishing at endpoints, the second variation
formula (index form) gives, when gamma is minimizing,
sum_i Int_0^{ell} ( |V_i'|^2 - Rm(V_i, gamma', gamma', V_i) ) ds >= 0.
(3.21)
Take three parallel orthonormal normal fields n_1, n_2, n_3 along gamma
(the normal bundle to a geodesic is trivial), and set
V_i(s) = sin(pi s / ell) n_i(s), i = 1, 2, 3.
Then V_i'(s) = (pi/ell) cos(pi s / ell) n_i(s), so |V_i'|^2 = (pi/ell)^2
cos^2(pi s/ell), and Rm(V_i, gamma', gamma', V_i) = sin^2(pi s/ell) *
Rm(n_i, gamma', gamma', n_i). Summing over i,
sum_{i=1}^{3} Rm(n_i, gamma', gamma', n_i) = Ric(gamma', gamma')
(3.22)
(because the three normals plus gamma' span the tangent space, and
Ric is the trace of Rm over an ON basis).
Plug into (3.21):
Int_0^{ell} [ 3 (pi/ell)^2 cos^2(pi s/ell)
- sin^2(pi s/ell) Ric(gamma', gamma') ] ds >= 0. (3.23)
Now Int_0^{ell} cos^2(pi s/ell) ds = ell/2, so the first term is
3 pi^2 / (2 ell). Split the second term using Ric = 3 g + (Ric - 3 g):
-Int sin^2(pi s/ell) Ric(gamma', gamma') ds
= -3 Int sin^2(pi s/ell) ds
- Int sin^2(pi s/ell) (Ric - 3 g)(gamma', gamma') ds
>= -3 (ell/2) - Int_0^{ell} H(gamma(s)) ds
>= -3 ell/2 - delta_j^{max} D_j. (3.24)
(Here we used sin^2 <= 1 and (3.20).)
Combine (3.23) and (3.24):
3 pi^2 / (2 ell) - 3 ell/2 - delta_j^{max} D_j >= 0,
i.e.,
3 pi^2 / (2 ell) >= 3 ell/2 - delta_j^{max} D_j. (3.25)
Since D_j <= 2 ell (from ell >= D_j/2), we get
3 pi^2 / (2 ell) >= (3/2 - 2 delta_j^{max}) ell. (3.26)
Multiply by ell and rearrange:
ell^2 (3/2 - 2 delta_j^{max}) <= 3 pi^2 / 2. (3.27)
For each j with 2 delta_j^{max} <= 3/4 (which happens for all large j by
(3.19)),
ell^2 <= (3 pi^2 / 2) / (3/4) = 2 pi^2. (3.28)
Thus ell <= pi sqrt(2), hence D_j <= 2 ell <= 2 pi sqrt(2). Set
D_0 = 3 pi (safely above 2 pi sqrt(2) ~ 8.89). This proves (3.2e).
3.12. Step 10: upper volume bound (upper half of (3.2d)).
Since Ric >= 0, Bishop-Gromov applied to Euclidean space gives
Vol B(x, r) <= omega_4 r^4 for any r, where omega_4 = pi^2/2 is the
volume of the Euclidean unit 4-ball. Take r = diam(M_j, g_j) = D_j
>= every distance in M_j. Then B(x, D_j) = M_j, so
V_j <= omega_4 D_j^4 <= omega_4 D_0^4.
Set V_0 = omega_4 D_0^4. This is the upper half of (3.2d).
3.13. End of the proof of Proposition 3.1.
Combining (3.2a) with Steps 4-10 gives every claim. [QED]
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4. TWO SOBOLEV INEQUALITIES
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4.1. Overview.
A Sobolev inequality bounds the L^4 norm of a function in terms of its
derivative plus its L^2 norm:
||f||_{L^4}^2 <= S ( ||nabla f||_{L^2}^2 + ||f||_{L^2}^2 ).
On any fixed smooth metric such an S exists; the interesting question
is whether S is uniform over our sequence (M_j, g_j). It is, and we
prove it in this section (initial metric) and Section 5.2 (metric
along a Ricci flow). Both come from the same elementary lemma, which
we prove first.
4.2. Truncation lemma.
LEMMA 4.1. Let (M, g) be a closed 4-manifold, a > 0, P >= 0 smooth.
Define
E(f) = Int_M ( a |nabla f|^2 + P f^2 ) dV. (4.1)
Suppose there is a constant C such that for every Lipschitz f and
every m = Vol{supp f},
||f||_{L^2}^2 <= C m^{1/2} E(f). (4.2)
Then for some C' = C'(C),
||f||_{L^4}^2 <= C' E(f) (4.3)
for every Lipschitz f.
Proof. Dyadic layers. For k in Z set
m_k = Vol{ |f| > 2^k }, f_k = min{ (|f| - 2^k)_+, 2^k }.
On { |f| > 2^{k+1} } the function f_k equals 2^k (its max), so
2^{2k} m_{k+1} <= ||f_k||_{L^2}^2. (4.4)
Applying (4.2) to f_k with support volume m_k gives
2^{2k} m_{k+1} <= C m_k^{1/2} E(f_k). (4.5)
The gradients of f_k live on disjoint dyadic annuli
{2^k < |f| < 2^{k+1}}, where |nabla f_k| = |nabla f|. So
sum_k ||nabla f_k||_{L^2}^2 = ||nabla f||_{L^2}^2. (4.6)
Also sum_k f_k^2 <= (4/3) f^2 pointwise (geometric series in k, since
f_k <= 2^k and only the contributions with 2^k < |f| count). Therefore
sum_k E(f_k) <= C_2 E(f) (4.7)
for a numerical constant C_2.
Dyadic sum. Put b_k = 2^{4k} m_k. Then (4.5) becomes
b_{k+1} = 2^{4(k+1)} m_{k+1} = 16 * 2^{4k} m_{k+1}
<= 16 * 2^{2k} C m_k^{1/2} E(f_k)
= 16 C b_k^{1/2} E(f_k). (4.8)
The partial sums B = sum_k b_k are comparable to ||f||_{L^4}^4:
writing K(x) for the integer with 2^{K(x)} < |f(x)| <= 2^{K(x)+1},
sum_{k : 2^k < |f(x)|} 2^{4k} <= (16/15) |f(x)|^4.
Integrating over M and using Fubini,
B = Int_M (sum_k 2^{4k} 1_{|f| > 2^k}) dV
<= (16/15) ||f||_{L^4}^4, (4.9)
and similarly B >= (1/16) ||f||_{L^4}^4 (the k-sum fails to be much
smaller than the leading term).
Summing (4.8) over k and bounding b_k^{1/2} <= B^{1/2}:
B <= 16 C_2 C B^{1/2} E(f). (4.10)
If B = 0 then f = 0 and (4.3) is trivial. Otherwise divide by B^{1/2}:
B^{1/2} <= 16 C_2 C E(f), so B <= (16 C_2 C)^2 E(f)^2.
Combine with (4.9):
||f||_{L^4}^2 <= C' E(f), C' = 16 C_2 C * sqrt(16/15). (4.11)
This is (4.3). ✓
4.3. The initial Sobolev inequality.
PROPOSITION 4.2. Let (M_j, g_j) be the normalized metrics from
Proposition 3.1. Then there is a uniform constant S such that
||f||_{L^4}^2 <= S Int_{M_j} (|nabla f|^2 + f^2) dV_j (4.12)
for every Lipschitz f.
Proof. We show (4.2) with a = P = 1, then apply Lemma 4.1. The
estimate on the operator kernel d(x,y)^{-3} drives the argument.
4.4. Kernel bound.
For Lipschitz f, let f̄ = (1/V) Int_M f dV. For any x,
|f(x) - f̄| <= (D^4 / (4 V)) Int_M |nabla f(y)| d(x,y)^{-3} dV(y).
(4.13)
Proof. Let gamma: [0, d(x,y)] -> M be the minimizing geodesic from x
to y. Then f(x) - f(y) = -Int_gamma (d/ds) f ds, so |f(x) - f(y)| <=
Int_gamma |nabla f|. Averaging over y:
|f(x) - f̄| <= (1/V) Int_M Int_gamma |nabla f| ds dV(y). (4.14)
In polar coordinates y = exp_x(r theta), dV(y) = J_x(r, theta) dr dtheta.
Under Ric >= 0 the comparison J_x(r, theta) <= (r/s)^3 J_x(s, theta)
holds for r >= s. Change the order of integration (fix s, integrate
r from s to cut(x, theta)):
Int_s^{cut} J_x(r, theta) dr
<= J_x(s, theta)/s^3 Int_s^D r^3 dr
<= J_x(s, theta) D^4 / (4 s^3). (4.15)
So the inner integral over M is at most
(D^4 / 4) Int_M |nabla f(y)| d(x,y)^{-3} dV(y). (4.16)
Divide by V: (4.13). ✓
4.5. Kernel size on a set.
LEMMA 4.3. Let A be a measurable set, Vol(A) = m > 0. Then
sup_x in A Int_A d(x,y)^{-3} dV(y) <= C m^{1/4}, (4.17)
and the same holds with x and y interchanged.
Proof. Fix x. Split at rho = m^{1/4}.
Inner part. On A ∩ B(x, rho), use Bishop-Gromov: total area of
geodesic sphere of radius r is at most the Euclidean surface area
4 omega_4 r^3. So
Int_{A ∩ B(x, rho)} d(x,y)^{-3} dV
<= Int_0^rho r^{-3} 4 omega_4 r^3 dr = 4 omega_4 rho. (4.18)
Outer part. On A \ B(x, rho), d(x,y) > rho, so d^{-3} < rho^{-3}:
Int_{A \ B(x, rho)} d(x,y)^{-3} dV <= rho^{-3} m = m^{1/4}. (4.19)
Sum: at most (4 omega_4 + 1) m^{1/4}. ✓
4.6. Operator norm.
LEMMA 4.4. Let T be the integral operator on L^2(A) with kernel
K(x,y) = d(x,y)^{-3} restricted to A x A. Then
||T||_{L^2(A) -> L^2(A)} <= C m^{1/4}. (4.20)
Proof. Schur test: for K >= 0, if both one-sided integrals are at
most M, then ||T|| <= M. Lemma 4.3 gives M <= C m^{1/4}. ✓
4.7. Endgame.
Let A = supp f, so Vol(A) = m. Since nabla f = 0 a.e. off A,
(4.13) reads (with |nabla f| supported on A):
|f(x) - f̄| <= (D^4 / (4V)) Int_A |nabla f(y)| d(x,y)^{-3} dV(y).
(4.21)
Take L^2(A) norms:
||f - f̄||_{L^2(A)}
<= (D^4 / (4V)) ||T(|nabla f|)||_{L^2(A)}
<= (C D^4 / (4V)) m^{1/4} ||nabla f||_{L^2(M)}. (4.22)
Decompose f = (f - f̄) + f̄ on A:
||f||_{L^2} <= (C D^4 / (2V)) m^{1/4} ||nabla f||_{L^2}
+ sqrt(m) |f̄|. (4.23)
And |f̄| <= (sqrt(m) / V) ||f||_{L^2}, so sqrt(m)|f̄| <= (m/V) ||f||_{L^2}.
Define C_4 = C/2. Then
||f||_{L^2} <= (C_4 D^4 / V) m^{1/4} ||nabla f||_{L^2}
+ (m/V) ||f||_{L^2}. (4.24)
Case 1: m/V <= 1/2. Then 1 - m/V >= 1/2, so
||f||_{L^2} <= (2 C_4 D^4 / V) m^{1/4} ||nabla f||_{L^2}. (4.25)
Square and use D <= D_0, V >= v_0:
||f||_{L^2}^2 <= C_5 m^{1/2} ||nabla f||_{L^2}^2
<= C_5 m^{1/2} E(f), (4.26)
with C_5 = (2 C_4 D_0^4 / v_0)^2. This is (4.2).
Case 2: m/V > 1/2. Then m >= V/2 >= v_0/2, so m^{1/2} >= (v_0/2)^{1/2}.
Also E(f) >= ||f||_{L^2}^2. So
||f||_{L^2}^2 <= E(f) <= (2/v_0)^{1/2} m^{1/2} E(f) = C_6 m^{1/2} E(f). (4.27)
Again (4.2).
Lemma 4.1 converts (4.26) and (4.27) into (4.12) with
S = 16 C_2 C max(C_5, C_6) * sqrt(16/15). ✓ [QED]
================================================================================
5. SMOOTHING WITH RICCI FLOW
================================================================================
5.1. Ricci flow: the heat equation for metrics.
Ricci flow is the geometric PDE
partial_t g(t) = -2 Ric(g(t)), g(0) = g. (5.1)
Intuition: the metric "diffuses" in the direction of its curvature,
like heat spreading on a metal plate. Curvature spikes smooth out;
over time the metric becomes more regular and more symmetric.
Short-time existence. Hamilton proved that on any closed manifold,
any smooth initial metric evolves smoothly and uniquely on some time
interval [0, T_max), with T_max > 0. T_max may depend on the initial
metric. Write T_max = T_j for the flow starting at g_j.
Scalar curvature evolution. Differentiating, using
partial_t g = -2 Ric, and the contracted second Bianchi identity,
partial_t R = Delta R + 2 |Ric|^2. (5.2)
The right side is a heat equation with a nonnegative source. The
maximum principle implies min R(t) >= min R(0). Because R(0) >= 0
by sec(g_j) >= 0, we have
R(g_j(t)) >= 0 pointwise for all t in existence. (5.3)
Volume evolution. The volume form satisfies partial_t dV = -R dV, so
d/dt V(t) = -Int R dV <= 0, V(t) <= V(0) = V_j. (5.4)
So volume never increases along the flow, and V(t) in (0, V_0].
Notation. Until a common lifespan is proved, every statement below
is conditional on 0 <= t < T_j. The symbol Int stands for Int_{M_j}
with respect to g_j(t); dV, |T|, norms are likewise with respect to
the current metric. We drop the j index when no confusion arises.
5.2. Weighted Sobolev from Perelman's entropy.
PROPOSITION 5.1 (Entropy-weighted Sobolev). There is a universal
constant S_1 such that, for every j and every
0 <= t < min{1, T_j}, every Lipschitz f satisfies
||f||_{L^4}^2 <= S_1 Int ( 4 |nabla f|^2 + (R+1) f^2 ) dV. (5.5)
The proof uses Perelman's monotone entropy. The key point is that no
knowledge of T_j is required: the argument only refers to existing
times, so it produces a-conditional-on-existence estimate that lifts
the initial Sobolev constant to the whole (short) flow.
5.2.1. Perelman entropy.
In dimension n = 4, Perelman's entropy is
mu(g, tau) = inf_{ ||f||_{L^2} = 1 } [
tau Int ( 4 |nabla f|^2 + R f^2 ) dV
- Int f^2 log f^2 dV
- 2 log(4 pi tau) - 4 ]. (5.6)
The infimum is over smooth positive f with unit L^2 norm; by
approximation it extends to nonnegative Lipschitz f, with 0 log 0 = 0.
The parameter tau > 0 is called the "entropy parameter"; it is not a
flow time, and mu(g, tau) is defined at any smooth (g, tau). Perelman's
monotonicity theorem states that mu(g(t), tau) is nondecreasing along
the flow when tau decreases at unit rate. Equivalently, for any
tau > 0 and any 0 <= t < T_j,
mu(g_j(t), tau) >= mu(g_j(0), tau + t). (5.7)
The right side uses the entropy parameter tau + t, not the later flow
time tau + t; only the flow at time 0 appears. This is why (5.7)
imposes no requirement on T_j.
5.2.2. Lower bound at initial time.
We claim the initial Sobolev inequality (4.12) implies
mu(g_j(0), s) >= -C for all 0 < s <= 2, (5.8)
with C a universal constant. To see this, take smooth positive f with
||f||_{L^2} = 1. Set y = Int |nabla f|^2 dV. Jensen's inequality and
the initial Sobolev inequality give
Int f^2 log f^2 dV <= log Int f^4 dV
= 2 log ||f||_{L^4}
<= 2 log( S (y+1) ), (5.9)
where S comes from (4.12). (Used ||f||_{L^4}^2 <= S(y + ||f||^2) =
S(y+1) and S(y+1) >= 1.)
Substitute into (5.6) at time 0 with parameter s:
s Int (4|nabla f|^2 + R f^2) - Int f^2 log f^2 - 2 log(4 pi s) - 4
>= 4 s y - 2 log(S (y+1)) - 2 log(4 pi s) - 4
= 4 s y - 2 log(s (y+1)) - 2 log(4 pi S) - 4. (5.10)
The only terms that depend on f are 4 s y - 2 log(s(y+1)). Write
z = s (y+1) so z runs over [s, infinity). Then
4 s y - 2 log(s (y+1)) = 4 (z - s) - 2 log z
= 4 z - 2 log z - 4 s. (5.11)
The auxiliary function z |--> 4z - 2 log z has minimum value 2 + 2 log 2
at z = 1/2, and 4z - 2 log z -> +infinity as z -> 0+ or z -> infinity.
For 0 < s <= 2, -4s is at least -8. So (5.11) is at least
2 + 2 log 2 - 8 = 2 log 2 - 6.
Therefore
inf_{||f|| = 1} ( 4 s y - 2 log(s(y+1)) ) >= 2 log 2 - 6,
and (5.10) is bounded below by a universal constant. So (5.8) holds
with C = 2 + 6 + 2 log(4 pi S) + 4, a universal constant.
5.2.3. Log-Sobolev on the flowing metric.
Combine (5.7) with (5.8). Since t < T <= 1/6 and 0 < tau <= 1, we have
tau + t < 1 + 1/6 < 2, so (5.8) applies at tau + t:
mu(g_j(t), tau) >= mu(g_j(0), tau + t) >= -C (5.12)
for every 0 < tau <= 1 and 0 <= t < T_j.
Now unpack (5.12) using the definition (5.6) of mu. For any smooth
positive f with ||f||_{L^2} = 1,
tau Int (4|nabla f|^2 + R f^2) dV - Int f^2 log f^2 dV
- 2 log(4 pi tau) - 4 >= -C.
Rearrange:
Int f^2 log f^2 dV <= tau E_0(f) + 2 log(4 pi tau) + 4 + C,
(5.13)
where E_0(f) = Int (4 |nabla f|^2 + R f^2) dV. Since
2 log(4 pi tau) = 2 log tau + 2 log(4 pi)
and 2 log(4 pi) < 4, we may write (renaming the constant)
Int f^2 log f^2 dV <= tau E_0(f) - 2 log(1/tau) + C'
(5.14)
for a new universal C', whenever 0 < tau <= 1 and ||f||_{L^2} = 1.
Lipschitz extension. Approximating any Lipschitz f with compact support
by smooth positive functions, (5.14) extends by continuity (0 log 0 = 0).
5.2.4. Support estimate.
Fix f with ||f||_{L^2} = 1 and support volume m = Vol{supp f} > 0. On
its support, f^2 integrates to 1, so Jensen's inequality on the support
gives
Int f^2 log f^2 dV >= -log m. (5.15)
(Minimum of Int g log g over probability densities g = f^2 with
support in a set of measure m is -log m, attained by g = 1/m on the
support.)
Choose tau = (1 + E_0(f))^{-1} in (0, 1]. Then (5.14) and (5.15) give
log(1/m) <= E_0(f)/(1+E_0(f)) + 2 log(1 + E_0(f)) + C'.
Since E_0/(1+E_0) <= 1, we obtain
log(1/m) <= 1 + 2 log(1 + E_0(f)) + C'. (5.16)
Exponentiate and rearrange: for a universal constant C_10,
1 <= C_10 m^{1/2} (1 + E_0(f)). (5.17)
5.2.5. Homogenization and Lemma 4.1.
(5.17) was derived for ||f||_{L^2} = 1. For general Lipschitz f with
||f||_{L^2} > 0, apply (5.17) to f/||f||:
1 <= C_10 m^{1/2} (1 + E_0(f)/||f||^2).
Multiply both sides by ||f||^2:
||f||_{L^2}^2 <= C_10 m^{1/2} ( ||f||_{L^2}^2 + E_0(f) ).
But E_0(f) + ||f||_{L^2}^2 = Int(4|nabla f|^2 + (R+1) f^2) dV, exactly
the energy in Lemma 4.1 with a = 4, P = R+1. So (4.2) holds with
C = C_10, and Lemma 4.1 gives (5.5) with S_1 = C_10 * 16 C_2 * sqrt(16/15),
all universal. ✓ [QED]
Remark. (5.5) is a "weighted" Sobolev inequality: there is an extra
f^2 coefficient R on the right. Section 5.3 removes it under the
"small tail" hypothesis.
5.3. The high-curvature tail stays small.
The idea, from Chan-Chen-Lee and related work, is to isolate the "high
part" of |Rm| above a moving threshold. If the tail is small in L^2 at
time 0, then it stays small along the flow, supplied by a special
quadratic reaction that matches the tail ODE.
5.3.1. Curvature evolution in moving frames.
Use orthonormal frames moving with the flow (partial_t e_i = Ric(e_i),
Section 2.6). In such frames the Riemann tensor evolves as
partial_t Rm = Delta Rm + Q(Rm), |Q(Rm)| <= C_q |Rm|^2,
(5.18)
where Delta is the connection Laplacian, Q is a specific quadratic
combination of Rm with itself, and C_q is a universal constant
(depending only on dimension 4). There is no linear Ric term in these
frames: it has been absorbed by the frame motion.
By Kato's inequality |nabla |T|| <= |nabla T| applied to T = Rm, the
pointwise norm u = |Rm| satisfies
(partial_t - Delta) u <= C_q u^2 wherever u > 0. (5.19)
5.3.2. The threshold function.
Define K: [0, T_theory) -> R by
K(0) = 12 C_0, K'(t) = C_q K(t)^2, (5.20)
where C_0 is from (2.3). The ODE has explicit solution
K(t) = 12 C_0 / (1 - 12 C_0 C_q t),
which blows up at time t_* = 1/(12 C_0 C_q). Fix
T = min{ 1/12, 1/(24 C_0 C_q) }. (5.21)
Then T < 1/(12 C_0 C_q) strictly. Case 1: T = 1/(24 C_0 C_q), so
12 C_0 C_q T = 1/2 and K(T) = 12 C_0 / (1 - 1/2) = 24 C_0. Case 2:
T = 1/12 (so C_0 C_q < 1/2), so 12 C_0 C_q T = C_0 C_q < 1/2 and
K(T) = 12 C_0 / (1 - C_0 C_q) < 24 C_0. In both cases K(t) <= 24 C_0
on [0, T].
5.3.3. The tail function.
For each j and each existing time t, define
w_j(t) = max{ 0, |Rm_{g_j(t)}| - K(t) }. (5.22)
Initial size. At time 0 the algebraic bound (2.3) gives
|Rm_{g_j(0)}| <= C_0 R_{g_j}. So
w_j(0) <= max{ 0, C_0 R_{g_j} - 12 C_0 }
= C_0 (R_{g_j} - 12)_+. (5.23)
Taking L^2 norms and using (3.2b),
||w_j(0)||_{L^2} <= C_0 ||R_{g_j} - 12||_{L^2}
<= 4 C_0 ||E_{g_j}||_{L^2} -> 0. (5.24)
So the initial tail shrinks to zero in L^2.
5.3.4. Statement and proof.
PROPOSITION 5.2 (Tail propagation and unweighted Sobolev). After
discarding finitely many indices, there are uniform constants S_2, C
such that, for every j and every t in [0, min{T, T_j}):
||f||_{L^4}^2 <= S_2 Int (|nabla f|^2 + f^2) dV
(5.25)
Int w_j(t)^2 dV <= e^{C t} Int w_j(0)^2 dV_{g_j}
(5.26)
Int |Rm_{g_j(t)}|^2 dV <= C. (5.27)
The Sobolev estimate holds for every Lipschitz f. In particular, the
supremum of ||w_j(t)||_{L^2} over these existing times tends to zero
as j -> infinity.
Proof.
Step 1. Small-tail to unweighted Sobolev. We work under the temporary
bootstrap assumption
||w_j(t)||_{L^2} <= eta for all t in [0, min{T, T_j}), (5.28)
with eta > 0 a small universal constant to be chosen below.
Under sec(g_j(t)) (any sign) we have the pointwise bound R <= C |Rm|
with a dimension-4 constant C, and by (5.3) R >= 0. So
R <= C (K(t) + w). (5.29)
K(t) is bounded on [0, T] by K_max := K(T) = 24 C_0. So
R + 1 <= C(K_max + 1) + C w, and the weighted Sobolev (5.5) reads
||f||_{L^4}^2 <= S_1 Int (4 |nabla f|^2 + C_1 f^2) dV
+ S_1 C Int w f^2 dV (5.30)
with C_1 = C(K_max + 1) a dimension-4 constant. Bound the last term
by Cauchy-Schwarz:
Int w f^2 dV <= ||w||_{L^2} ||f||_{L^4}^2 <= eta ||f||_{L^4}^2.
(5.31)
Therefore
(1 - S_1 C eta) ||f||_{L^4}^2
<= S_1 Int (4 |nabla f|^2 + C_1 f^2) dV. (5.32)
Choose eta <= 1/(2 S_1 C). Then (1 - S_1 C eta) >= 1/2, so
||f||_{L^4}^2 <= 2 S_1 Int (4 |nabla f|^2 + C_1 f^2) dV. (5.33)
This is (5.25) with S_2 = 2 S_1 max(4, C_1). Note S_2 does not depend
on eta.
Step 2. Tail energy. Denote Y(t) = Int w_j(t)^2 dV_j(t). By definition
of w_j and (5.19)-(5.20), on {u > K(t)}:
(partial_t - Delta) w
= (partial_t - Delta)(u - K)
= (partial_t - Delta)u - K'
<= C_q u^2 - C_q K^2
= C_q (u^2 - K^2)
= C_q (w^2 + 2 K w). (5.34)
Because w = 0 on {u <= K}, inequality (5.34) holds everywhere
(Lipschitz truncation; Kato for the positive part).
Differentiate Y(t):
d/dt Y(t) = Int (2 w partial_t w) dV + Int w^2 (partial_t dV)
= Int (2 w partial_t w) dV - Int R w^2 dV
<= Int [2 w (Delta w + C_q w^2 + 2 C_q K w)] dV - Int R w^2 dV
= -2 Int |nabla w|^2 dV
+ 2 C_q Int w^3 dV
+ 4 C_q K Int w^2 dV
- Int R w^2 dV. (5.35)
The last term is <= 0 because R >= 0. So
Y'(t) <= -2 Int |nabla w|^2
+ 2 C_q Int w^3
+ 4 C_q K_max Int w^2. (5.36)
Step 3. Absorb the cubic term. Hölder with exponents 2 and 2:
Int w^3 = Int w^2 w <= ||w||_{L^4}^2 ||w||_{L^2}
<= eta ||w||_{L^4}^2, by (5.28). (5.37)
Sobolev (5.25) gives ||w||_{L^4}^2 <= S_2 Int (|nabla w|^2 + w^2) dV.
Combining:
Int w^3 <= eta S_2 Int (|nabla w|^2 + w^2) dV. (5.38)
Plug into (5.36):
Y'(t) <= -(2 - 2 C_q eta S_2) Int |nabla w|^2 dV
+ (2 C_q eta S_2 + 4 C_q K_max) Int w^2 dV.
Choose eta also small enough that 2 C_q eta S_2 <= 1. Then
Y'(t) <= -(1) Int |nabla w|^2 + (1 + 4 C_q K_max) Int w^2
<= C Y(t), (5.39)
where C = 1 + 4 C_q K_max is a universal constant.
Step 4. Gronwall and bootstrap closure. By Step 5 below (independent),
Y(0) = Int w_j(0)^2 dV_{g_j} <= 16 C_0^2 ||E_j||_{L^2}^2 -> 0. So for
large j we have Y(0) e^{C T} < eta^2.
Gronwall applied to (5.39) between 0 and any t in [0, min{T, T_j}),
under the bootstrap (5.28), gives
Y(t) <= e^{C t} Y(0) <= e^{C T} Y(0) < eta^2. (5.40)
Strict inequality everywhere on [0, min{T, T_j}). By continuity of
Y, we cannot first touch Y = eta^2 without violating the strict
inequality, so (5.28) holds throughout the interval for large j. The
bootstrap closes.
Therefore for large j, Y(t) <= e^{C t} Y(0) on [0, min{T, T_j}).
This is (5.26).
Step 5. Initial tail estimate used in Step 4. From (5.23),
Y(0) = Int w_j(0)^2 dV_{g_j}
<= C_0^2 Int (R_{g_j} - 12)_+^2 dV_{g_j}
<= C_0^2 ||R_{g_j} - 12||_{L^2}^2
<= 16 C_0^2 ||E_j||_{L^2}^2 -> 0. (5.41)
Step 6. Energy (5.27). Since |Rm| <= K + w and (a + b)^2 <= 2a^2 + 2b^2,
Int |Rm|^2 dV <= 2 Int K^2 dV + 2 Int w^2 dV
<= 2 K_max^2 V_0 + 2 e^{C T} Y(0). (5.42)
The first term is a universal constant, the second -> 0 by (5.41).
So the integral is bounded by a universal C, giving (5.27). ✓ [QED]
5.4. Noncollapse.
The Sobolev inequality (5.25) is strong enough to force a uniform lower
bound on the volume of small balls, hence to rule out collapse.
LEMMA 5.3 (Noncollapse). There is a universal constant v_* > 0 such
that, for every j, every existing t in [0, min{T, T_j}), every
x in M_j, and every r in (0, 1],
Vol_{g_j(t)} B_{g_j(t)}(x, r) >= v_* r^4. (5.43)
Proof. Fix one smooth time slice. Write V_x(r) = Vol B(x, r). Insert
into (5.25) the geodesic cutoff phi that is 1 on B(x, r/2), vanishes
outside B(x, r), and has |nabla phi| <= 2/r. Then
V_x(r/2)^{1/2} = ( Int_{B(x, r/2)} phi^4 dV )^{1/2}
<= ( Int_M phi^4 dV )^{1/2}
= ||phi||_{L^4}^2
<= S_2 Int_M (|nabla phi|^2 + phi^2) dV
<= S_2 (4/r^2 + 1) V_x(r). (5.44)
Since r <= 1, 4/r^2 + 1 <= 5/r^2. So
V_x(r/2)^{1/2} <= 5 S_2 V_x(r) / r^2. (5.45)
Define a(r) = V_x(r) / r^4. Then V_x(r/2) = a(r/2) (r/2)^4 = a(r/2)
r^4/16 and V_x(r)/r^4 = a(r). So (5.45) is
sqrt(a(r/2) r^4/16) <= 5 S_2 a(r) r^4 / r^2, i.e.
a(r/2)^{1/2} · r^2/4 <= 5 S_2 a(r) r^2, i.e.
a(r) >= c a(r/2)^{1/2}, c = 1/(20 S_2). (5.46)
Iterate k times:
a(r) >= c^{2 - 2^{1-k}} a(2^{-k} r)^{2^{-k}}. (5.47)
Proof by induction: k = 1 is (5.46). Step: a(r) >= c a(r/2)^{1/2} >=
c (c a(r/4)^{1/2})^{1/2} = c^{1+1/2} a(r/4)^{1/4}. Continue.
Since the metric is smooth at any fixed time, a(s) -> omega_4 as
s -> 0+ (Euclidean comparison). So a(2^{-k} r)^{2^{-k}} -> omega_4^0 = 1.
Letting k -> infinity in (5.47) gives
a(r) >= c^2. (5.48)
So V_x(r) >= c^2 r^4. Set v_* = c^2 > 0. ✓
5.5. A uniform smoothing interval.
The two outputs of Proposition 5.2 (unweighted Sobolev and small tail)
plus Lemma 5.3 (noncollapse) are now enough to prove that the flows
do not die before time T.
PROPOSITION 5.4 (Uniform lifespan and curvature bound). For all large
j, the flow g_j(t) exists on [0, T]. Moreover, with A = 8 K_max T + 4
(fixed by T and K_max, both universal), for every existing t with
0 < t < T,
max_{M_j} |Rm_{g_j(t)}| <= A / t. (5.49)
Proof.
Step 1. Choose A. Fix A > 2 such that A/T >= 1 and A/T >= 4 K_max.
For instance A = 4 T K_max + 4 suffices. (A/T >= 4 K_max from
A >= 4 T K_max + 4; A/T >= 1 from A >= T.)
Step 2. Suppose a crossing. Assume, for contradiction, that for some
j there is an existing time t < T at which the quantity
s |--> s max_{M_j} |Rm_{g_j(s)}| first reaches the value A. Denote
Q = max_{M_j} |Rm_{g_j(t)}| = A / t. (5.50)
We have Q >= A/T >= 4 K_max >= 4 K(t). Also Q = A/t >= 4/t > 0, and
since A > 2, Q >= 2/t.
Step 3. Curvature bound on the preceding slice. For s in
[t - Q^{-1}, t], the first-crossing hypothesis says
s max |Rm_{g_j(s)}| < A, so max |Rm_{g_j(s)}| < A/s. (5.51)
Now s >= t - Q^{-1}. Since Q^{-1} = t/A <= t/2 (because A >= 2),
t - Q^{-1} >= t/2. Hence for s in [t - Q^{-1}, t],
max |Rm_{g_j(s)}| < A / (t - Q^{-1}) <= A / (t/2) = 2 A / t = 2 Q. (5.52)
So the curvature is at most 2Q on the space-time slab
M_j x [t - Q^{-1}, t].
Step 4. Shi derivative estimate. Rescale by Q: g_tilde(s) = Q g(t - s/Q)
for s in [0, 1]. Then g_tilde is again a Ricci flow on a closed 4-
manifold, with |Rm_{g_tilde(s)}| = |Rm_g(t - s/Q)| / Q <= 2. Shi's
derivative estimate on the unit spacetime slab gives
|nabla_{g_tilde} Rm_{g_tilde}| <= C for s in [1/2, 1].
Equivalently, at the original time t (s = 1), |nabla_{g} Rm_{g}|(x) <=
C Q^{3/2} for all x in M_j.
Key point: Shi's derivative estimate does not require a positive lower
bound on injectivity radius, nor a bound on curvature derivatives;
the short-time local form (see Morgan-Tian, Theorem 3.27, applied to
the entire closed manifold) suffices.
Step 5. Concentration contradiction. Let x realize the maximum of
|Rm_{g_j(t)}|. Then |Rm_{g_j(t)}|(x) = Q. By Step 4, for y at
distance <= c Q^{-1/2} from x (with c small universal, say c chosen so
C c <= 1/4),
|Rm_{g_j(t)}|(y) >= Q - (C Q^{3/2})(c Q^{-1/2})
= Q - C c Q
>= 3 Q / 4. (5.53)
Since Q >= 4 K(t), we have w_j(t)(y) = |Rm|(y) - K(t) >= 3Q/4 - Q/4 =
Q/2 on this ball. Also Q >= 4/t. Since t < T <= 1/6, Q >= 4/t > 24 > 1,
so c Q^{-1/2} <= c <= 1 and Lemma 5.3 applies at radius c Q^{-1/2}:
Vol B(x, c Q^{-1/2}) >= v_* (c Q^{-1/2})^4 = v_* c^4 / Q^2.
(5.54)
Therefore
Int w_j(t)^2 dV >= (Q/2)^2 · (v_* c^4 / Q^2) = v_* c^4 / 4. (5.55)
On the other hand, Proposition 5.2 says sup_{[0, min{T, T_j})}
||w_j(t)||_{L^2} -> 0. For large j, this supremum is smaller than
(v_* c^4/4)^{1/2}, contradicting (5.55). So for large j no crossing
occurs.
Hence for large j and every existing t in (0, T),
t · max_{M_j} |Rm_{g_j(t)}| < A, (5.56)
which is (5.49).
Step 6. Extension of lifespan. Suppose T_j <= T for some j. Then by
(5.56) the curvature on [T_j/2, T_j) is bounded by 2A/T_j. Curvature
is already bounded on [0, T_j/2] by compactness of the closed time
interval and smoothness of the individual flow. So curvature is
bounded on [0, T_j) uniformly. By the Ricci flow curvature
continuation criterion (Hamilton), the flow extends past T_j, a
contradiction. Hence T_j > T. ✓ [QED]
================================================================================
6. EINSTEIN AND SECTIONAL-CURVATURE DEFECTS
================================================================================
6.1. Overview.
With the common interval [0, T] in hand, we now propagate two "defect"
quantities. These are the only pieces of information at time 0 that
survive the smoothing and identify the limit:
(a) The Einstein defect L = Ric - lambda(t) g, where
lambda(t) = 3/(1-6t). At t = 0 this is Ric - 3g, whose L^2
norm -> 0 by (3.2c). We prove it stays small up to time T.
(b) The negative-sectional-curvature defect F(Rm), which is 0
exactly when sec >= 0. Initially 0 by hypothesis. We prove
Int F(Rm(t)) <= C t.
The mechanism in both cases: the high-curvature tail w_j(t) is uniformly
small (Prop 5.2), and the flow acts as a heat equation that damps the
defects. The estimates are integrated in time and closed via Gronwall.
6.2. The Einstein defect.
Throughout, lambda(t) = 3/(1 - 6t) for t in [0, T]. Since
T <= 1/12 (from T = min{1/12, 1/(24 C_0 C_q)}), we have
1 - 6T >= 1 - 6/12 = 1/2, so lambda is smooth on [0, T] with
0 < lambda(t) <= 6.
Note lambda(0) = 3 and lambda'(t) = 2 lambda(t)^2. Define the tensor
L = Ric - lambda(t) g. (6.1)
At time 0, L = Ric - 3g and ||L_j(0)||_{L^2} -> 0 by (3.2c).
PROPOSITION 6.1. After discarding finitely many indices,
sup_{ 0 <= t < T } ||L_{g_j(t)}||_{L^2(g_j(t))} -> 0. (6.2)
Proof.
6.2.1. Evolution of L in moving frames.
Work in moving orthonormal frames (2.4) with partial_t e_i = Ric(e_i).
In such frames the metric components are delta_ij (constant). The Ricci
tensor evolves as
partial_t Ric = Delta Ric + 2 Rm(Ric), (6.3)
where for a symmetric 2-tensor B, (Rm(B))_{ij} = sum_{k,l} R_ikjl B_kl
(summed over k, l in an ON frame), and Rm(g) = Ric.
The -2 Ric² term that appears in the fixed-frame Ricci evolution is
exactly cancelled by the frame motion; the components of g in moving
frames being constant means Delta g = 0 in these frames.
Differentiating L = Ric - lambda g with respect to t:
partial_t L
= partial_t Ric - lambda' g - lambda (partial_t g)
= (Delta Ric + 2 Rm(Ric)) - 2 lambda^2 g - lambda * 0 (g const)
= Delta Ric + 2 Rm(Ric) - 2 lambda^2 g. (6.4)
Meanwhile, using Delta g = 0 and Rm(g) = Ric:
Delta L = Delta Ric,
2 Rm(L) = 2 Rm(Ric) - 2 lambda Ric,
2 lambda L = 2 lambda Ric - 2 lambda^2 g.
Summing:
Delta L + 2 Rm(L) + 2 lambda L = Delta Ric + 2 Rm(Ric) - 2 lambda^2 g. (6.5)
Comparing (6.4) and (6.5) gives the evolution:
partial_t L = Delta L + 2 Rm(L) + 2 lambda L. (6.6)
6.2.2. Bounding the reaction term.
On [0, T], lambda is bounded by 18. Also |Rm| <= K(t) + w_j(t) with
K(t) <= K_max = 24 C_0. So
|2 lambda L| <= 36 |L|, |2 Rm(L)| <= 2 |Rm| |L| <= C (K_max + w) |L|.
Combining: |2 Rm(L) + 2 lambda L| <= C (1 + w) |L|, with C universal.
6.2.3. L^2 energy.
Denote ||L|| = ||L||_{L^2(g_j(t))}. Differentiate:
d/dt ||L||^2 = Int 2 <L, partial_t L> dV + Int |L|^2 (partial_t dV)
= Int 2 <L, partial_t L> dV - Int R |L|^2 dV. (6.7)
Since R >= 0, the last term is <= 0. Substituting (6.6) and
Cauchy-Schwarz:
d/dt ||L||^2
<= Int [ 2 <L, Delta L> + 2 C (1 + w) |L|^2 ] dV
= -2 Int |nabla L|^2 dV + 2 C Int |L|^2 dV + 2 C Int w |L|^2 dV.
(6.8)
6.2.4. Absorb the w-term.
Hölder with exponents 2 and 2:
Int w |L|^2 dV <= ||w||_{L^2} ||L^2||_{L^2} = ||w||_{L^2} ||L||_{L^4}^2.
Sobolev (5.25) applied to |L| (which is Lipschitz if L is:
||L||_{L^4}^2 <= S_2 Int (|nabla |L||^2 + |L|^2) dV <= S_2 Int (|nabla L|^2 + |L|^2) dV,
using Kato |nabla |L|| <= |nabla L|). Let eta_j = sup_{[0, T)} ||w_j(t)||_{L^2},
so eta_j -> 0 by Proposition 5.2. Then
Int w |L|^2 dV <= eta_j S_2 Int (|nabla L|^2 + |L|^2) dV. (6.9)
Plug into (6.8):
d/dt ||L||^2 <= -(2 - 2 C eta_j S_2) Int |nabla L|^2
+ (2 C eta_j S_2 + 2 C) Int |L|^2. (6.10)
For large j, 2 C eta_j S_2 <= 1. Then
d/dt ||L||^2 <= -Int |nabla L|^2 + C ||L||^2 <= C ||L||^2. (6.11)
6.2.5. Gronwall.
||L_j(t)||^2 <= e^{C t} ||L_j(0)||^2 <= e^{C T} ||L_j(0)||^2.
By (3.2c), ||L_j(0)||_{L^2(g_j)} = ||Ric_j - 3 g_j||_{L^2} -> 0. So
sup_{[0, T)} ||L_j(t)||^2 -> 0. This is (6.2). ✓ [QED]
6.3. The sectional-curvature defect.
Define the function F on the space V of algebraic curvature tensors on
R^4 by
F(B) = max{ 0, sup_{orthonormal X, Y in R^4} (-B(X, Y, X, Y) ) }. (6.12)
Then F(Rm(g)) is: 0 if sec(g) >= 0, otherwise the magnitude of the
most negative sectional curvature. F is computed in any orthonormal
frame; the value is frame-independent.
Properties of F.
(i) F >= 0, F(0) = 0;
(ii) F is a supremum of linear functionals and 0, hence convex;
(iii) F is invariant under the orthogonal group O(4);
(iv) F is positively homogeneous of degree 1: F(cB) = |c| F(B);
(v) F is 1-Lipschitz: |F(B) - F(B')| <= ||B - B'||.
Mollification. Let rho_delta be a smooth nonneg radial mollifier on V
supported in the ball of radius delta about 0, with total integral 1.
Set F_delta = F * rho_delta (convolution on V).
Then F_delta is:
(i') smooth on V, nonneg;
(ii') convex (convolution of convex functions);
(iii') O(4)-invariant (radial mollifier);
(iv') |DF_delta| <= 1;
(v') |F_delta(B) - F(B)| <= C_11 delta for a universal C_11.
PROPOSITION 6.2. There is a universal C such that
Int_{M_j} F(Rm_{g_j(t)}) dV <= C t, 0 <= t < T. (6.13)
Proof.
6.3.1. Chain rule along the flow.
Let F_delta(Rm) denote the scalar function on M_j formed by composing
F_delta with Rm(g_j(t)) (which at each point is an algebraic curvature
tensor). Then F_delta(Rm) is smooth on M_j.
Chain rule for the heat equation on the tensor bundle. In moving frames,
partial_t F_delta(Rm) = DF_delta(Rm)[partial_t Rm],
Delta F_delta(Rm) = DF_delta(Rm)[Delta Rm] + sum_a D^2 F_delta(Rm)[nabla_a Rm, nabla_a Rm],
where Delta is the connection Laplacian, nabla_a is covariant
derivative in the e_a direction, and the sum runs over a = 1, 2, 3, 4.
Subtracting:
(partial_t - Delta) F_delta(Rm)
= DF_delta(Rm)[(partial_t - Delta) Rm]
- sum_a D^2 F_delta(Rm)[nabla_a Rm, nabla_a Rm].
By (5.18), (partial_t - Delta) Rm = Q(Rm) with |Q(Rm)| <= C_q |Rm|^2.
The first term is bounded: |DF_delta| |Q(Rm)| <= C_q |Rm|^2.
The second term is <= 0: convexity of F_delta means D^2 F_delta(Rm) is
positive semidefinite as a quadratic form on V, so each
D^2 F_delta(Rm)[nabla_a Rm, nabla_a Rm] >= 0.
Therefore
(partial_t - Delta) F_delta(Rm) <= C_q |Rm|^2. (6.14)
6.3.2. L^1 energy.
Differentiate:
d/dt Int F_delta(Rm) dV
= Int (partial_t F_delta(Rm)) dV + Int F_delta(Rm) (partial_t dV)
= Int (partial_t - Delta) F_delta(Rm) dV
- Int R F_delta(Rm) dV (Delta is closed, so ∫Delta = 0)
<= Int C_q |Rm|^2 dV - Int R F_delta(Rm) dV. (6.15)
The last term is <= 0 (R >= 0, F_delta >= 0). So
d/dt Int F_delta(Rm) dV <= C_q Int |Rm|^2 dV <= C_q C, (6.16)
where the last inequality is (5.27).
6.3.3. Integration and limit.
Integrating (6.16) from 0 to t:
Int F_delta(Rm(t)) dV <= Int F_delta(Rm(0)) dV + C_q C t. (6.17)
At time 0, sec(g_j) >= 0 so F(Rm(0)) = 0, hence
0 <= F_delta(Rm(0)) <= C_11 delta. Therefore
Int F_delta(Rm(0)) dV <= C_11 delta V_j <= C_11 delta V_0. (6.18)
Combining (6.17) and (6.18):
Int F_delta(Rm(t)) dV <= C_11 delta V_0 + C_q C t.
Let delta -> 0. Since F_delta -> F pointwise and F_delta(Rm) is dominated
(for each fixed tensor Rm, |F_delta - F| <= C_11 delta -> 0), the
dominated convergence theorem gives
Int F(Rm(t)) dV <= C_q C t. (6.19)
This is (6.13) with C = C_q C. ✓ [QED]
6.4. The fixed-time scaling.
Only one thing remains to understand before extracting the limit: how
the sectional-curvature defect (6.13) passes to a limit that is
shrinking (not static). We defer this to Section 7.4, where the
scaling property of F plays a role.
================================================================================
7. THE COMPACT EINSTEIN LIMIT
================================================================================
7.1. Setting.
Let t_* = T/2. This is a positive time strictly inside the interval [0, T].
By Proposition 5.4, every g_j(t) exists on [0, T], so t_* is valid for
all j. Also t_* < 1/6 since T < 1/6.
We use the uniform bounds at time t_*:
(a) |Rm_{g_j(t_*)}| <= A / t_* (Prop 5.4);
(b) Vol B(x, r) >= v_* r^4 for r in (0, 1] (Lemma 5.3);
(c) Vol(M_j, g_j(t_*)) <= V_0 (Prop 5.2 and volume monotonicity).
From these we extract precise bounds on diameter and injectivity radius
at time t_*, then apply Hamilton's compactness theorem.
7.2. Diameter bound at t_*.
Fix a point x_j in M_j. Cover B(x_j, D) by balls of radius 1/2 centred
at a maximal set of points at mutual distance >= 1/2. Each such ball has
volume at least v_* (1/2)^4 = v_*/16, and the union has volume at most
V_0. So the number of points is at most 16 V_0 / v_*, and hence
diam(M_j, g_j(t_*)) <= 2 * 16 V_0 / v_* = 32 V_0 / v_* =: D_*. (7.1)
(The bound also follows from a simple "walk along a minimizing geodesic
and collect 1/2-separated points" argument.)
7.3. Injectivity radius bound at t_*.
The Cheeger-Gromov-Taylor injectivity radius estimate (in "Finite
propagation speed, kernel estimates for functions of the Laplace operator,
and the geometry of complete Riemannian manifolds", 1982, Theorem 4.7)
says: for a smooth Riemannian 4-manifold with
|Rm| <= K, Vol B(p, 1) >= v,
there is a universal i_0 = i_0(K, 1, v) > 0 such that inj(p) >= i_0.
Apply with K = A/t_* and v = v_*. Then
inj(M_j, g_j(t_*)) >= i_0 > 0 (7.2)
with i_0 uniform.
7.4. Hamilton compactness.
We have at time t_*:
|Rm| <= A/t_*,
inj >= i_0,
diam <= D_*.
Hamilton's compactness theorem for Ricci flow (Hamilton, "A compactness
property for solutions of the Ricci flow", 1995) gives, after passing
to a subsequence, a smooth complete Ricci flow (N, h(t)) on some
manifold N, for t ranging over some interval containing t_*, and a
sequence of time-independent embeddings
Phi_j: (N, h(t_*)) -> (M_j, g_j(t_*)) (7.3)
such that (Phi_j)^* g_j(t) converges to h(t) smoothly on compact
subsets of N x I, for any compact interval I containing t_*.
We may in fact apply the theorem simultaneously on any sequence of
compact time intervals of the form [t_* - eps, t_* + eps] with eps <
t_*, take a diagonal subsequence, and obtain smooth convergence on
N x I for every compact I subset of (0, T). This uses the bound
(5.49): |Rm| <= A/t on the whole time axis, so no local curvature
explosion occurs.
7.5. N is compact and simply connected.
N is compact. If N were complete but noncompact, then for every L > 0
it would contain a point q at distance exactly L from a base point p.
Choose L > 2 D_* (with D_* from (7.1)). For large j, the convergence
embedding Phi_j is defined on a neighborhood of B(p, L) in N and
distorts lengths by at most factor 2 (from smooth convergence). Then
any curve in M_j from Phi_j(p) to Phi_j(q) must have length
>= L/2. But Phi_j(p), Phi_j(q) are points in M_j whose distance is at
most D_* = diam(M_j, g_j(t_*)), contradiction with L/2 > D_*. Hence
N is compact.
N is simply connected. The M_j are closed simply connected manifolds,
and the time-t_* slice (Phi_j)^* g_j(t_*) converges to h(t_*)
smoothly on all of N (by compactness of N, the embeddings are
eventually defined on all of N). Since Phi_j is an embedding and its
image is open and closed in M_j (M_j connected), it must be a
diffeomorphism M_j -> N. Since M_j is simply connected, so is N.
Eventually Phi_j is a diffeomorphism. Because N is compact, Phi_j is
defined on all of N for j large enough; by the inverse function
theorem, its image is open; by compactness of M_j, it is closed; since
M_j is connected, it is all of M_j; hence Phi_j is a diffeomorphism.
7.6. The limit is Einstein.
By Proposition 6.1, for each j:
sup_{0 <= t < T} ||Ric_{g_j(t)} - lambda(t) g_j(t)||_{L^2} -> 0
(7.4)
as j -> infinity. Since (Phi_j)^* g_j(t) -> h(t) smoothly on compact
subsets (and N is compact, so on all of N), the L^2 limit is exactly
the limit metric's L^2 norm:
Ric_{h(t)} - lambda(t) h(t) = 0 for every t in (0, T). (7.5)
So the limit flow is Einstein: Ric_{h(t)} = lambda(t) h(t).
Integrate the Ricci flow ODE along the limit. Since h(t) is a Ricci
flow, partial_t h(t) = -2 Ric_{h(t)} = -2 lambda(t) h(t). Both sides
are proportional to h(t) at every t, so h(t) is homothetically
shrinking: h(t) = c(t) h(t_*) with c satisfying c' = -2 lambda c,
c(t_*) = 1.
Solve for c. Since lambda(t) = 3/(1 - 6t),
d/dt log c(t) = -2 lambda(t) = -6/(1 - 6t).
Integrating: log c(t) = log(1 - 6t) + const. Using c(t_*) = 1: const
= -log(1 - 6 t_*). So
c(t) = (1 - 6t) / (1 - 6 t_*). (7.6)
Set
h = (1 / (1 - 6 t_*)) h(t_*). (7.7)
Then h(t) = (1 - 6t) h for all t in (0, T). We verify that (N, h)
is Einstein.
Ricci under constant scaling. For any smooth metric g and constant
c > 0, the Ricci tensor is invariant: Ric_{c g} = Ric_g. The reason:
the Christoffel symbols Gamma^k_{ij} = (1/2) g^{kl}(d_i g_{jl} + d_j
g_{il} - d_l g_{ij}) are invariant under g -> c g (numerator scales
by c, inverse metric by c^{-1}, product unchanged); the (3,1) Riemann
tensor is built from derivatives of Gamma and products Gamma Gamma,
so also invariant; and the Ricci tensor is a trace of it, so also
invariant.
Apply to h(t) = (1-6t) h. From Ric_{h(t)} = lambda(t) h(t) and
h(t) = (1-6t) h,
lambda(t) (1-6t) h = Ric_{(1-6t) h} = Ric_h.
So Ric_{h(t)} = Ric_{(1-6t)h} = Ric_h.
Therefore Ric_h = lambda(t) (1 - 6t) h = 3/(1-6t) * (1-6t) h = 3h.
So (N, h) is Einstein with Ric_h = 3h. ✓ (7.8)
7.7. The limit has nonnegative sectional curvature.
We use Proposition 6.2 and the explicit scaling h(t) = (1-6t) h.
The function F on algebraic curvature tensors was defined in (6.12) by
F(B) = max{ 0, sup_{orthonormal X, Y} -B(X, Y, X, Y) },
where |X|, |Y| refer to the metric g at the point where B = Rm(g) is
being evaluated. This normalization matters when we scale the metric.
Scaling property. Let g' = c g with c > 0. Then (as computed in
Section 2.2) the (0,4)-curvature tensor scales as Rm_{g'} = c Rm_g
at the same geometric point. But F is evaluated at an ON frame for
g', which is the frame {e_i / sqrt(c)} if {e_i} was ON for g. In
that frame, the components of Rm_{g'} are
Rm_{g'}(e'_i, e'_j, e'_k, e'_l) = c Rm_g(e'_i, e'_j, e'_k, e'_l)
= c (1/sqrt c)^4 Rm_g(e_i, e_j, e_k, e_l)
= c^{-1} Rm_g(e_i, e_j, e_k, e_l).
Similarly the volume form satisfies dV_{g'} = c^2 dV_g in 4 dimensions.
Combining, at each point:
F(Rm_{c g}) = c^{-1} F(Rm_g), dV_{c g} = c^2 dV_g. (7.9)
Apply (7.9) to h(t) = (1-6t) h with c = 1 - 6t:
F(Rm_{h(t)}) dV_{h(t)}
= (1-6t)^{-1} F(Rm_h) · (1-6t)^2 dV_h
= (1-6t) F(Rm_h) dV_h. (7.10)
On the other hand, by Proposition 6.2 and smooth convergence,
Int_N F(Rm_{h(t)}) dV_{h(t)}
= lim_j Int_{M_j} F(Rm_{g_j(t)}) dV_{g_j(t)}
<= C t. (7.11)
Combine (7.10) and (7.11):
(1 - 6t) Int_N F(Rm_h) dV_h <= C t, 0 < t < T. (7.12)
Let t -> 0+. Then (1 - 6t) -> 1, Ct -> 0, so
Int_N F(Rm_h) dV_h = 0. (7.13)
Since F >= 0 is continuous and Rm_h is smooth on the compact manifold
N, F(Rm_h) is a continuous nonnegative function whose integral is
zero. Therefore F(Rm_h) = 0 pointwise at every point of N.
By the definition (6.12) of F, this means all sectional curvatures of
h are nonnegative:
sec_h >= 0 everywhere. (7.14)
7.8. The packaged conclusion.
Statement of the compactness result, independent of any classification
assumption.
PROPOSITION 7.1 (Compact Einstein limit). Let (M_j, g_j) be a sequence
as in (3.1), normalized as in Proposition 3.1, with flows g_j(t) and
uniform lifespan T as in Proposition 5.4. Then there is a subsequence,
a connected simply connected closed smooth 4-manifold N, a smooth
metric h on N, and diffeomorphisms Phi_j: N -> M_j, with:
(i) (Phi_j)^* g_j(t) -> (1-6t) h smoothly on N x I,
for every compact I subset of (0, T);
(ii) Ric_h = 3h;
(iii) sec_h >= 0.
Proof. Summary of Sections 7.1-7.7. (i) from Hamilton compactness
plus eventual diffeomorphism of embeddings; (ii) from Proposition
6.1 and the integration of the homothetic flow in Section 7.6;
(iii) from Proposition 6.2 and the scaling (7.9) in Section 7.7. ✓
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8. PROVING THE GAP THEOREM
================================================================================
We now assemble the pieces.
THEOREM 8.1 (L^2 Einstein gap). There is a universal constant epsilon_0 > 0
with the following property. If (M, g) is a connected closed simply
connected Riemannian 4-manifold with
sec_g >= 0 and D(g) = Int_M |E_g|^2 dV_g < epsilon_0,
then M is diffeomorphic, with no prescribed orientation, to one of
S^4, CP^2, S^2 x S^2.
Proof. Suppose no such epsilon_0 exists. Then for each j = 1, 2, 3, ...
there is a connected closed simply connected 4-manifold (M_j, g_j)
with sec(g_j) >= 0, D(g_j) < 1/j, and M_j NOT diffeomorphic to any of
S^4, CP^2, or S^2 x S^2.
Step 1: normalize. Rescale each g_j by a positive constant c_j to
arrange (1/V_j) Int R_{j} dV_j = 12. This is legal because D is
scale-invariant in dimension 4. All bounds derived in Proposition 3.1
hold: v_0 <= V_j <= V_0, diam(M_j, g_j) <= D_0, ||Ric - 3g||_{L^2} -> 0,
||R - 12||_{L^2} <= 4 ||E||_{L^2} -> 0.
Step 2: uniform Ricci flow. By Proposition 5.4, there is a universal
T > 0 such that every maximal Ricci flow g_j(t) starting at g_j(0) =
g_j exists on [0, T], and the curvature bound |Rm_{g_j(t)}| <= A/t
holds on (0, T).
Step 3: compact Einstein limit. By Proposition 7.1, after passing to
a subsequence, there are a closed simply connected smooth 4-manifold
N, a smooth metric h on N, and diffeomorphisms Phi_j: N -> M_j with:
Ric_h = 3h,
sec_h >= 0.
Step 4: classification. By Assumption 1.1, N is diffeomorphic (indeed,
since it is simply connected, isometric up to conjugating by the
universal cover identification) to one of S^4, CP^2, S^2 x S^2.
Step 5: contradiction. Each Phi_j: N -> M_j is a diffeomorphism. Since
N is diffeomorphic to one of the three standard manifolds, so is M_j.
This contradicts the choice of M_j. So the epsilon_0 exists. ✓ [QED]
Remarks on the proof.
(i) Epsilon_0 is not explicit. It is "the smallest positive
eigenvalue of the defect operator on the space of bad metrics",
shown positive by the contradiction above. In the full
argument, one could make it effective by making every
constant explicit.
(ii) Assumption 1.1 is used exactly once, only in Step 4, only on
an exact Einstein metric. Everything before Step 4 is
classification-independent.
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9. WHAT IS PROVEN AND WHAT IS ASSUMED
================================================================================
Unconditional. Proposition 7.1 and every step that produces it --
normalization (Section 3), Sobolev (Section 4), entropy, tail,
noncollapse, and lifespan (Section 5), the two defects (Section 6),
and compactness (Section 7) -- is proved in full without any
classification input.
Assumed. Assumption 1.1 (the classification of exact Einstein limits,
from [ZeroPlane]) is used exactly once, in Step 4 of Theorem 8.1, on
an exact Einstein metric. No stability of Einstein metrics, no
approximate version, and no bounds on derivatives of curvature near
the limit are assumed. If a different proof of the classification
exists, substitute it in Step 4 and Theorem 8.1 follows unchanged.
Not preserved along the flow. sec >= 0 is a hypothesis at time 0 only.
Ricci flow can create negative curvature, and Proposition 6.2 bounds
the total negative amount by C t. The shrinking h(t) = (1-6t) h of
Section 7.6 converts that C t bound into a zero bound in the limit,
so the proof does not require sec_j(t) >= 0 for t > 0.
================================================================================
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END OF DOCUMENT
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