diff 999db1f..b34550f univalent_inverse_power_v2.txt
diff --git a/univalent_inverse_power_v2.txt b/univalent_inverse_power_v2.txt index ebbdd54..a92cbeb 100644 --- a/univalent_inverse_power_v2.txt +++ b/univalent_inverse_power_v2.txt @@ -2,54 +2,6 @@ A STRICT INVERSE-FIRST-POWER BOUND FOR UNIVALENT FUNCTIONS a complete, self-contained account -WHAT THIS IS - -This is a complete proof of one theorem about the shapes that a -complex-differentiable (holomorphic) map can produce. It is written so -that a student who knows complex numbers and one-variable calculus can -follow every step. Nothing is left as an exercise and nothing is -handed to a "standard lemma" without saying exactly what the lemma -says. - -Three results from analysis are used without proof. Each is named -every time it is used, and each is stated in words: - - (C1) COMPACTNESS. A family of injective holomorphic functions that - is normalized at one point (value and derivative pinned) is - compact in the sense of locally uniform convergence, and the - limits are again injective. This is Montel's theorem plus - Hurwitz's theorem. - - (C2) SARD. For a smooth map of the plane, the set of images of - points where the derivative fails to be invertible has area - zero. - - (C3) INDEX COUNTING. A simple closed curve has turning number one. - For a smooth real function of two variables, the index of an - isolated critical point is +1 if the Hessian determinant is - negative (a peak) and -1 if it is positive (a saddle); a - logarithmic pole has index +1. - -You will need: complex numbers, partial derivatives, the area of a -plane region, and the idea of averaging over a collection of maps. -You will not need measure theory beyond the two phrases "area zero" -and "average of a nonnegative quantity". - - -CONTENTS - - Part 0 The problem and the answer - Part 1 A compact model, and one operator that measures growth - Part 2 A weighted law of maps, when the operator grows too fast - Part 3 A potential, and why it cannot run off to infinity - Part 4 Critical points, and pairing each saddle with a peak - Part 5 Transport: the pairing forces an average to be <= 0 - Part 6 The harmonic barrier, and the uniform gap - Part 7 What the proof does and does not say - Appendix Notation and where each constant comes from - - -================================================================ PART 0. THE PROBLEM AND THE ANSWER ================================================================