Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Proof of the Toponogov Conjecture on Complete Surfaces

Brendan Guilfoyle, Wilhelm Klingenberg

Abstract

We prove a conjecture of Toponogov on complete convex planes, namely that such planes must contain an umbilic point, albeit at infinity. Our proof is indirect. It uses Fredholm regularity of an associated Riemann-Hilbert boundary value problem and an existence result for holomorphic discs with Lagrangian boundary conditions, both of which apply to a putative counterexample. Corollaries of the main theorem include a Hawking-Penrose singularity-type theorem, as well as the proof of a conjecture of Milnor's from 1965 in the convex case.

Proof of the Toponogov Conjecture on Complete Surfaces

Abstract

We prove a conjecture of Toponogov on complete convex planes, namely that such planes must contain an umbilic point, albeit at infinity. Our proof is indirect. It uses Fredholm regularity of an associated Riemann-Hilbert boundary value problem and an existence result for holomorphic discs with Lagrangian boundary conditions, both of which apply to a putative counterexample. Corollaries of the main theorem include a Hawking-Penrose singularity-type theorem, as well as the proof of a conjecture of Milnor's from 1965 in the convex case.

Paper Structure

This paper contains 18 sections, 42 theorems, 295 equations.

Table of Contents

  1. Introduction and Results
  2. Complete convex planes after Toponogov
  3. Fredholm Regularity of the Associated Riemann-Hilbert Problem
  4. Neutral Kähler Geometry of $TS^2$
  5. The neutral Kähler surface and submanifolds
  6. The second fundamental form of a positive surface
  7. Angles between positive surfaces
  8. Mean Curvature Flow in $TS^2$
  9. The unparameterized flow
  10. Evolution equations for a positive surface
  11. Boundary conditions
  12. Existence Results for the I.B.V.P.
  13. Short-time existence
  14. Long-time existence
  15. Interior gradient estimate
  16. ...and 3 more sections

Key Result

Theorem 1

Let $P \subset{{\mathbb R}^3}$ be the image of a $C^{3,\alpha}$--smooth embedding of ${\mathbb R}^2$ into ${\mathbb R}^3$. Then the following three properties cannot hold simultaneously: Here $\kappa_1,\kappa_2$ are the principal curvatures of $P$.

Theorems & Definitions (88)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 8
  • proof
  • Proposition 9
  • proof
  • Proposition 10
  • proof
  • Definition 11
  • ...and 78 more