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Proof of the Caratheodory Conjecture

Brendan Guilfoyle, Wilhelm Klingenberg

TL;DR

The paper proves Carathéodory’s conjecture for $C^{3+\alpha}$-smooth closed convex surfaces in ${\mathbb E}^3$ by translating the problem into the neutral Kähler geometry of $TS^2$. It shows that an umbilic point on a convex surface corresponds to a complex point on a Lagrangian surface in $TS^2$, and that a hypothetical surface with a single umbilic would force a totally real Lagrangian hemisphere without a holomorphic disc boundary, which is contradicted by constructing a holomorphic disc via mean curvature flow with boundary in $TS^2$. The strategy combines a Banach-manifold setup for holomorphic discs, transversality arguments, and detailed analysis of mean curvature flow in the neutral metric, including edge/boundary control and compactness results. Consequently, the reformulated conjecture holds for $C^{3+\alpha}$-smooth convex surfaces, with broad implications for the interplay between complex, symplectic, and Riemannian geometry in neutral signature settings.

Abstract

A well-known conjecture of Caratheodory states that the number of umbilic points on a closed convex surface in ${\mathbb E}^3$ must be greater than one. In this paper we prove this for $C^{3+α}$-smooth surfaces. The Conjecture is first reformulated in terms of complex points on a Lagrangian surface in $TS^2$, viewed as the space of oriented geodesics in ${\mathbb E}^3$. Here complex and Lagrangian refer to the canonical neutral Kaehler structure on $TS^2$. We then prove that the existence of a closed convex surface with only one umbilic point implies the existence of a totally real Lagrangian hemisphere in $TS^2$, to which it is not possible to attach the edge of a holomorphic disc. The main step in the proof is to establish the existence of a holomorphic disc with edge contained on any given totally real Lagrangian hemisphere. To construct the holomorphic disc we utilize mean curvature flow with respect to the neutral metric. Long-time existence of this flow is proven by a priori estimates and we show that the flowing disc is asymptotically holomorphic. Existence of a holomorphic disc is then deduced from Schauder estimates.

Proof of the Caratheodory Conjecture

TL;DR

The paper proves Carathéodory’s conjecture for -smooth closed convex surfaces in by translating the problem into the neutral Kähler geometry of . It shows that an umbilic point on a convex surface corresponds to a complex point on a Lagrangian surface in , and that a hypothetical surface with a single umbilic would force a totally real Lagrangian hemisphere without a holomorphic disc boundary, which is contradicted by constructing a holomorphic disc via mean curvature flow with boundary in . The strategy combines a Banach-manifold setup for holomorphic discs, transversality arguments, and detailed analysis of mean curvature flow in the neutral metric, including edge/boundary control and compactness results. Consequently, the reformulated conjecture holds for -smooth convex surfaces, with broad implications for the interplay between complex, symplectic, and Riemannian geometry in neutral signature settings.

Abstract

A well-known conjecture of Caratheodory states that the number of umbilic points on a closed convex surface in must be greater than one. In this paper we prove this for -smooth surfaces. The Conjecture is first reformulated in terms of complex points on a Lagrangian surface in , viewed as the space of oriented geodesics in . Here complex and Lagrangian refer to the canonical neutral Kaehler structure on . We then prove that the existence of a closed convex surface with only one umbilic point implies the existence of a totally real Lagrangian hemisphere in , to which it is not possible to attach the edge of a holomorphic disc. The main step in the proof is to establish the existence of a holomorphic disc with edge contained on any given totally real Lagrangian hemisphere. To construct the holomorphic disc we utilize mean curvature flow with respect to the neutral metric. Long-time existence of this flow is proven by a priori estimates and we show that the flowing disc is asymptotically holomorphic. Existence of a holomorphic disc is then deduced from Schauder estimates.

Paper Structure

This paper contains 27 sections, 62 theorems, 413 equations.

Table of Contents

  1. Introduction
  2. Strategy and Proof
  3. Reformulation of the Conjecture in $TS^2$
  4. The manifold of holomorphic discs with boundary
  5. Proof of the Conjecture
  6. Mean Curvature Flow of Compact Spacelike Submanifolds
  7. Immersed spacelike submanifolds
  8. Multi-angles
  9. Multi-angles
  10. The height functions
  11. The I.V.P.
  12. Neutral Kähler Geometry of $TS^2$
  13. The neutral Kähler surface and submanifolds
  14. The second fundamental form of a positive surface
  15. Angles between positive surfaces
  16. ...and 12 more sections

Key Result

Proposition 1

A surface $\Sigma$ in $TS^2$ is Lagrangian iff there exists a surface S in ${\mathbb E}^3$ which is orthogonal to the oriented lines of $\Sigma$.

Theorems & Definitions (126)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Example 4
  • Definition 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Definition 8
  • ...and 116 more