[Paper Review] Proof of the Caratheodory Conjecture by Mean Curvature Flow in the Space of Oriented Affine Lines

Brendan Guilfoyle, Wilhelm Klingenberg|arXiv (Cornell University)|Aug 6, 2008

Geometric Analysis and Curvature Flows7 references10 citations

TL;DR

This paper proves the Carathéodory conjecture by showing that the index of any isolated umbilic point on a $C^3$-smooth surface in $\mathbb{E}^3$ is at most one, using mean curvature flow in the space of oriented affine lines. The key innovation is constructing stable holomorphic discs with boundary on a Lagrangian surface in $TS^2$, which implies the Keller-Maslov index is at least one, thereby establishing that closed convex surfaces must have more than one umbilic point.

ABSTRACT

We prove that the index of an isolated umbilic point on a $C^3$-smooth surface in Euclidean 3-space ${\mathbb E}^3$ is less than or equal to one. As a corollary, we establish the Caratheodory conjecture, that the number of umbilic points on a closed convex surface in ${\mathbb E}^3$ must be greater than one. We do this by first reformulating the problem in terms of the index of an isolated complex point on a Lagrangian surface in $TS^2$, viewed as the space of oriented geodesics in ${\mathbb E}^3$. The main step in the proof is to establish the existence of stable holomorphic discs with boundary contained on the Lagrangian surface enclosing the complex point. We first show that the existence of such discs implies that the Keller-Maslov index must be greater than or equal to one, which for topological reasons, places a bound on the index of the isolated complex point on the Lagrangian surface. To construct the holomorphic disc we utilize mean curvature flow with respect to the canonical neutral Kaehler metric on $TS^2$. We prove long-time existence of this flow by a priori estimates and show that, for small enough initial disc, the flowing disc is asymptotically holomorphic. Convergence to a bubbled holomorphic disc is then proven by a version of compactness for J-holomorphic discs with boundary contained in a totally real surface. Continuity up to the boundary assures that the Keller-Maslov index is retained in the limit and this establishes our main result.

Motivation & Objective

• To resolve the long-standing Carathéodory conjecture regarding the minimum number of umbilic points on closed convex surfaces in $\mathbb{E}^3$.
• To establish a topological bound on the index of isolated umbilic points via the Keller-Maslov index in a Lagrangian setting.
• To construct stable holomorphic discs with boundary on a Lagrangian surface in $TS^2$ to constrain the index of complex points.
• To apply mean curvature flow with respect to the canonical neutral Kähler metric on $TS^2$ to evolve initial discs toward holomorphic limits.
• To prove convergence of the flow to a bubbled holomorphic disc while preserving the Keller-Maslov index.

Proposed method

• Reformulate the umbilic point index problem as a complex point index problem on a Lagrangian surface in $TS^2$, the space of oriented geodesics in $\mathbb{E}^3$.
• Use the canonical neutral Kähler metric on $TS^2$ to define a mean curvature flow for discs with boundary on the Lagrangian surface.
• Establish long-time existence of the flow via a priori estimates, ensuring the evolving discs remain well-behaved.
• Prove that for small initial discs, the flowing discs become asymptotically holomorphic under the flow.
• Apply a compactness theorem for $J$-holomorphic discs with boundary on a totally real surface to extract a limit disc.
• Use continuity up to the boundary to show the Keller-Maslov index is preserved in the limit, implying it is at least one.

Experimental results

Research questions

• RQ1 Can the index of an isolated umbilic point on a $C^3$-smooth surface in $\mathbb{E}^3$ be bounded above by one?
• RQ2 Does the existence of stable holomorphic discs with boundary on a Lagrangian surface imply a lower bound on the Keller-Maslov index?
• RQ3 Can mean curvature flow in the space of oriented affine lines produce a holomorphic limit disc from an initial disc with boundary on the Lagrangian surface?
• RQ4 Is the Keller-Maslov index preserved under the limit of the mean curvature flow in this geometric setting?
• RQ5 Does the existence of such a limit disc imply that closed convex surfaces in $\mathbb{E}^3$ must have more than one umbilic point?

Key findings

• The index of any isolated umbilic point on a $C^3$-smooth surface in $\mathbb{E}^3$ is at most one.
• The existence of stable holomorphic discs with boundary on the Lagrangian surface implies the Keller-Maslov index is at least one.
• Mean curvature flow with respect to the neutral Kähler metric on $TS^2$ exists for all time and preserves the boundary condition.
• The flowing discs converge to a bubbled holomorphic disc, and the Keller-Maslov index is preserved in the limit.
• The Carathéodory conjecture is confirmed: every closed convex surface in $\mathbb{E}^3$ has at least two umbilic points.
• The topological constraint from the index bound implies that the number of umbilic points on a closed convex surface is greater than one.