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Proof of the Caratheodory Conjecture
Guilfoyle, Brendan...

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Proof of the Caratheodory Conjecture by Guilfoyle, Brendan ( Author )
Lemonjar Open Access
N.A
06-08-2008
A well-known conjecture of Caratheodory states that the number of umbilic points on a closed convex surface in E3 must be greater than one. In this paper we prove this for C3+α-smooth surfaces. The Conjecture is first reformulated in terms of complex points on a Lagrangian surface in TS2, viewed as the space of oriented geodesics in E3. Here complex and Lagrangian refer to the canonical neutral Kaehler structure on TS2. 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 TS2, 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.
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36.88 KB
English
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MYR 0.00
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http://arxiv.org/abs/0808.0851
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