Proof of the Caratheodory Conjecture
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2008
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| _version_ | 1866917894028263424 |
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| author | Guilfoyle, Brendan Klingenberg, Wilhelm |
| author_facet | Guilfoyle, Brendan Klingenberg, Wilhelm |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_0808_0851 |
| institution | arXiv |
| publishDate | 2008 |
| record_format | arxiv |
| spellingShingle | Proof of the Caratheodory Conjecture Guilfoyle, Brendan Klingenberg, Wilhelm Differential Geometry Analysis of PDEs Symplectic Geometry 53A05 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. |
| title | Proof of the Caratheodory Conjecture |
| topic | Differential Geometry Analysis of PDEs Symplectic Geometry 53A05 |
| url | https://arxiv.org/abs/0808.0851 |