Quasiconformal Gauss maps and the Bernstein problem for Weingarten multigraphs
We prove that any complete, uniformly ellipticWeingarten surface in Euclidean 3-space whose Gauss map image omits an open hemisphere is a cylinder or a plane. This generalizes a classical theorem by Hoffman, Osserman and Schoen for constant mean curvature surfaces. In particular, this proves that pl...
| Autores: | , , |
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| Formato: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2020 |
| País: | España |
| Recursos: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/115303 |
| Acesso em linha: | https://hdl.handle.net/11441/115303 |
| Access Level: | acceso abierto |
| Palavra-chave: | Weingarten surfaces Fully nonlinear elliptic equations Bernstein problem Multigraphs Curvature estimates Quasiconformal Gauss map |
| Resumo: | We prove that any complete, uniformly ellipticWeingarten surface in Euclidean 3-space whose Gauss map image omits an open hemisphere is a cylinder or a plane. This generalizes a classical theorem by Hoffman, Osserman and Schoen for constant mean curvature surfaces. In particular, this proves that planes are the only complete, uniformly elliptic Weingarten multigraphs. We also show that this result holds for a large class of non-uniformly elliptic Weingarten equations. In particular, this solves in the affirmative the Bernstein problem for entire graphs for that class of elliptic equations. To obtain these results, we prove that planes are the only complete multigraphs with quasiconformal Gauss map and bounded second fundamental form. |
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