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arXiv · 0907.2284

Orientability of linear Weingarten surfaces, spacelike CMC-1 surfaces and maximal surfaces

Masatoshi Kokubu·Masaaki Umehara
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Abstract

We prove several topological properties of linear Weingarten surfaces of Bryant type, as wave fronts in hyperbolic 3-space. For example, we show the orientability of such surfaces, and also co-orientability when they are not flat. Moreover, we show an explicit formula of the non-holomorphic hyperbolic Gauss map via another hyperbolic Gauss map which is holomorphic. Using this, we show the orientability and co-orientability of CMC-1 faces (i.e., constant mean curvature one surfaces with admissible singular points) in de Sitter 3-space. (CMC-1 faces might not be wave fronts in general, but belong to a class of linear Weingarten surfaces with singular points.) Since both linear Weingarten fronts and CMC-1 faces may have singular points, orientability and co-orientability are both nontrivial properties. We also remark on some properties of non-orientable maximal surfaces in Lorentz-Minkowski 3-space, comparing the corresponding properties of CMC-1 faces in de Sitter 3-space.

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Masatoshi Kokubu, Masaaki Umehara. 2011-07-08. Orientability of linear Weingarten surfaces, spacelike CMC-1 surfaces and maximal surfaces. https://arxiv.org/abs/0907.2284

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