Orientability of linear Weingarten surfaces, spacelike CMC-1 surfaces and maximal surfaces
Orientability of linear Weingarten surfaces, spacelike CMC-1 surfaces and maximal surfaces
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线性 Weingarten 曲面、类空间 CMC-1 曲面和最大曲面的定向性
DOI：
10.1002/mana.200910176
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
M.Kokubu and M.Umehara
中科院分区：
文献类型：
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作者：
Lothar Gottsche;中島啓、吉岡康太;M.Kokubu and M.Umehara
我们证明了Bryant型线性Weingarten曲面在双曲三维空间中作为波前的若干拓扑性质。例如，我们展示了这种表面的可定向性，以及当它们不平坦时的共定向性。此外，我们还通过另一个全纯双曲高斯映射给出了非全纯双曲高斯映射的显式公式。利用这一点，我们证明了在de Sitter 3 -空间中CMC - 1面（即具有可容许奇点的常平均曲率曲面）的可定向性和共定向性。（CMC‐1面可能不是一般的波前，而是属于一类具有奇点的线性Weingarten面。）由于线性Weingarten面和CMC‐1面都可能有奇异点，所以可定向性和共定向性都是非平凡性质。进一步证明了Bryant型线性Weingarten曲面的基本群之字形表示是平凡的。我们还讨论了洛伦兹-闵可夫斯基3空间中不可定向极大面的一些性质，并比较了德西特3空间中CMC - 1面的相应性质。©2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim
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. Furthermore, we show that the zig‐zag representation of the fundamental group of a linear Weingarten surface of Bryant type is trivial. 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. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim