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

Abstract

AbstractWe 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