Gauss map harmonicity and mean curvature of a hypersurface in a homogeneous manifold
The Gauss map of an orientable hypersurface in a homogeneous manifold with an invariant Riemannian metric is defined. The main aim of this paper is to extend to this setting some results on the Gauss map of a constant mean curvature hypersurface in the Euclidean space, namely the Ruh-Vilm theorem relating the harmonicity of the Gauss map and the constancy of the mean curvature, and the Hoffman-Osserman-Schoen theorem characterizing the plane and the circular cylinder as the only complete constant mean curvature surfaces whose Gauss image is contained in a closed hemisphere of the sphere. For example, the following result is proved: Let \(M\) be an orientable hypersurface in a homogeheous manifold \(G/H\) and \(N:M\to S^{n+k}\) the Gauss map of \(M\). There is the equivalence between the following assertions: (i) \(M\) has constant mean curvature; (ii) The Gauss map \(N\) is harmonic; (iii) \(N\) satisfies the following equation \(\Delta N(z)=-(\|B\|^2+\|B^i\|^2+(n+k)\text{Ric}(l_x^{-1}(\eta (z))))N(z)\), where \(\eta\) is a normal vector field to \(M\) and \(\text{Ric}(w)\) is the Ricci curvature of \(G\) with respect to \(w\in TG\). Moreover, the following result is proved: Assume that \(\dim G/H=3\). Let \(M\) be a complete constant mean curvature surface of \(G/H\) and \(N:M\to S^{3+k}\) the Gauss map of \(M\). Suppose that \(\text{Ric}>0\) or \(\|B^i\|>0\) or \(M\) is not totally geodesic. Then \(N(M)\) is contained in a hemisphere of the unit sphere in the Lie algebra \(g\) of \(G\) if and only if \(M\) is invariant under a one-parameter subgroup of isometries of \(G\).
scientific article; zbMATH DE number 5125697