Integral formulas for compact space-like hypersurfaces in de Sitter space: Applications to the case of constant higher order mean curvature
\textit{A. J. Goddard} conjectured in 1977 [Math. Proc. Camb. Philos. Soc. 82, 489-495 (1977; Zbl 0386.53042)] that the only complete spacelike hypersurfaces in the de Sitter space \(S^{n+1}_1\) with constant mean curvature \(H\) should be the totally umbilical ones. \textit{K. Akutagawa} [Math. Z. 196, 13-19 (1987; Zbl 0611.53047)] and \textit{S. Montiel} [Indiana Univ. Math. J. 37, 909-917 (1988; Zbl 0677.53067)] showed that in some cases this is true. On the other hand \textit{Q.-M. Cheng} and \textit{S. Ishikawa} [Manuscr. Math. 95, No. 4, 499-505 (1998; Zbl 0913.53022)] showed that here \(H\) can be replaced by the scalar curvature \(S\). In the present paper these results are generalized for the \(r\)th mean curvatures \(H_r\), introduced by \(\text{det}(tI- A)= \sum^n_{r=0} \left(\begin{smallmatrix} n\\ r\end{smallmatrix}\right) H_rt^{n-r}\), where \(A\) is the shape operator. Here \(H= H_1\) and \(S= n(n-1)(1- H_2)\). Some integral formulas for compact spacelike hypersurfaces in \(S^{n+1}_1\) are developed, called the Minkowski formulas. As applications some theorems are proved to characterize the totally umbilical round spheres, among them the following. The only compact spacelike hypersurfaces in \(S^{n+1}_1\) having \(H_r\) and \(H_{r+1}\) both constant, with \(0\leq r\leq n-2\), are the totally umbilical round spheres; here \(H_0= 1\). The only compact spacelike hypersurfaces in \(S^{n+1}_1\) with constant \(r\)th mean curvature \(H_r\), \(2\leq r\leq n\), which are contained in the chronological future (or past) of an equator of \(S^{n+1}_1\) are the totally umbilical round spheres.
scientific article; zbMATH DE number 1356235
scientific article