Integral formulas for compact spacelike \(n\)-submanifolds in de Sitter spaces. Applications to the parallel mean curvature vector case
The authors introduce a new method to study compact spacelike \(n\)-submanifolds in de Sitter spaces \(S^{n+ p}_q\) by means of certain integral formulas. Several interesting results are then established. Theorem 3.1. Let \(M^n\) be a complete maximal submanifold in \(S^{n+ p}_q\), \(1\leq q\leq p\), such that \(\text{Ric }\geq n- 1\). Then \(M^n\) is totally geodesic in \(S^{n+ q}_q\). For surfaces: The only complete maximal surfaces in \(S^{2+ p}_q\), \(1\leq q\leq p\), with Gaussian curvature \(\geq 1\) are the totally geodesic ones. Several applications are given to the parallel mean curvature case; for instance Theorem 5.2: Let \(M^2\) be a compact spacelike surface with non-zero parallel mean curvature vector \(H\) in \(S^{2+ p}_q\), \(1\leq q\leq p\). If \(M^2\) is a topological sphere and its Gaussian curvature satisfies \(K\geq 1+ \langle H, H\rangle\), then \(M^2\) is totally umbilical.
scientific article; zbMATH DE number 824168