An integral inequality for compact maximal surfaces in \(n\)-dimensional de Sitter space and its applications (Q1891969)

The authors prove an integral inequality for the Gaussian curvature of compact maximal surfaces in \(n\)-dimensional de Sitter space: Let \(x : M^2 \to S^n_1 \subset L^{n+1}\) be a compact maximal surface in \(n\) dimensional de Sitter space \(n \geq 3\), then \(\int_M (K-1) dv \leq 0\), and equality holds if and only if the immersion \(n\) is totally geodesic. Some applications are given such as the following Bernstein type result for maximal surfaces in \(S^n_1\): The only complete maximal surfaces in \(S^n_1\) with Gaussian curvature \(K \geq 1\) are the totally geodesic ones.