Two-ended Hypersurfaces with Zero Scalar Curvature

Abstract

In many ways hypersurfaces with null r-th curvature function Hr behave much like the minimal ones (H1 = 0). One such manifestation is the following result to be proved in this paper, which extends to scalar-flat hypersurfaces (H2 = 0) a well-known theorem of R. Schoen. Theorem. The only complete scalar-flat embeddings Mn ⊂ ℝn+1, free of flat points, which are regular at infinity and have two ends, are the hypersurfaces of revolution. The main tools in the proof of this and other related results presented in this paper are the Maximum Principle for the non-linear equation Hr (graph u) = 0 and the property that any height function h of a hypersurface with null Hr satisfies the intrinsic linear equation div (Tr-1∇h) = 0, where Tr-1 denotes the Newton tensor of the hypersurface shape operator.