Constant scalar curvature hypersurfaces with spherical boundary in Euclidean space

Abstract

It is still an open question whether a compact embedded hypersurface in the Euclidean space with constant mean curvature and spherical boundary is necessarily a hyperplanar ba1l or a spherical cap, even in the simplest case of a compact constant mean curvature surface in R3 bounded by a circle. In this paper we prove that this is true for the case of the scalar curvature. Specifica1ly we prove that the only compact embedded hypersurfaces in the Euclidean space with constant scalar curvature and spherical boundary are the hyperplanar round balIs (with zero scalar curvature) and the spherical caps (with positive constant scalar curvature). The same applies in general to the case of embedded hypersurfaces with constant r-mean curvature, with r>=2.