The Gauss–Kronecker curvature of minimal hypersurfaces in four-dimensional space forms

Abstract

Let \({{\mathbb{Q}^4}(c)}\) be a four-dimensional space form of constant curvature c. In this paper we show that the infimum of the absolute value of the Gauss–Kronecker curvature of a complete minimal hypersurface in \({\mathbb{Q}^4(c), c\leq 0}\), whose Ricci curvature is bounded from below, is equal to zero. Further, we study the connected minimal hypersurfaces M 3 of a space form \({{\mathbb{Q}^4}(c)}\) with constant Gauss–Kronecker curvature K. For the case c ≤ 0, we prove, by a local argument, that if K is constant, then K must be equal to zero. We also present a classification of complete minimal hypersurfaces of \({\mathbb{Q}^4(c)}\) with K constant.