A curvatura de Gauss-Kronecker de hipersuperfícies mínimas em espaços forma quadridimensionais

Abstract

In this work, we study the results obtained by Asperti et al. [1] and Hasanis et al. [17] involving the Gauss-Kronecker curvature of minimal hypersurfaces in four-dimensional space forms. We present concepts related to the study of Riemannian manifolds, as well as the orthonormal frame field technique used by both articles. Among the results of [1], a local version of the result obtained by Cheng [4] stands out for the Euclidean and hyperbolic cases. In the spherical case, we obtain an isometry between the image of a minimal immersion of a complete hypersurface with non-zero constant Gauss-Kronecker curvature and the Clifford torus. We also present two theorems referring to the classification of complete minimal hypersurfaces in four-dimensional space forms, in addition to developing the results found in [17].