Mean curvature of extrinsic spheres in submanifolds of real space forms

Abstract

Given a submanifold P m with the Hilbert-Schmidt norm of its second fundamental form bounded from above, in a real space form of constant curvature b, K n (b), we have obtained a lower bound for the norm of the mean curvature normal vector field of extrinsic spheres with sufficiently small radius in P m in terms of the mean curvature of the geodesic spheres in K m (b), with same radius, and the mean curvature of P m . Introduction.In the paper [5] it was obtained a lower bound for the norm of the mean curvature normal vector field of the boundary of some well-characterized domains, the extrinsic balls, defined in a hypersurface of the real space forms with constant sectional curvature b, that we shall denote as K n (b), in terms of the mean curvature of the geodesic spheres of these space forms and the mean curvature of the hypersurface. The purpose of this work is to present the same lower bound for the norm of the mean curvature normal vector field defined on the boundary of the extrinsic balls, when these extrinsic domains are placed on a submanifold with any codimension in the real space forms, and we have the additional restriction that the norm of the second fundamental form of this submanifold is bounded from above, i.e., the submanifold is not too extrinsically curved.This objective takes place in the more general setting which deals with to what extent information about the extrinsic curvatures of the extrinsic spheres, (the boundary of the extrinsic balls), in a submanifold P m can give us an idea about the way the whole submanifold P m is immersed in the ambient manifold or about some other global geometric properties of P m . In fact, there are in the literature a number of works where other geometric quantities defined on the extrinsic balls and spheres, such like its volume, have been used for similar purposes, (see e.g. the paper [2], where a characterization of totally geodesic embeddings is obtained from an asymptotic formula for the volume of an extrinsic ball of small radius in a submanifold of the n-dimensional euclidean space, or the papers [3] and [4], which deal with the behaviour of the quotient between the volumes of the extrinsic