The volume of an extrinsic ball in a minimal submanifold has a well defined lower bound when the ambient manifold has an upper bound on its sectional curvatures, see e.g.  and . When this upper bound is non-positive, the second named author has shown an isoperimetric inequality for such domains, see . This result again gives the comparison result for volumes alluded to above together with a characterization of the totally geodesic submanifolds of hyperbolic space forms. In the present paper we find a corresponding sharp isoperimetric inequality for minimal submanifolds in spaces with sectional curvatures bounded from above by any constant. As a corollary we find again a characterization of the totally geodesic submanifolds of spherical space forms.
|Journal||Journal fuer die Reine und Angewandte Mathematik|
|Publication status||Published - 2002|