Isoperimetric inequalities and volume growth of extrinsic balls in minimalsubmanifolds
by
Vicente Palmer
Universitat Jaume I de Castelló, Spain
Coauthors: Steen Markvorsen (The Technical University of Denmark, Denmark)

Poster

Given a submanifold P immersed in a riemannian manifold M, an extrinsic ball with center p in P is defined as the connected component containing p of the intersection between a geodesic ball of the ambient space centered at p and the submanifold P. S.Y. Cheng, P. Li and S.T. Yau proved that the volume of an extrinsic ball in a minimal submanifold of a real space form has a well defined lower bound. This result was extended by S. Markvorsen for minimal submanifolds of a riemannian manifold with just an upper bound on the sectional curvature.

We are going to present some results about the volume of these extrinsic domains that generalizes the above mentioned. Namely, when the sectional curvatures of the ambient manifold are bounded from above, it can be proved two isoperimetric inequalities satisfied by the extrinsic balls in a minimal submanifold, getting as a corollary the volume comparison results above alluded, together with a characterization of the totally geodesic submanifolds of hyperbolic and spherical space forms. On the other hand, we have compared, using these isoperimetric inequalities, the volume growth of the extrinsic balls with the volume growth of balls and spheres in the space forms of constant curvature.

Date received: September 8, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-45.