Isoperimetric inequality and area growth of surfaces with bounded mean curvature

Abstract

ISOPERIMETRIC INEQUALITY AND AREA GROWTH OF SURFACES WITH BOUNDED MEAN CURVATURE MAY 2014 DECHANG CHEN Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor William H. Meeks III In this thesis, we give a lower bound on the areas of small geodesic balls in an immersed hypersurface M contained in a Riemannian manifold N . This lower bound depends only on an upper bound for the absolute mean curvature function of M , an upper bound of the absolute sectional curvature of N and a lower bound for the injectivity radius of N . As a consequence, we prove that if M is a noncompact complete surface of bounded absolute mean curvature in Riemannian manifold N with positive injectivity radius and bounded absolute sectional curvature, then the area of geodesic balls of M must grow at least linearly in terms of their radius. In particular, this result implies the classical result of Yau that a complete minimal hypersurface in R must have infinite area. We also attain partial results on the conjecture: If M is a compact immersed surface in hyperbolic 3-space H, and the absolute mean curvature function of M is bounded from above by 1, then Area(M) ≤ (Length(∂M)) 2 4π .