Off-campus UMass Amherst users: To download campus access dissertations, please use the following link to log into our proxy server with your UMass Amherst user name and password.
Non-UMass Amherst users: Please talk to your librarian about requesting this dissertation through interlibrary loan.
Dissertations that have an embargo placed on them will not be available to anyone until the embargo expires.
Author ORCID Identifier
Open Access Dissertation
Doctor of Philosophy (PhD)
Year Degree Awarded
Month Degree Awarded
William H. Meeks III
Geometry and Topology
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 Rn must have infinite area. We also attain partial results on the conjecture: If M is a compact immersed surface in hyperbolic 3-space H3, and the absolute mean curvature function of M is bounded from above by 1, then Area(M)<=1/4*(Length(boundary M))2.
Chen, Dechang, "Isoperimetric inequality and area growth of surfaces with bounded mean curvature" (2014). Doctoral Dissertations. 62.