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Author ORCID Identifier



Open Access Dissertation

Document Type


Degree Name

Doctor of Philosophy (PhD)

Degree Program


Year Degree Awarded


Month Degree Awarded


First Advisor

William H. Meeks III

Subject Categories

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.