Filling Volume Minimality and Boundary Rigidity of Metrics Close to a Negatively Curved Symmetric Metric

Abstract

We investigate the relation between boundary data of a compact manifold and its interior geometry. A compact Riemannian manifold D with smooth boundary partial(D) is boundary rigid if its interior geometry is uniquely determined by partial(D) and distances between points on partial(D). D is a minimal filling if for any D' with partial(D')=partial(D), having larger distances between points on partial(D) implies that Vol(D') is greater or equal to Vol(D).

In this thesis, we generalize D. Burago and S. Ivanov's work on filling volume minimality and boundary rigidity of almost real hyperbolic metrics. We show that regions with metrics close to a negatively curved symmetric metric are strict minimal fillings and hence boundary rigid. This includes perturbations of real, complex, quaternionic and Cayley hyperbolic metrics.