| In this thesis we study issues related to the convergence theory of Riemannian manifolds with boundary. First, we establish a compactness theorem for the class of compact, uniformly mean convex Riemannian manifolds with boundary that satisfy bounds on diameter, area of the boundary, and curvature quantities.;Next, we establish a compactness theorem for manifolds with boundary that have controlled volume growth and integral bounds on curvature quantities. In dimension three, we replace the volume growth assumption with a simpler volume condition at the boundary, provided that an integral norm of the ambient curvature is small.;We use the convergence theory to prove 'geometric stability theorems' for Riemannian 3-manifolds whose ambient curvatures are small. The first theorem applies to 3-manifolds that have Ricci curvature close to 0 (in the pointwise sense) and whose boundaries are Gromov-Hausdorff close to a fixed metric on S2 with positive curvature. Such manifolds are close (in an appropriate Holder topology) to the region enclosed by a Weyl embedding of the fixed boundary metric into Euclidean space. This can be thought of as a generalization of a rigidity theorem of Cohn-Vossen--Pogorelov. We then establish stability theorems corresponding to the rigidity theorems of Hopf and Almgren.;The stability theorems have corresponding statements when the Ricci curvature is small in an appropriate integral norm. In particular, we establish a theorem that applies to compact 3-manifolds that have boundary close to the round metric on the sphere and Ricci curvature close to 0 in the L2 sense. Such manifolds are close (in an appropriate Sobolev space topology) to the unit ball in Euclidean space. |
