UC Santa Barbara

In this work, we study interactions between the curvature of a Riemannian manifold and the geometry of its submanifolds. In particular, we consider manifolds with intermediate Ricci curvature bounded below and manifolds with integral curvature bounds.

First, we develop the tools for studying manifolds with intermediate Ricci curvature bounds. In particular, we prove a comparison theorem for the Hessian of the distance function to a submanifold based on a lower bound for the k-Ricci curvature.

The main result is a generalization of the inequality of E. Heintze and H. Karcher for the volume of tubes around minimal submanifolds to an inequality based on integral bounds for k-Ricci curvature. Even in the case of a pointwise bound, this generalizes the classical inequality by replacing a sectional curvature bound with a k-Ricci bound. This theorem is motivated by the estimates of Petersen-Shteingold-Wei for the volume of tubes around a geodesic and generalizes their result.

Finally, we give several applications of these comparison theorems to the geometry and topology of submanifolds in spaces with curvature bounded below. The first is a uniform lower bound on the volume of minimal submanifolds in spaces with integral curvature bounds. We then bound the relative growth of the fundamental group of a closed minimal submanifold in terms of the growth of the fundamental group of the embedding space when the latter has nonnegative intermediate Ricci curvature. We conclude with an application of the comparison theory for intermediate Ricci curvature to certain geometric inequalities which are motivated by questions in general relativity.