Immersions with bounded second fundamental form

TL;DR

This paper establishes compactness results for immersions of compact manifolds with bounded second fundamental form in L^p, extending classical theorems and enabling applications to geometric flow convergence.

Contribution

It generalizes Langer's classical compactness theorem to arbitrary dimensions and codimensions, and introduces a localized version suitable for geometric flow analysis.

Findings

Proves compactness for immersions with bounded L^p second fundamental form.

Provides a localized compactness result for practical applications.

Facilitates convergence proofs in geometric flow studies.

Abstract

We first consider immersions on compact manifolds with uniform $L^p$-bounds on the second fundamental form and uniformly bounded volume. We show compactness in arbitrary dimension and codimension, generalizing a classical result of J. Langer. In the second part, this result is used to deduce a localized version, being more convenient for many applications, such as convergence proofs for geometric flows.