Immersions with bounded second fundamental form

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.

References in corpus (2)

• A Compactness Theorem for the Second Fundamental Form
• Existence of Integral $m$-Varifolds minimizing $\int |A{|}^p$ and $\int |H{|}^p$, $p>m$, in Riemannian Manifolds