Compactness of immersions with local Lipschitz representation

Abstract

We consider immersions admitting uniform representations as a λ-Lipschitz graph. In codimension 1, we show compactness for such immersions for arbitrary fixed $\lambda <\infty$ and uniformly bounded volume. The same result is shown in arbitrary codimension for $\lambda ⩽\frac{1}{4}$.