Immersions with local Lipschitz representation
Patrick Breuning · FreiDok plus (Universitätsbibliothek Freiburg) · 2011
First we consider immersions admitting uniform bounds on the second fundamental form and the volume. Here the manifolds on which the immersions are defined are not required to be compact. It shall be shown compactness for such kind of immersions, using local graph representations over the affine tangent space. As a corollary we show a convergence result for measures defined by such immersions. Inspired by these results, in the second part we like to take immersions with uniform graph representations as our starting point. We consider immersions admitting a local representation as an L-Lipschitz graph over an appropriately chosen m-space. In the case of hypersurfaces with uniformly bounded volume, we show compactness for arbitrary fixed Lipschitz constant L. In particular, up to diffeomorphism, there are only finitely many manifolds admitting such an immersion. The same result is shown in arbitrary codimension for L sufficiently small. Finally, in the third part, we assume only a bound for some weaker norm to be satisfied by the graph functions. More precisely, we investigate immersions with continuous graph functions with small supremum norm. The graph functions are shown to be differentiable and to be Lipschitz with small Lipschitz constant.