Compactness of immersions with local Lipschitz representation

Abstract

We consider immersions admitting uniform representations as an L-Lipschitz graph. In codimension 1, we show compactness for such immersions for arbitrary fixed finite L and uniformly bounded volume. The same result is shown in arbitrary codimension for L less than or equal to 1/4

Immersions with Bounded Second Fundamental Form

“…Acknowledgement: I would like to thank my advisor Ernst Kuwert for his support. Moreover I would like to thank Manuel Breuning for proofreading my dissertation [6], where the results of this paper were established first.…”

$C^1$-regularity for local graph representations of immersions

“…Acknowledgement: I would like to thank my advisor Ernst Kuwert for his support. Moreover I would like to thank Manuel Breuning for proofreading my dissertation [1], where the result of this paper was established first.…”

Two-dimensional curvature functionals with superquadratic growth

“…Compactness results. We start by quoting the fundamental compactness theorem of J. Langer (see also [6]). E…”

“…The prescribed mean curvature system (3.2) for the u k fulfills the assumption of We now apply a localized version of Langer's theorem from [6] to obtain a proper immersion f 0 : Σ 0 → R n , such that thef k converge to f 0 locally in C 1,β up to diffeomorphisms. Weak lower semicontinuity of W p implies H f0 = 0.…”

“…We now argue that the Gauß map converges to a constant locally uniformly on Σ 0 = R 2 , contradicting the definition of r k . More precisely, from the compactness theorem in [6] we know thatf k • φ k → f 0 locally in C 1 and moreover 0)) is diffeomorphic. Nowf k (q k ) = 0 by construction, therefore there exists a p k ∈ U k with φ k (p k ) = q k .…”