@misc{Scharrer2016,
  type      = {Master Thesis},
  author    = {Scharrer, Christian},
  title     = {Relating diameter and mean curvature for varifolds},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-97013},
  school      = {Universit{\"a}t Potsdam},
  pages     = {42},
  year      = {2016},
  abstract  = {The main results of this thesis are formulated in a class of surfaces (varifolds) generalizing closed and connected smooth submanifolds of Euclidean space which allows singularities. Given an indecomposable varifold with dimension at least two in some Euclidean space such that the first variation is locally bounded, the total variation is absolutely continuous with respect to the weight measure, the density of the weight measure is at least one outside a set of weight measure zero and the generalized mean curvature is locally summable to a natural power (dimension of the varifold minus one) with respect to the weight measure. The thesis presents an improved estimate of the set where the lower density is small in terms of the one dimensional Hausdorff measure. Moreover, if the support of the weight measure is compact, then the intrinsic diameter with respect to the support of the weight measure is estimated in terms of the generalized mean curvature. This estimate is in analogy to the diameter control for closed connected manifolds smoothly immersed in some Euclidean space of Peter Topping. Previously, it was not known whether the hypothesis in this thesis implies that two points in the support of the weight measure have finite geodesic distance.},
  language  = {en}
}
@article{Menne2016,
  author    = {Menne, Ulrich},
  title     = {Weakly Differentiable Functions on Varifolds},
  series = {Indiana University mathematics journal},
  volume    = {65},
  journal   = {Indiana University mathematics journal},
  publisher = {Indiana University, Department of Mathematics},
  address   = {Bloomington},
  issn      = {0022-2518},
  doi       = {10.1512/iumj.2016.65.5829},
  pages     = {977 -- 1088},
  year      = {2016},
  abstract  = {The present paper is intended to provide the basis for the study of weakly differentiable functions on rectifiable varifolds with locally bounded first variation. The concept proposed here is defined by means of integration-by-parts identities for certain compositions with smooth functions. In this class, the idea of zero boundary values is realised using the relative perimeter of superlevel sets. Results include a variety of Sobolev Poincare-type embeddings, embeddings into spaces of continuous and sometimes Holder-continuous functions, and point wise differentiability results both of approximate and integral type as well as coarea formulae. As a prerequisite for this study, decomposition properties of such varifolds and a relative isoperimetric inequality are established. Both involve a concept of distributional boundary of a set introduced for this purpose. As applications, the finiteness of the geodesic distance associated with varifolds with suitable summability of the mean curvature and a characterisation of curvature varifolds are obtained.},
  language  = {en}
}