Relating diameter and mean curvature for varifolds

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.

Level thesis:degree_level

master

Grantor dc:publisher

Universität Potsdam

Year

2016

Author dc:creator

• Scharrer, Christian

Contributors dc:contributor

• Menne, Ulrich

• varifold
• rectifiable varifold
• indecomposable varifold
• first variation
• mean curvature
• isoperimetric inequality
• density of a measure
• geodesic distance
• intrinsic diameter
• Varifaltigkeit
• rektifizierbare Varifaltigkeit
• unzerlegbare Varifaltigkeit
• erste Variation
• mittlere Krümmung
• isoperimetrische Ungleichung
• Dichte eines Maßes
• geodätischer Abstand
• intrinsischer Diameter

Statement dc:rights

• Keine öffentliche Lizenz: Unter Urheberrechtsschutz

Repository record source_url

https://publishup.uni-potsdam.de/frontdoor/index/index/docId/9701

OAI identifier oai:identifier

oai:kobv.de-opus4-uni-potsdam:9701