{"id":{"repo_id":"potsdam-thes","oai_identifier":"oai:kobv.de-opus4-uni-potsdam:9701"},"canonical_url":"https://search.dev.ndltd.org/etd/potsdam-thes/oai:kobv.de-opus4-uni-potsdam:9701","repository":{"repo_id":"potsdam-thes","name":"Universität Potsdam - Thes","base_url":"https://publishup.uni-potsdam.de/opus4-ubp/oai"},"display":{"title":"Relating diameter and mean curvature for varifolds","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Scharrer, Christian"],"institution":"Universität Potsdam","degree_name":null,"degree_level":"master","degree_discipline":null,"degree_department":null,"school":null,"contributors":["Menne, Ulrich"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-09-26","date_published":"2016-09-26","updated_at":"2026-07-24T03:52:09Z","subjects":["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"],"languages":[],"rights":["Keine öffentliche Lizenz: Unter Urheberrechtsschutz"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://publishup.uni-potsdam.de/frontdoor/index/index/docId/9701","outbound_label":"Repository record","outbound_source":"source_url"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Menne, Ulrich"]},{"key":"dc:creator","label":"Author","values":["Scharrer, Christian"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:publisher","label":"Institution","values":["Universität Potsdam"]},{"key":"dc:type","label":"Dc Type","values":["masterThesis"]},{"key":"thesis:degree_level","label":"Degree Level","values":["master"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Universität Potsdam"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["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"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["Keine öffentliche Lizenz: Unter Urheberrechtsschutz"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.","Die wichtigsten Ergebnisse dieser Arbeit sind formuliert für eine Klasse von Oberflächen (Varifaltigkeiten), welche geschlossene glatte Untermannigfaltigkeiten des euklidischen Raums verallgemeinern und Singularitäten erlauben. Gegeben sei eine mindestens zwei-dimensionale unzerlegbare Varifaltigkeit im euklidischen Raum, sodass die erste Variation lokal beschränkt ist, die totale Variation absolut stetig bezüglich dem Gewichtsmaß ist, die Dichte des Gewichtsmaßes außerhalb einer Nullmenge mindesten eins ist, und die verallgemeinerte mittlere Krümmung bezüglich dem Gewichtsmaß lokal summierbar zu einer natürlichen Potenz (Dimension der Varifaltigkeit minus eins) ist. Es wird die Menge, wo die untere Dichte klein ist, durch das ein-dimensionale Hausdorff-Maß abgeschätzt. Das Ergebnis ist eine neue, stark verbesserte untere Dichte-Schranke. Ist der Träger des Gewichtsmaßes kompakt, so wird der intrinsische Diameter des Trägers des Gewichtsmaßes abgeschätzt durch ein Integral der verallgemeinerten mittleren Krümmung. Diese Ungleichung ist analog zu der Ungleichung von Peter Topping für geschlossene zusammenhängende Mannigfaltigkeit, welche durch eine glatte Immersion in den euklidischen Raum eingebettet sind. Bisher war nicht bekannt, dass die oben genannten Annahmen an die Varifaltigkeit implizieren, dass der geodätische Abstand zweier Punkte aus dem Träger des Gewichtsmaßes endlich ist."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Relating diameter and mean curvature for varifolds","Relativer Diameter und mittlere Krümmung für Varifaltigkeiten"]}]}],"canonical_facts":{"dc:contributor":["Menne, Ulrich"],"dc:creator":["Scharrer, Christian"],"dc:description.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.","Die wichtigsten Ergebnisse dieser Arbeit sind formuliert für eine Klasse von Oberflächen (Varifaltigkeiten), welche geschlossene glatte Untermannigfaltigkeiten des euklidischen Raums verallgemeinern und Singularitäten erlauben. Gegeben sei eine mindestens zwei-dimensionale unzerlegbare Varifaltigkeit im euklidischen Raum, sodass die erste Variation lokal beschränkt ist, die totale Variation absolut stetig bezüglich dem Gewichtsmaß ist, die Dichte des Gewichtsmaßes außerhalb einer Nullmenge mindesten eins ist, und die verallgemeinerte mittlere Krümmung bezüglich dem Gewichtsmaß lokal summierbar zu einer natürlichen Potenz (Dimension der Varifaltigkeit minus eins) ist. Es wird die Menge, wo die untere Dichte klein ist, durch das ein-dimensionale Hausdorff-Maß abgeschätzt. Das Ergebnis ist eine neue, stark verbesserte untere Dichte-Schranke. Ist der Träger des Gewichtsmaßes kompakt, so wird der intrinsische Diameter des Trägers des Gewichtsmaßes abgeschätzt durch ein Integral der verallgemeinerten mittleren Krümmung. Diese Ungleichung ist analog zu der Ungleichung von Peter Topping für geschlossene zusammenhängende Mannigfaltigkeit, welche durch eine glatte Immersion in den euklidischen Raum eingebettet sind. Bisher war nicht bekannt, dass die oben genannten Annahmen an die Varifaltigkeit implizieren, dass der geodätische Abstand zweier Punkte aus dem Träger des Gewichtsmaßes endlich ist."],"dc:format.medium":["application/pdf"],"dc:publisher":["Universität Potsdam"],"dc:rights":["Keine öffentliche Lizenz: Unter Urheberrechtsschutz"],"dc:subject":["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"],"dc:title":["Relating diameter and mean curvature for varifolds","Relativer Diameter und mittlere Krümmung für Varifaltigkeiten"],"dc:type":["masterThesis"],"thesis:degree_level":["master"],"thesis:institution_name":["Universität Potsdam"]},"updated_at":"2026-07-24T03:52:09Z"}