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Alexandrov's Theorem for Integral Varifolds and Applications to Geometric Inequalities
Abstract
In this paper, we use optimal mass transportation to derive an isoperimetric inequality and an interior Michael-Simon-Sobolev inequality for weakly twice differentiable integral varifolds of dimension and codimension at least $2$, with locally bounded first variation and L2loc mean curvature. This generalizes a result of Brendle and Eichmair. In both inequalities the corresponding constants are optimal in codimension 2. As part of this process we also generalize Alexandrov's theorem for convex functions to this setting. This necessitates the use of measure and distribution-valued solutions to certain geometric differential operators, in addition to the theory of Lp Taylor expansions started by Calderón and Zygmund.
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BibTeX
@misc{gaudet2026alexandrovs,
title = {{Alexandrov's Theorem for Integral Varifolds and Applications to Geometric Inequalities}},
author = {Mitchell Gaudet},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20628},
eprint = {2609.20628},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20628},
}
RIS
TY - UNPB
TI - Alexandrov's Theorem for Integral Varifolds and Applications to Geometric Inequalities
AU - Mitchell Gaudet
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20628
UR - https://arxiv.org/abs/2609.20628
AB - In this paper, we use optimal mass transportation to derive an isoperimetric inequality and an interior Michael-Simon-Sobolev inequality for weakly twice differentiable integral varifolds of dimension and codimension at least $2$, with locally bounded first variation and L^2_loc mean curvature. This generalizes a result of Brendle and Eichmair. In both inequalities the corresponding constants are optimal in codimension 2. As part of this process we also generalize Alexandrov's theorem for convex functions to this setting. This necessitates the use of measure and distribution-valued solutions to certain geometric differential operators, in addition to the theory of L^p Taylor expansions started by Calderón and Zygmund.
ER -