发表机构
University of Toronto（多伦多大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过最优质量输运，为积分varifolds建立了等周和Michael-Simon-Sobolev不等式，推广了Brendle-Eichmair结果，并推广了Alexandrov定理，常数在余维2时最优。
AI 中文摘要
本文利用最优质量输运，推导了维数和余维数至少为2、具有局部有界一阶变分和L^2_loc平均曲率的弱二次可微积分varifolds的等周不等式和内部Michael-Simon-Sobolev不等式。这推广了Brendle和Eichmair的一个结果。在这两个不等式中，相应的常数在余维数2时是最优的。作为该过程的一部分，我们还将凸函数的Alexandrov定理推广到这一设置。这需要使用某些几何微分算子的测度和分布值解，以及由Calderón和Zygmund开创的L^p Taylor展开理论。
英文摘要
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.
Comments36 pages, Comments Welcome!
