varifolds的最优等周不等式
The optimal isoperimetric inequality for varifolds
AI总结：
本文证明了Brendle等周不等式对有限质量与全变差的varifolds成立，允许奇异部分，仅需可积平均曲率，常数在余维数一、二时最优。
AI中文摘要：
我们证明了Brendle等周不等式对于具有有限质量和有限全一阶变差的varifolds成立，允许任意奇异部分。分布平均曲率仅假设可积。该结果适用于积分varifolds，更一般地，适用于具有正下密度界且权测度集中在可数多个$C^2$子流形上的可求积varifolds。证明使用了最优传输和环境凸函数的切向拉普拉斯算子的局部性定理。在余维数一和二中等周常数是尖锐的。
英文摘要：
We prove the Brendle isoperimetric inequality for varifolds of finite mass and finite total first variation, allowing an arbitrary singular part. The distributional mean curvature is assumed only integrable. The result applies to integral varifolds and, more generally, to rectifiable varifolds with a positive lower density bound and weight measure concentrated on countably many $C^2$ submanifolds. The proof uses optimal transport and a locality theorem for the tangential Laplacian of an ambient convex function. The isoperimetric constant is sharp in codimensions one and two.
