AI 中文总结
该研究在Ricci曲率有下界的非光滑空间中引入函数水平集的平均曲率概念，定义Willmore泛函并证明其满足尖锐Willmore不等式，还将尖锐等容容量不等式推广到非光滑情形。
AI 中文摘要
本研究的目标是在Ricci曲率有下界的非光滑空间中，为函数的水平集引入平均曲率的概念，并证明其满足尖锐几何不等式。更确切地说，我们在Sobolev函数上定义了合适的Willmore泛函$\boldsymbol{\text{W}}$，其有限性定义域在任意$1\boldsymbol{\text{≤}}p\boldsymbol{<}\boldsymbol{\text{∞}}$的$L^p$空间中是稠密的。对于任何具有有限Willmore能量的函数，我们证明其几乎所有水平集都存在平均曲率向量，满足关于切散度的自然分部积分公式。作为主要应用，我们证明在具有欧氏体积增长的$\boldsymbol{\text{RCD}}(0,N)$空间中，静电势的几乎每个水平集都具有上述意义下的平均曲率向量。此外，我们证明该向量满足与光滑情形相同的尖锐Willmore不等式，同时给出刚性与近刚性结论。最后，作为技术工具，我们将尖锐等容容量不等式推广到非光滑情形。
英文摘要
The goal of this work is to introduce a notion of mean curvature for level sets of functions in non-smooth spaces with Ricci curvature bounded below, and to prove that it satisfies sharp geometric inequalities. More precisely, we define a suitable Willmore functional $\mathcal{W}$ on Sobolev functions, whose domain of finiteness is dense in $L^p$ for any $1\le p<\infty$. For any function with finite Willmore energy, we show that almost all of its level sets admit a mean curvature vector satisfying the natural integration by parts formula with respect to the tangential divergence. As a main application, we show that in ${\rm RCD}(0,N)$ spaces with Euclidean volume growth, almost every level set of the electrostatic potential possesses a mean curvature vector in the above sense. Furthermore, we prove that this vector satisfies the same sharp Willmore inequality as in the smooth setting, alongside rigidity and almost-rigidity statements. Finally, as a technical tool, we generalize the sharp isocapacitary inequality to the non-smooth setting.
Comments30 pages. Comments welcome
