论文标题
无负曲率曲率的空间中的尖锐等等和Sobolev不平等现象
Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature
论文作者
论文摘要
通过使用最佳的质量传输理论，我们证明了$ {\ sf cd}（0，n）$度量空间的尖锐等等不等式，假设无穷大的渐近体积增长。我们的结果将最近被证明的等等范围扩展到了非平滑框架的规范空间和Riemannian歧管。在具有非负RICCI曲率的$ n $维riemannian流形的情况下，我们概述了S. Brendle（2021）的刚性结果的替代证明。作为等级不平等的应用，我们在riemannian歧管中建立了具有明显尖锐常数的Sobolev和Rayleigh-Faber-Krahn不平等现象，具有非负RICCI曲率；在这里，我们使用适当的对称技术和最佳体积非汇编属性。后者不平等中的平等案例也表明，当且仅当Riemannian歧管与欧几里得空间等均衡时，存在足够平滑的非零极函数。
By using optimal mass transport theory we prove a sharp isoperimetric inequality in ${\sf CD} (0,N)$ metric measure spaces assuming an asymptotic volume growth at infinity. Our result extends recently proven isoperimetric inequalities for normed spaces and Riemannian manifolds to a nonsmooth framework. In the case of $n$-dimensional Riemannian manifolds with nonnegative Ricci curvature, we outline an alternative proof of the rigidity result of S. Brendle (2021). As applications of the isoperimetric inequality, we establish Sobolev and Rayleigh-Faber-Krahn inequalities with explicit sharp constants in Riemannian manifolds with nonnegative Ricci curvature; here we use appropriate symmetrization techniques and optimal volume non-collapsing properties. The equality cases in the latter inequalities are also characterized by stating that sufficiently smooth, nonzero extremal functions exist if and only if the Riemannian manifold is isometric to the Euclidean space.
