论文标题
在非负RICCI曲率下等等不平等的刚性
Rigidities of Isoperimetric inequality under nonnegative Ricci curvature
论文作者
论文摘要
具有非阴性的ricci曲率和欧几里得体积增长的非压缩riemannian流形的尖锐等等不平等，在多种贡献中使用不同的方法提高了一般性[Arxiv：1812.05022，arxiv，arxiv：2012.09490，arxiv：2009.139.139.139.139.11.139.1179.11] Balogh和Kristaly [Arxiv：2012.11862]也涵盖了M.M.S.在Lott，Sturm和Villani的合成意义上验证非负RICCI曲率条件。与阳性RICCI曲率的紧凑型情况形成鲜明对比的是，对于包括加权riemannian歧管在内的大量空间，文献中没有完全表征平等案例。 本说明的范围是通过证明[Arxiv：2012.11862]的一般性来解决此问题，即仅通过度量球才能实现等量不平等的平等。每当发生这种情况时，从理论上讲，空间就被迫成为锥体。 我们的结果适用于产生的不同框架作为定性的新刚性结果：它扩展到加权的riemannian流动[arxiv：2009.13717]的刚性结果，它扩展到一般$ \ $ \ shtssf {rcd} $ space space space space space [arxiv：2201.04916]的刚性结果，以及eucties to euc in for euc in for euc in for euclidean the euclidean nist in n yuclidean the in Ind in t yuclidean the in the in Ind in n yuclidean the in the in for n yuclidean th欧几里得锥的各向异性和加权等数不平等必然是沃尔夫形状。
The sharp isoperimetric inequality for non-compact Riemannian manifolds with non-negative Ricci curvature and Euclidean volume growth has been obtained in increasing generality with different approaches in a number of contributions [arXiv:1812.05022, arXiv:2012.09490, arXiv:2009.13717, arXiv:2103.08496] culminated by Balogh and Kristaly [arXiv:2012.11862] covering also m.m.s.'s verifying the non-negative Ricci curvature condition in the synthetic sense of Lott, Sturm and Villani. In sharp contrast with the compact case of positive Ricci curvature, for a large class of spaces including weighted Riemannian manifolds, no complete characterisation of the equality cases is present in the literature. The scope of this note is to settle this problem by proving, in the same generality of [arXiv:2012.11862], that the equality in the isoperimetric inequality can be attained only by metric balls. Whenever this happens the space is forced, in a measure theoretic sense, to be a cone. Our result applies to different frameworks yielding as corollaries new rigidity results: it extend to weighted Riemannian manifold the rigidity results of [arXiv:2009.13717], it extend to general $\mathsf{RCD}$ spaces the rigidity results of [arXiv:2201.04916] and finally applies also to the Euclidean setting by proving that that optimisers in the anisotropic and weighted isoperimetric inequality for Euclidean cones are necessarily the Wulff shapes.
