非负里奇曲率下等周不等式的刚性
Rigidities of Isoperimetric inequality under nonnegative Ricci curvature
AI总结：
本文在非负里奇曲率度量测度空间的一般框架下证明等周不等式等号仅由度量球取得，并由此导出加权黎曼流形、一般RCD空间及欧氏锥体中的新刚性结果。
摘要：
对于具有非负里奇曲率和欧氏体积增长的非紧黎曼流形，尖锐等周不等式已在一系列贡献中以越来越大的普适性和不同方法获得[arXiv:1812.05022, arXiv:2012.09490, arXiv:2009.13717, arXiv:2103.08496]，并由Balogh和Kristaly[arXiv:2012.11862]最终完成，其结论还覆盖了在Lott、Sturm和Villani的合成意义下满足非负里奇曲率条件的度量测度空间。与正里奇曲率的紧致情形形成鲜明对比的是，对于包括加权黎曼流形在内的一大类空间，文献中尚无等号情形的完整刻画。本文的目的在于解决这一问题，在[arXiv:2012.11862]相同的普适性下证明：等周不等式中的等号只能由度量球取得。每当等号成立时，该空间在测度论意义上被迫成为锥体。我们的结果适用于不同框架，并作为推论给出新的刚性结果：它将[arXiv:2009.13717]的刚性结果推广到加权黎曼流形，将[arXiv:2201.04916]的刚性结果推广到一般$\mathsf{RCD}$空间，最后还适用于欧氏情形，证明欧氏锥体的各向异性和加权等周不等式中的最优形状必然是Wulff形状。
英文摘要：
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.
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