论文标题
Sobolev不平等的稳定性在RICCI曲率下限的Riemannian歧管上的稳定性
Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds
论文作者
论文摘要
我们研究了两类Sobolev不平等现象的定性稳定性。在阳性曲率的情况下，我们证明尖锐的Sobolev不平等的几乎极端功能接近圆形球体的极端功能。在非负RICCI曲率和欧几里得体积增长的情况下，与欧几里得Sobolev不平等中的极端功能相比，我们显示出类似的结果。作为应用程序，我们推断出最小化Yamabe指标的稳定性结果。这些论点依赖于在不同空间上的广义狮子的浓度紧凑性以及奇异空间上Sobolev不平等的僵化结果。
We study the qualitative stability of two classes of Sobolev inequalities on Riemannian manifolds. In the case of positive Ricci curvature, we prove that an almost extremal function for the sharp Sobolev inequality is close to an extremal function of the round sphere. In the setting of non-negative Ricci curvature and Euclidean volume growth, we show an analogous result in comparison with the extremal functions in the Euclidean Sobolev inequality. As an application, we deduce a stability result for minimizing Yamabe metrics. The arguments rely on a generalized Lions' concentration compactness on varying spaces and on rigidity results of Sobolev inequalities on singular spaces.
