Ricci flows for non-smooth spaces, monotonic quantities, and rigidity

The focus of this project is to develop synthetic notions of Ricci flow for non-smooth spaces. In the general framework of time evolutions of metric measure spaces we will use ideas of geometric analysis and optimal transport to give several synthetic characterizations of Ricci flow and analyze the behavior of such flows.A second major goal will be to exhibit functionals that behave monotonically under this notion of Ricci flow and to establish rigidity results characterizing extremal evolutions. In particular, we will investigate natural non-smooth analogues of Perelman's W entropy and L length and we will explore their relation to generalized notions of Ricci solitons.Finally, we aim to establish sharp geometric and analytic comparison results for time-dependent metric measure spaces under Ricci flow which extend to a dynamic setting the powerful results developed recently for static spaces with synthetic lower Ricci bounds.

该项目的重点是为非平滑空间开发RICCI流的合成概念。在公制度量空间的时间演变的一般框架中，我们将使用几何分析和最佳传输的想法来提供几种ricci流的合成特征，并分析此类流动的行为。第二个主要目标是展览功能，这些功能在ricci流的概念上单调性均低于Ricci流动和建立刚性的极端效果。特别是，我们将研究Perelman的W熵和L长度的自然非平滑类似物，我们将探索它们与Ricci孤子的广义概念的关系。在本文中，我们的目的是建立在RICCI流动下为稳定的稳定范围而稳定的稳定范围的稳定度量的时间依赖度度量空间的尖锐的几何和分析结果，从而使稳定的结果与稳定的结果相关。