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

这个项目的重点是发展非光滑空间的Ricci流的合成概念。在度量度量空间的时间演化的一般框架下，我们将利用几何分析和最优传输的思想给出Ricci流的几个综合刻画，并分析这种流的行为。第二个主要目标是展示在这种Ricci流概念下表现单调行为的泛函，并建立刻画极值演化的刚性结果。特别地，我们将研究Perelman的W熵和L长度的自然非光滑类似，并探索它们与广义Ricci孤子概念的关系。最后，我们的目标是建立含时度量空间在Ricci流下的几何和解析比较结果，这些结果推广到动态环境中，这些结果是最近在具有合成Ricci下界的静态空间中得到的强有力的结果。

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.