标题： 里奇极限流和弱解 显示英文标题

标题： Ricci limit flows and weak solutions

摘要： 在本文中，我们通过Kleiner-Lott、Haslhofer-Naber和Bamler近年来提出的几种不同的处理奇点的Ricci流方法进行了调和。具体来说，我们证明了Bamler预紧性定理所提供的Ricci流的非坍塌极限，以及Kleiner-Lott的奇异Ricci流，都是Haslhofer-Naber意义上的弱解。我们还将Haslhofer-Naber的所有路径空间估计推广到非坍塌Ricci极限流的设置中。建立这些结果的关键步骤是针对布朗运动的新 hitting 估计。一个根本性的困难在于，与文献中所有先前的hitting估计形成鲜明对比的是，在Ricci流下缺乏下界热核估计。为克服这一点，我们引入了一种新的hitting估计方法，该方法通过利用时空的热核几何来弥补缺乏下界热核估计的不足。 显示英文摘要

摘要： In this paper we reconcile several different approaches to Ricci flow through singularities that have been proposed over the last few years by Kleiner-Lott, Haslhofer-Naber and Bamler. Specifically, we prove that every noncollapsed limit of Ricci flows, as provided by Bamler's precompactness theorem, as well as every singular Ricci flow from Kleiner-Lott, is a weak solution in the sense of Haslhofer-Naber. We also generalize all path-space estimates from Haslhofer-Naber to the setting of noncollapsed Ricci limit flows. The key step to establish these results is a new hitting estimate for Brownian motion. A fundamental difficulty, in stark contrast to all prior hitting estimates in the literature, is the lack of lower heat kernel bounds under Ricci flow. To overcome this, we introduce a novel approach to hitting estimates that compensates for the lack of lower heat kernel bounds by making use of the heat kernel geometry of space-time.