论文标题
无尺寸的harnack不平等现象，用于共轭加热方程及其在几何流量上的应用
Dimension-free Harnack inequalities for conjugate heat equations and their applications to geometric flows
论文作者
论文摘要
让$ m $成为一个可区分的流形，它具有完整的Riemannian指标$ g（t）$在几何流量下在时间间隔$ [0，t [$的几何流量下演变而来的。在本文中，我们为$ m $的相应共轭半群的衍生物提供了一个概率表示，该分子是由Schrödinger类型运营商生成的。在这个衍生产品公式的帮助下，我们在不断发展的riemannian流形的环境中得出了基本的harnack型不平等。特别是，我们建立了无维的harnack不平等，并显示如何在移动指标的设置中使用它来实现热内核上限。此外，通过共轭半群的超扣除率，我们获得了一个规范的log-sobolev不平等。在所谓的修改后的RICCI流以及一般的几何流量的情况下，我们讨论并应用了这些结果。
Let $M$ be a differentiable manifold endowed with a family of complete Riemannian metrics $g(t)$ evolving under a geometric flow over the time interval $[0,T[$. In this article, we give a probabilistic representation for the derivative of the corresponding conjugate semigroup on $M$ which is generated by a Schrödinger type operator. With the help of this derivative formula, we derive fundamental Harnack type inequalities in the setting of evolving Riemannian manifolds. In particular, we establish a dimension-free Harnack inequality and show how it can be used to achieve heat kernel upper bounds in the setting of moving metrics. Moreover, by means of the supercontractivity of the conjugate semigroup, we obtain a family of canonical log-Sobolev inequalities. We discuss and apply these results both in the case of the so-called modified Ricci flow and in the case of general geometric flows.
