Anton Thalmaier

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.

We prove a completely new integral criterion for the existence and completeness of the wave operators $W_{\pm }(-{\Delta }_h,-{\Delta }_g,I_{g,h})$ corresponding to the (unique self-adjoint realizations of) the Laplace-Beltrami operators $-{\Delta }_j$, $j=1,2$, that are induced by two quasi-isometric complete Riemannian metrics $g$ and $h$ on an open manifold $M$. In particular, this result provides a criterion for the absolutely continuous spectra of $-{\Delta }_g$ and $-{\Delta }_h$ to coincide. Our proof relies on estimates that are obtained using a probabilistic Bismut type formula for the gradient of a heat semigroup. Unlike all previous results, our integral criterion only requires some lower control on the Ricci curvatures and some upper control on the heat kernels, but no control at all on the injectivity radii. As a consequence, we obtain a stability result for the absolutely continuous spectrum under a Ricci flow.