Scattering theory without injectivity radius assumptions and spectral stability for the Ricci flow

TL;DR

This paper introduces a new integral criterion for the spectral analysis of Laplace-Beltrami operators on manifolds, avoiding injectivity radius assumptions, and demonstrates spectral stability under Ricci flow using probabilistic heat kernel estimates.

Contribution

It provides the first spectral stability result for the Ricci flow without relying on injectivity radius bounds, using probabilistic methods and integral criteria.

Findings

Spectral stability under Ricci flow established

Integral criterion for wave operators derived

No injectivity radius control needed

Abstract

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.