Elliptic PDEs on compact Ricci limit spaces and applications

TL;DR

This paper investigates elliptic partial differential equations on limit spaces of Riemannian manifolds with Ricci curvature bounds, establishing continuity of key geometric quantities and advancing differential calculus on these spaces.

Contribution

It introduces new continuity results for solutions and spectral invariants of elliptic PDEs on Ricci limit spaces, enabling further geometric analysis.

Findings

Continuity of solutions to Poisson's equations

Continuity of eigenvalues of Schrödinger operators and Hodge Laplacian

Application to second-order differential calculus on limit spaces

Abstract

In this paper we study elliptic PDEs on compact Gromov-Hausdorff limit spaces of Riemannian manifolds with lower Ricci curvature bounds. In particular we establish continuities of geometric quantities, which include solutions of Poisson's equations, eigenvalues of Schrodinger operators, generalized Yamabe constants and eigenvalues of the Hodge Laplacian, with respect to the Gromov-Hausdorff topology. We apply these to the study of second-order differential calculus on such limit spaces.