Cheeger-Gromov convergence in a conformal setting

TL;DR

This paper establishes conditions under which sequences of conformally related Riemannian manifolds with boundary converge smoothly, extending Cheeger-Gromov compactness results using elliptic estimates and the 'flatzoomer' method.

Contribution

It introduces a new convergence criterion for conformal satellites of manifolds with boundary, based on eigenfunctions of elliptic operators.

Findings

Sequences have smoothly Cheeger-Gromov convergent subsequences.

Conformal factors as principal eigenfunctions ensure convergence.

Provides a Cheeger-Gromov compactness result for manifolds with boundary.

Abstract

For a sequence $\{(M_i, g_i, x_i)\}$ of pointed Riemannian manifolds with boundary, the sequence $\{(M_i,\tilde g_i,x_i)\}$ is its conformal satellite if the metric $\tilde g_i$ is conformal to $g_i$, that is, $\tilde g_i=u^{\frac{4}{n-2}}_ig_i$. Assuming the manifolds $(M_i,g_i,x_i)$ have uniformly bounded geometry, we show that both sequences have smoothly Cheeger-Gromov convergent subsequences provided the conformal factors $u_i$ are principal eigenfunctions of an appropriate elliptic operator. Part of our result is a Cheeger-Gromov compactness for manifolds with boundary. We use stable versions of classical elliptic estimates and inequalities found in the recently established 'flatzoomer' method.