Cheeger-Gromov convergence in a conformal setting

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, ${\tilde{g}}_i=u_i^{\frac{4}{n-2}}{g}_i$. 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.

25 pages. The authors discovered a mistake in the paper. In particular, the claim of Theorem B does not hold, however Theorem A still true, and, by insistence of the first author, Theorem A will be published by the second author in a separate paper