Compactness of conformal metrics with integral bounds on Ricci curvature

Conghan Dong and Yuxiang Li

Vol. 316 (2022), No. 1, 65–79

DOI: 10.2140/pjm.2022.316.65

Abstract

Let $(M,g)$ be a closed Riemannian manifold with dimension $n>2$, and

$$\{g_k=u_k^{\frac{4}{(n-2)}}g\}$$

be a noncollapsing conformal metric sequence with fixed volume. We prove that $\{\log <!--FUNCTION\; APPLICATION-->u_k\}$ is compact in $C^{0,\alpha }(M)$ if $\parallel \mathrm{Ric}<!--FUNCTION\; APPLICATION-->(g_k){\parallel }_{L^p(M,g_k)}$ is bounded, where $p>\frac{n}{2}$.

Keywords

conformal metric, integral Ricci curvature, blow-up

Mathematical Subject Classification

Primary: 53C21, 58J05

Milestones

Received: 2 April 2020

Revised: 12 November 2021

Accepted: 20 November 2021

Published: 26 February 2022