Compactness of Conformal Metrics with \(L^p\)-Bounded \(Q\)-Curvature on Closed Smooth Riemannian Manifolds

TL;DR

This paper proves a compactness result for conformal metrics on closed Riemannian manifolds with bounded Q-curvature in L^p, positive Yamabe invariant, and a lower bound on the first Laplacian eigenvalue.

Contribution

It establishes precompactness of conformal metrics with bounded Q-curvature, volume normalization, and spectral bounds on manifolds of dimension at least five.

Findings

Metrics form a precompact set in the Hölder topology.

Bounded Q-curvature in L^p implies geometric compactness.

Spectral bounds contribute to the compactness result.

Abstract

Let \((M^n,g)\) be a smooth closed Riemannian manifold of dimension \(n \ge 5\) with positive Yamabe invariant and semi-positive \(Q\)-curvature. We establish a precompactness result in the \(C^{\alpha}\)-H\"older topologie on the space of Riemannian metrics, for some \(\alpha>0\), for the set of metrics \(\tilde{g}\) conformal to \(g\), with volume equal to that of the standard sphere \(\mathbb{S}^n\), whose \(Q\)-curvature is nonnegative and uniformly bounded in \(L^p(M,\tilde{g})\) for some \(p > \frac{n}{4}\), and whose first positive eigenvalue of the Laplace-Beltrami operator satisfies \( \lambda_1(M,\tilde{g}) \ge n + \frac{1}{\Lambda} \) for some positive constant \(\Lambda\).