On Isospectral compactness in conformal class for 4-manifolds

TL;DR

This paper proves that in certain 4-manifolds with positive Yamabe invariant and small Weyl curvature, the set of conformal metrics sharing the same spectrum as a near-constant scalar curvature metric is compact.

Contribution

It establishes the compactness of isospectral metrics within a conformal class under specific geometric conditions on 4-manifolds.

Findings

Set of isospectral conformal metrics is compact in $C^ abla$ topology.

Results apply to 4-manifolds with positive Yamabe invariant and small Weyl curvature.

Provides new insights into spectral geometry and conformal invariants.

Abstract

Let $(M, g_0)$ be a closed 4-manifold with positive Yamabe invariant and with $L^2$-small Weyl curvature tensor. Let $g_1 \in [g_0]$ be any metric in the conformal class of $g_0$ whose scalar curvature is $L^2$-close to a constant. We prove that the set of Riemannian metrics in the conformal class $[g_0]$ that are isospectral to $g_1$ is compact in the $C^\infty$ topology.