Compactness of isospectral conformal metrics on 4-manifolds

Abstract

Let a sequence of conformal Riemannian metrics $\{g_k=u_k^2g_0\}$ be isospectral to $g_0$ over a compact boundaryless smooth 4-dimension manifold $(M,g_0)$. We prove that the subsequence of conformal factors $\{u_k\}$ converges to $u$ weakly in $W^{2,p}_{loc}(M\setminus \mathcal{S})$ for some $p<2$, where $\mathcal{S}$ is a finite set of points and $u\in W^{2,p}(M,g_0)$. Moreover, if the isospectral invariant $\frac{\int_M R(g_k)dV_{g_k}}{6\sqrt{\mathrm{Vol}(M,g_k)}}$ is strictly smaller than the Yamabe constant of the standard sphere $\mathbb{S}^4$, then the subsequence of distance functions $\{d_k\}$ defined by $\{g_k\}$ uniformly converges to $d_u$ and the subsequence of metric spaces $\{(M,d_k)\}$ converges to the metric space $(M,d_u)$ in the Gromov-Hausdorff topology, where $d_u$ is the distance function defined by $u^2g_0$.