Compactness of isospectral conformal metrics and isospectral potentials on a 4-manifold
Let \((M,g_0)\) be a compact Riemannian manifold of dimension 4 with constant scalar curvature \(s_0\). Let \(g_1\) be a metric which is conformal to \(g_0\) and let \({\mathcal G}\) be the set of all metrics \(g_2\) which are conformal to \(g_0\) and isospectral to \(g_1\); i.e. \(\text{spec} (\Delta(g_1)) = \text{spec} (\Delta (g_2))\) where \(\Delta\) is the scalar Laplacian. Let \(q\) be a given potential function and let \(\text{Iso} (q)\) be the set of all potential functions \(q_1\) which are isospectral to \(q\), i.e. \(\text{spec} (\Delta(g_0)-q) = \text{spec} (\Delta (g_0)-q_1)\). Let \(C_s\) be the Sobolev constant with respect to the metric \(g_0\). Theorem: 1) Let \(s_0\leq 0\). Assume that \[ \begin{multlined} \int_Ms(g_1)d \text{vol} (g_1)\leq 6/C_s \text{vol} (M, g_1)^{1/2},\\ \int_M s(g_1)^2d \text{vol} (g_1) - \Bigl\{ \int_Ms(g_1)d \text{vol} (g_1) \Bigr\}^2/ \text{vol} (M,g_1)\leq 11^2/52^2C^2_s. \end{multlined} \] Then \({\mathcal G}\) is compact in the \(C^\infty \) topology. 2) Assume that \(\int_M\{q-\text{vol} (M)^{-1} \int_M q d \text{vol} (g_0)\}^2d \text{vol}(g_0)\leq 1/18^2C^2_s\), then \(\text{Iso}(q)\) is compact in the \(C^\infty\) topology.
scientific article; zbMATH DE number 922155
scientific article