On compactness of isospectral conformal metrics of \(4\)-sphere
Let \((S^4, g_0)\) be the 4-dimensional sphere with the standard metric. Let \(g= u^2 g_0\) be a conformally equivalent metric. The author shows that if the scalar curvature of \(g\) is near constant in the \(L^2\) sense, then the set of conformal metrics which are isospectral to \(g\) is compact in the \(C^\infty\) topology modulo gauge equivalence. The conformal group is noncompact, so it is necessary to work modulo gauge equivalence. The author notes that the condition that the scalar curvature of \(g\) is near constant does not imply that \(g\) is \(C^0\) close to the standard metric modulo gauge equivalence. This paper generalizes work of Osgood, Phillips, and Sarnak in dimension 2 and of Brooks, Chang, Perry, and Yang in dimension 3.
scientific article; zbMATH DE number 856920