Compactness Of Isospectral Conformal Metrics On 4-manifolds

Two Riemannian metrics on a closed manifold are said to be isospectral if their associated Laplace-Beltrami operators have the identical spectrum.According to the expansion of the heat kernel,we can get spectral invariants which give information about curvatures,then we can study the compactness of isospectral Riemannian metrics.In this thesis,we prove that the compactness of metric spaces induced by 4 dimensional isospectral conformal Riemannian metrics in the Gromov-Hausdorff topology sense.Let a sequence of conformal Riemannian metrics {gk=uk2g0} be isospectral to g0 over a compact boundaryless smooth 4-dimensional manifold(M,g0),up to a subsequence,conformal factors {uk} converges to u weakly in Wloc2,p(M\S)for some p<2,where S is a finite set of points and u∈W2,p(M,g0).Moreover,up to subsequence,{dk}defined by {gk} uniformly converges to du over any compact subset in M\S,where du is defined by gu=u2g0.If we construct bubble trees at singular points,the sequence of metric spaces(M,dk)defined by gk converges to a connected metric space(M∞,d∞)in the Gromov-Hausdorff topology sense and we can get the structure of the limit space.Furthermore,(?)Vol(M,gk)=H4(M∞,d∞),where H4 is the 4 dimensional Hausdorff measure defined by d∞.Moreover,if the isospectral invariant(?)is strictly smaller than the Yamabe constant of the standard sphere S4,then the subsequence of distance functions {dk} defined by ｛gk} uniformly converges to du and the subsequence of metric spaces {(M,dk)}converges to the metric space(M,du)in the Gromov-Hausdorff topology,where du is the distance function defined by u2g0.