Critical Riemannian metrics

[摘要] Let $(M,g)$ be a compact oriented n-dimensional smooth Riemannian manifold. Consider the following quadratic Riemannian functional$$SR(g) = intsb{M} vert Rsb{ijkl}(g)vert sp{2}dmu$$which is homogeneous of degree ${nover2}-2,$ where $Rsb{ijkl}$(g) is the curvature tensors of $(M,g)$ and $dmu$ is the volume element measured by g. A critical point of $SR(g)$ is called a critical metric on M, that is, the Ricci tensor satisfies the critical equations grad$SRsb{g}$ = 0. In particular, for a compact 4-manifold M, every Einstein metric is a critical metric for SR on M. In this thesis, we propose an extension of the compactness property for Einstein metrics to critical metrics on a compact smooth Riemannian 4-manifold M. More precisely, first we consider the subspace $G(M)$ of all critical metrics on M with the injectivity radius bounded from below by a constant $isb{0} >$ 0 and diameter bounded from above by d. Then we are able to prove that $G(M)$ is compact as a subset of moduli space of critical metrics in the $Csp{infty}$-topology (Theorem 6.1). Second, we replaced the injectivity radius lower bound by the local volume bound, then we get a compact 4-dimensional critical orbifold (Theorem 7.1). Furthermore, by using the fundamental equations of Riemannian submersions with totally geodesic fibers, we construct some critical Riemannian 4-manifolds.

[发布机构] Rice University