Critical Riemannian metrics

• Main Author: Chang, Shun-Cheng
• Other Authors: Gao, L. Zhiyong
• Format: Others
• Language: English
• Published: 2009
• Subjects: Mathematics
• Online Access: http://hdl.handle.net/1911/16328

Let $(M,g)$ be a compact oriented n-dimensional smooth Riemannian manifold. Consider the following quadratic Riemannian functional$$SR(g) = \int\sb{M}\ \vert R\sb{ijkl}(g)\vert \sp{2}d\mu$$which is homogeneous of degree ${n\over2}-2,$ where $R\sb{ijkl}$(g) is the curvature tensors of $(M,g)$ and $d\mu$ 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$SR\sb{g}$ = 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 $i\sb{0} >$ 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 $C\sp{\infty}$-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.