Conference on Geometrical Analysis

초록

Bach-flat metric was introduced in the study of a conformally invariant gravitational theory and has played important roles in general relativity and geometry. This metric is the most natural generalization of an Einstein metric. Important examples of Bach-flat metrics are Einstein metrics, conformally flat metrics and self-dual Einstein metrics. Einstein metrics and Bach-flat metrics share many important properties. When the curvature of a given Einstein metric $(M,g)$ is sufficiently close to that of the constant curvature space, in $L_{\frac{n}{2}}$ sense, it is known that $(M,g)$ is isometric to a quotient of the constant curvature space. In this talk, we present rigidity phenomena on noncompact complete Bach-flat manifolds and the warped product construction of Bach-flat metrics.