UC Santa Cruz

In this thesis, we introduce a new type of geometric flow of Riemannian metrics based on Bach tensor and the gradient of Weyl curvature functional and coupled with an elliptic equation which preserves a constant scalar curvature along with this flow. We named this flow by conformal Bach flow. In this thesis, we first establish the short-time existence of the conformal Bach flow and its regularity. After that, some evolution equations of curvature tensor along this flow are derived and we use them to obtain the $L^2$ estimates of the curvature tensors. After that, these estimates help us characterize the finite-time singularity. We also prove a compactness theorem for a sequence of solutions with uniformly bounded curvature norms. Finally, some singularity model is investigated.