Curvature flows on four manifolds with boundary

Abstract

Given a compact four dimensional smooth Riemannian manifold (M, g) with smooth boundary, we consider the evolution equation by Q-curvature in the interior keeping the T-curvature and the mean curvature to be zero and the evolution equation by T-curvature at the boundary with the condition that the Q-curvature and the mean curvature vanish. Using integral method, we prove global existence and convergence for the Q-curvature flow (resp T-curvature flow) to smooth metric conformal to g of prescribed Q-curvature (resp T-curvature) under conformally invariant assumptions.