Conformal Curvature Flows on Compact Manifold of Negative Yamabe Constant

Abstract. We study some conformal curvature flows related to prescribed curvature problems on a smooth compact Riemannian manifold (M, g0) with or without boundary, which is of negative (generalized) Yamabe constant, including scalar curvature flow and conformal mean curvature flow. Using such flows, we show that there exists a unique conformal metric of g0 such that its scalar curvature in the interior or mean curvature curvature on the boundary is equal to any prescribed negative smooth function, which partially recovers the results of Kazdan-Warner, Aubin and Escobar. We also study the soliton to some Yamabe-type flow on a compact manifold with smooth boundary.