In section 1, we study the prescribed curvature flow on the compact surfaces with negative Euler characteristics. In particular, we recover the result of Kazdan and Warner using the prescribed curvature flow by proving that any negative function on a surface with negative Euler characteristic can be realized as the Gaussian curvature of some metric. Similar result is obtained for surfaces with smooth boundary. More precisely, we prove that given any negative function on the boundary of a surface with negative Euler characteristic, it can be realized as the geodesic curvature of some metric. In section 2, we study the Q-curvature flow on the standard sphere Sn and prove that the flow converges exponentially to the metric of constant sectional curvature for all initial data. In section 3, we prove the long time existence of the CR Yamabe flow on the compact strictly pseudoconvex CR manifold with positive CR invariant. We also prove the convergence of the CR Yamabe flow on the sphere by proving that: The contact form which is CR equivalent to the standard contact form on the sphere converges exponentially to a contact form of constant pseudo-Hermitian sectional curvature. We also show that the eigenvalues of some geometric operators are non-decreasing under the unnormalized CR Yamabe flow provided that the pseudo-Hermitian scalar curvature satisfies certain conditions.
