Some Prescribing Curvature Problems On Manifolds

The prescribing curvature problems on manifolds in conformal geometry often mean that if there exists a conformal metric in the conformal class of a given Rieman-nian metric, such that the curvature with respect to it is the prescribed functions. For example, the classical Yamabe problem is to study if there is a metric conformal to the background metric of a Riemannian manifold, such that its scalar curvature equals to a constant. More generally, the prescribing k-curvature problem is the pre-scribing curvature problem on the k-curvature defined by the elementary symmetric function on the eigenvalues of Schouten or modified Schouten tensors for general of a manifold. Call it a prescribing k-Yamabe problem if the prescribed function is a constant. In fact, the k-Yamabe problem is just the initial Yamabe problem when k=1. There are many results on the k-Yamabe problems of Schouten tensors, see. e.g.[CGY1, CGY2, Gl, GW1, GV2, HS1, J, LjS, LL1, STW, V2], etc..There are also many results on the κ-Yamabe problems of the modified Schouten curvature. Let (Mn,g),n≥3be a smooth Riemannian manifold, ATg(τ∈R) be its modified Schouten tensor. For the compact manifolds without boundary, Gursky-Viaclovsky [GV1] proved that the prescribing κ-curvature problem has unique so-lution for any smooth negative function when T<1and the κ-curvature of ATg is negative. Li-Sheng [LjS] also obtained this result by a parabolic approach. For T> n-1and the κ-curvature of ATg is positive, Sheng-Zhang [ShZ] proved the pre-scribing κ-curvature problem has unique solution for any smooth positive function on a compact manifold without boundary; Li-Sheng [LqS] studied the correspond-ing Dirichlet problem on a compact manifold with boundary. He-Sheng[HS3] got the local estimates for a more general fully nonlinear elliptic equations about the modified Schouten tensor for T<1and T> n-1, respectively, on a compact manifold with boundary. In this paper, we study the Neumann problems corre-sponding to [GV1] and [ShZ] by a parabolic approach, that is, we prove that the prescribing k-Yamabe problem has a unique solution on a compact manifold with totally geodesic boundary for both T<1, the κ-curvature of ATg is negative and T> n-1, the κ-curvature of ATg is positive; and the solution metric preserves the boundary totally geodesic. Moreover, using a elliptic approach, we prove that if the mean curvature of a manifold with umblic boundary with respect to its inner normal vector is nonnegative, there exist conformal metric such that the boundary becomes totally geodesic and its modified Schouten tensor satisfies a class of more general elliptic equations for T> n-1and the κ-curvature of ATg positive; and we have the same result for T<1and the κ-curvature of ATg negative.In order to study the diffusion operator on Riemannian manifold, Bakry-Emery [BE1] introduced the Bakry-Emery Ricci tensor in1985. It may be regarded as a generalization of Ricci tensor. In the recent30years, many authors shown that a lot of geometric and topological results on Ricci tensors are can be generalized to Bakry-Emery Ricci tensors, such as the Gromov-Hausdorff convergence theorem of measured space, volume comparison theorems, the splitting theorem, the rigid-ity theorem etc., see e.g.[BE2, Lo, SZ, WW]. Bakry-Emery Ricci tensor occurs naturally in the Ricci flow, measured Gromov-Hausdorff spaces and many different subjects [Lo, LV, St, P]. It displayed the fundamental importance in geometric analysis, diffusion procedure and the global Riemannian geometry.In this paper, we also call the function defined by the elementary symmetric function on the eigenvalues of Bakry-Emery Ricci tensor a κ-curvature of it. We prove that if its initial κ-curvature is negative, then its prescribing κ-curvature prob-lem has unique solution on a compact manifold without boundary. By considering the corresponding Dirichlet problem to the prescribing κ-curvature problem, we get the complete solution on a compact manifold with boundary. When Bakry-Emery Ricci tensor reduce to Ricci tensor, our results also right for Ricci tensor.