M-Convex Hypersurfaces with Prescribed Shifted Gaussian Curvature in Warped Product Manifolds

In this paper, we first give a notion of M -convexity, and then under suitable settings related to this superconvexity, we can obtain the existence of solutions to prescribed shifted Gaussian curvature equations in warped product manifolds of special type by the standard degree theory based on the a priori estimates for the solutions. This is to say that the existence of M -convex, closed hypersurface (which is graphic with respect to the base manifold and whose shifted Gaussian curvature satisfies some constraint) in a given warped product manifold of special type can be assured. Besides, different from prescribed Weingarten curvature problems in space forms, due to the M -convexity of hypersurfaces in the warped product manifold considered, we do not need to impose a sign condition for the radial derivative of the prescribed function in the shifted Gaussian curvature equation to prove the existence of solutions.