The Mean Curvature Estimates For The Level Sets Of Solutions Of The Monge-Ampere Equation In Space Forms

The convexity problem of the level sets of the solutions to elliptic partial d-ifferential equations is the important content in the study of partial differential equations. Monge-Ampere equation is the most important fully nonlinear partial differential equations. In this paper, for the fully nonlinear elliptic Monge-Ampere equation det V2;u= 1 with homogeneous Dirichlet boundary value condition, we are concerned with studing the mean curvature estimates for the level sets of the strict-ly convex solutions of the problem on the complete constant curvature Riemannian manifolds, i.e. space forms.Basic viewpoints:For the strictly convex solutions of the elliptic Monge-Ampere equation with homogeneous Dirichlet boundary value condition, we find an auxil-iary function which is related with curvature of the level sets of u and satisfies a differential inequality. By the maximum principle, we can prove the fact that the function attains its maximum on the boundary. Moreover, we can obtain the mean curvature estimates for the level sets of the strictly convex solutions of the problem.