Curvature Estimate For The Level Sets Of The Solution Of The Monge-Ampere Equation In 2-Dimensional Riemannian Manifolds

In this paper, we mainly concern the Dirichlet problem of Monge-Ampere e-quation in the 2-dimensional Riemannian Manifolds. We give the auxilary function which can attain its maximum on the boundary ??Moreover, for x € ?\?', we obtain the upper bounded estimate for the curva-ture of the level lines for the solution to the Monge-Ampere equation in Riemannian manifolds where ?''={x??|u(x)<c,c ?(min?u,0) is a constant}, k is the curvature of the level lines at a point.