Curvature estimation of the level sets of solutions of a class of elliptic partial differential equation

The curvature of the level set of elliptic partial differential equation solutions is always an important content in the study of convexity. Curvature is an important invariant of surface, which characterizes the degree of curve bending, is the important basis of differential geometry. Curvature is widely used in machining. In this paper, we study the completely nonlinear elliptic Monge-Ampère equation det D2u = eu with 0 boundary value Dirichlet condition in four-dimensional Euclidean space. It is proved that the auxiliary function obtains the maximum value at the boundary, and then the mean curvature and Gauss curvature of the level sets of the solutions of the equation are estimated quantitatively.