Curvature estimates for the level sets of solutions of the Monge-Amp\`{e}re equation $\det D^2 u=1$

TL;DR

This paper introduces new curvature functions for solutions of the Monge-Ampère equation, establishing boundary maxima and providing upper bounds for Gauss and mean curvatures of level sets.

Contribution

It develops auxiliary curvature functions that reach their maxima on the boundary and derives upper bounds for the curvatures of level sets of solutions.

Findings

Auxiliary curvature functions attain maximum on the boundary.

Upper bounds established for Gauss and mean curvatures.

Results contribute to understanding geometric properties of solutions.

Abstract

For the Monge-Amp\`{e}re equation $\det D^2 u=1$, we find new auxiliary curvature functions which attain respective maximum on the boundary. Moreover, we obtain the upper bounded estimates for the Gauss curvature and mean curvature of the level sets for the solution to this equation.