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

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