Sharp global Alexandrov estimates and entire solutions of Monge-Ampère equations

This paper continues our work [19] on sharp Alexandrov estimates. We obtain a sharp global uniform distance estimate from a convex function to the class of unimodular convex quadratic polynomials in terms of the total variation of its Monge-Ampère defect measure relative to Lebesgue measure. The estimate has an explicit optimal constant, and the inequality is strict in the regime of positive finite defect mass. In this regime we further prove asymptotic rigidity at infinity: every such convex function admits a unique quadratic asymptote with an explicit convergence rate, and satisfies a sharp affine invariant global Alexandrov estimate with equality if and only if the function solves the isolated singularity problem or the hyperplane obstacle problem. Standard subsolution methods are not well suited to this measure-theoretic setting and typically do not yield sharp constants, while the sharp Alexandrov estimates developed in our earlier work [19] play a central role here. As an application, for entire solutions of Monge-Ampère equations with multiple (possibly infinitely many) isolated singularities, we give an explicit quantitative mass-separation condition ensuring strict convexity and hence smoothness away from the set of the isolated singularities.

💡 Research Summary

The paper advances the quantitative theory of convex solutions to the Monge‑Ampère equation by establishing sharp global Alexandrov estimates that relate a convex function’s deviation from the class of unimodular quadratic polynomials to the total variation of its Monge‑Ampère defect measure.

Main Result – Sharp Global Distance Bound (Theorem 1.1).
For any entire convex function u on ℝⁿ (n ≥ 3) the infimum of the L∞‑distance between u and any quadratic polynomial of the form
 Q(x)=½ xᵀA x + b·x + c, A∈Aₙ (det A=1),
satisfies
 ‖u−Q‖{∞} ≤ 2^{−2/n} d{n,0} ωₙ^{−1/n} |μ|(ℝⁿ)^{2/n},
where μ = M_u − L is the signed Monge‑Ampère defect measure, |μ|(ℝⁿ) its total mass, ωₙ the volume of the unit ball, and
 d_{n,0}=Γ(1+1/n) Γ((n−2)/n) /