Sharp global Alexandrov estimates and entire solutions of Monge-Amp\`ere equations

TL;DR

This paper advances sharp global Alexandrov estimates for convex functions related to Monge-Ampère equations, providing explicit bounds, asymptotic rigidity, and applications to entire solutions with isolated singularities.

Contribution

It introduces a sharp global uniform distance estimate with an explicit optimal constant and applies it to analyze entire solutions with isolated singularities.

Findings

Established a sharp global distance estimate with an explicit constant.

Proved asymptotic rigidity and unique quadratic asymptotes for convex functions.

Provided conditions ensuring strict convexity and smoothness of solutions away from singularities.

Abstract

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-Amp\`ere 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-Amp\`ere 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.