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

Tianling Jin, Xushan Tu, Jingang Xiong

TL;DR

This work yields sharp global Alexandrov-type bounds for entire convex solutions of Monge-Ampère equations by measuring the L∞-distance to unimodular quadratic polynomials in terms of the Monge-Ampère defect mu = Mu − L. It proves an explicit, optimal constant bound, establishes asymptotic rigidity with a unique quadratic asymptote, and provides a sharp affine-invariant global estimate with equality characterized by isolated singularities or hyperplane obstacle problems. The authors also develop the measure-theoretic existence and uniqueness theory for det D^2 u = 1 + mu with finite total variation |mu| and derive a quantitative mass-separation condition ensuring strict convexity away from isolated singularities in the multi-singularity setting. Together, these results connect global geometric control with measure-theoretic Monge-Ampère equations, yielding precise regularity conclusions away from singular sets and a framework for understanding how singularities influence global behavior.

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-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.

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

TL;DR

This work yields sharp global Alexandrov-type bounds for entire convex solutions of Monge-Ampère equations by measuring the L∞-distance to unimodular quadratic polynomials in terms of the Monge-Ampère defect mu = Mu − L. It proves an explicit, optimal constant bound, establishes asymptotic rigidity with a unique quadratic asymptote, and provides a sharp affine-invariant global estimate with equality characterized by isolated singularities or hyperplane obstacle problems. The authors also develop the measure-theoretic existence and uniqueness theory for det D^2 u = 1 + mu with finite total variation |mu| and derive a quantitative mass-separation condition ensuring strict convexity away from isolated singularities in the multi-singularity setting. Together, these results connect global geometric control with measure-theoretic Monge-Ampère equations, yielding precise regularity conclusions away from singular sets and a framework for understanding how singularities influence global behavior.

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-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.
Paper Structure (4 sections, 14 theorems, 159 equations)

This paper contains 4 sections, 14 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Alexandrov estimates
  3. A quantitative decay estimate
  4. Proof of the main results

Key Result

Theorem 1.1

Suppose $n \ge 3$. Then for every entire convex function $u:\mathbb{R}^n\to\mathbb{R}$, we have Moreover, the inequality is strict whenever $0<|\mathscr{M} u-\mathcal{L}|(\mathbb{R}^n)<\infty$, and the constant $2^{-\frac{2}{n}} d_{n,0}$ is sharp for each fixed value of $|\mathscr{M} u-\mathcal{L}|(\mathbb{R}^n)$ in $(0,\infty)$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1: jin2025abp
  • Theorem 2.2
  • Theorem 2.3: jin2025abp
  • Theorem 2.4: jin2025abp
  • Lemma 3.1: jin2025abp
  • ...and 12 more