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Green--Wasserstein Inequality on Compact Surfaces

Maja Gwozdz

TL;DR

The paper addresses whether the 2D Green–Wasserstein inequality can be sharpened by removing the $\sqrt{\log n}$ factor while keeping the unrenormalized off-diagonal Green energy term. It combines a finite second-moment bound for the Green energy, via a degenerate $U$-statistic framework and a careful near-diagonal analysis, with the semi-discrete matching asymptotics of Ambrosio–Glaudo to argue by contradiction. The main finding is that no universal inequality of the proposed exact form exists on any compact connected surface, as the 2D Green energy contributes a $\log n$-type growth that cannot be suppressed. This establishes a sharp limitation on the unrenormalized energy approach in 2D and points to the necessity of renormalized energy notions or alternative forms for tight transport bounds on manifolds.

Abstract

Let $(M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary. In this note, we answer a question posed by Steinerberger: can one remove the $\sqrt{\log n}$ factor in the two-dimensional Green--Wasserstein inequality while keeping the unrenormalized off-diagonal Green term? We show that this is impossible on any compact connected surface: there is no inequality of the same form that holds uniformly over point sets with an $O(n^{-1/2})$ remainder for all $n$. We argue by contradiction and combine a second-moment estimate for the random Green energy of i.i.d. samples with the semi-discrete random matching asymptotics of Ambrosio--Glaudo.

Green--Wasserstein Inequality on Compact Surfaces

TL;DR

The paper addresses whether the 2D Green–Wasserstein inequality can be sharpened by removing the factor while keeping the unrenormalized off-diagonal Green energy term. It combines a finite second-moment bound for the Green energy, via a degenerate -statistic framework and a careful near-diagonal analysis, with the semi-discrete matching asymptotics of Ambrosio–Glaudo to argue by contradiction. The main finding is that no universal inequality of the proposed exact form exists on any compact connected surface, as the 2D Green energy contributes a -type growth that cannot be suppressed. This establishes a sharp limitation on the unrenormalized energy approach in 2D and points to the necessity of renormalized energy notions or alternative forms for tight transport bounds on manifolds.

Abstract

Let be a compact connected two-dimensional Riemannian manifold without boundary. In this note, we answer a question posed by Steinerberger: can one remove the factor in the two-dimensional Green--Wasserstein inequality while keeping the unrenormalized off-diagonal Green term? We show that this is impossible on any compact connected surface: there is no inequality of the same form that holds uniformly over point sets with an remainder for all . We argue by contradiction and combine a second-moment estimate for the random Green energy of i.i.d. samples with the semi-discrete random matching asymptotics of Ambrosio--Glaudo.
Paper Structure (3 sections, 4 theorems, 39 equations)

This paper contains 3 sections, 4 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. A second-moment bound for the Green energy
  3. Main result

Key Result

Lemma 1

Let $(M,g)$ be a compact connected $2$-dimensional Riemannian manifold without boundary, and let $G$ be the symmetric mean-zero Green function of $-\Delta$. It then holds that $G\in L^2(M\times M, dx\otimes dx)$, and so

Theorems & Definitions (13)

  • Remark 1: The diagonal
  • Definition 1: Mean-zero Green function
  • Remark 2: Local singularity
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 3: U-statistics
  • ...and 3 more