# A Wasserstein Inequality and Minimal Green Energy on Compact Manifolds

## TL;DR

This paper establishes a Wasserstein inequality relating the transportation cost of discrete measures to the Green's function on compact manifolds, leading to optimal convergence rates for minimal Green and Coulomb energy configurations.

## Contribution

It introduces a new Wasserstein bound involving Green's functions and demonstrates optimal convergence rates for energy minimizers on manifolds.

## Key findings

- Wasserstein bounds are derived using Green's functions on manifolds.
- Minimizers of Green energy converge at the optimal rate of n^{-1/d}.
- Results extend to Coulomb energy minimizers on spheres.

## Abstract

Let $M$ be a smooth, compact $d-$dimensional manifold, $d \geq 3,$ without boundary and let $G: M \times M \rightarrow \mathbb{R} \cup \left\{\infty\right\}$ denote the Green's function of the Laplacian $-\Delta$ (normalized to have mean value 0). We prove a bound on the cost of transporting Dirac measures in $\left\{x_1, \dots, x_n\right\} \subset M$ to the normalized volume measure $dx$ in terms of the Green's function of the Laplacian $$ W_2\left( \frac{1}{n} \sum_{k=1}^{n}{\delta_{x_k}}, dx\right) \lesssim_M \frac{1}{n^{1/d}} + \frac{1}{n} \left| \sum_{k, \ell=1 \atop k \neq \ell}^{n}G(x_k, x_{\ell})\right|^{1/2}.$$ We obtain the same result for the Coulomb kernel $G(x,y) = 1/\|x-y\|^{d-2}$ on the sphere $\mathbb{S}^d$, for $d \geq 3$, where we show that $$ W_2\left(\frac{1}{n} \sum_{k=1}^{n}{ \delta_{x_k}}, dx\right) \lesssim \frac{1}{n^{1/d}} + \frac{1}{n} \left| \sum_{k, \ell=1 \atop k \neq \ell}^{n}{\left(\frac{1}{\|x_k - x_{\ell}\|^{d-2}} - c_d \right)} \right|^{\frac{1}{2}},$$ where $c_d$ is the constant that normalizes the Coulomb kernel to have mean value 0. We use this to show that minimizers of the discrete Green energy on compact manifolds have optimal rate of convergence $W_2\left( \frac{1}{n} \sum_{k=1}^{n}{\delta_{x_k}}, dx\right) \lesssim n^{-1/d}$. The second inequality implies the same result for minimizers of the Coulomb energy on $\mathbb{S}^d$ which was recently proven by Marzo & Mas.