A Wasserstein Inequality and Minimal Green Energy on Compact Manifolds

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.