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Let $M$ be a smooth, compact $d-$dimensional manifold, $d \geq 3,$ withoutboundary 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 transportingDirac measures in $\left\{x_1, \dots, x_n\right\} \subset M$ to the normalizedvolume 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 theCoulomb 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 thatnormalizes the Coulomb kernel to have mean value 0. We use this to show thatminimizers of the discrete Green energy on compact manifolds have optimal rateof 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 forminimizers of the Coulomb energy on $\mathbb{S}^d$ which was recently proven byMarzo & Mas.

A Wasserstein Inequality and Minimal Green Energy on Compact Manifolds