Let M be a smooth, compact d−dimensional manifold, $d\ge 3$, without boundary and let $G:M\times M\to \mathbb{R}\cup \{\infty \}$ denote the Green's function of the Laplacian −Δ (normalized to have mean value 0). We prove a bound on the cost of transporting Dirac measures in $\{x_1,\dots ,x_n\}\subset M$ to the normalized volume measure dx in terms of the Green's function of the Laplacian$W_2(\frac{1}{n}\sum _{k=1}^n{\delta }_{x_k},dx){\lesssim }_M\frac{1}{n^{1/d}}+\frac{1}{n}{|\sum _{\genfrac{}{}{0ex}{}{k,\ell =1}{k\ne \ell }}^{n}G(x_k,x_{\ell })|}^{1/2}.$ We obtain the same result for the Coulomb kernel $G(x,y)=1/{\Vert x-y\Vert }^{d-2}$ on the sphere ${\mathbb{S}}^d$, for $d\ge 3$, where we show that$W_2(\frac{1}{n}\sum _{k=1}^n{\delta }_{x_k},dx)\lesssim \frac{1}{n^{1/d}}+\frac{1}{n}{\left|\sum _{\genfrac{}{}{0ex}{}{k,\ell =1}{k\ne \ell }}^n(\frac{1}{{\Vert x_k-x_{\ell }\Vert }^{d-2}}-c_d)\right|}^{\frac{1}{2}},$ 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 $W_2(\frac{1}{n}{\sum }_{k=1}^n{\delta }_{x_k},dx)\lesssim n^{-1/d}$. The second inequality implies the same result for minimizers of the Coulomb energy on ${\mathbb{S}}^d$ which was recently proven by Marzo & Mas.

令M为光滑的紧d维流形，$d\ge 3$，无国界，让 $G：\mathrm{中号}\times \mathrm{中号}\to \mathbb{[R}\cup \{\infty \}$表示拉普拉斯算子-Δ的格林函数（归一化为平均值0）。我们证明了运输Dirac措施的成本是有限的$\{X_{\mathrm{1个}}，\dots ，X_{ñ}\}\subset \mathrm{中号}$根据拉普拉斯算子的格林函数对归一化体积度量值dx进行计算${\mathrm{w\; ^}}_{\mathrm{2个}}（\frac{\mathrm{1个}}{ñ}\sum _{ķ=\mathrm{1个}}^{ñ}{\delta }_{X_{ķ}}，dX）{\lesssim }_{\mathrm{中号}}\frac{\mathrm{1个}}{{ñ}^{\mathrm{1个}/d}}+\frac{\mathrm{1个}}{ñ}{|\sum _{\genfrac{}{}{0ex}{}{ķ，\ell =\mathrm{1个}}{ķ\ne \ell }}^{ñ}G（X_{ķ}，X_{\ell }）|}^{\mathrm{1个}/\mathrm{2个}}。$ 对于库仑内核，我们获得了相同的结果 $G（X，ÿ）=\mathrm{1个}/{\Vert X-ÿ\Vert }^{d-\mathrm{2个}}$ 在球上 ${\mathbb{小号}}^d$， 为了 $d\ge 3$，在那里我们表明${\mathrm{w\; ^}}_{\mathrm{2个}}（\frac{\mathrm{1个}}{ñ}\sum _{ķ=\mathrm{1个}}^{ñ}{\delta }_{X_{ķ}}，dX）\lesssim \frac{\mathrm{1个}}{{ñ}^{\mathrm{1个}/d}}+\frac{\mathrm{1个}}{ñ}{\left|\sum _{\genfrac{}{}{0ex}{}{ķ，\ell =\mathrm{1个}}{ķ\ne \ell }}^{ñ}（\frac{\mathrm{1个}}{{\Vert X_{ķ}-X_{\ell }\Vert }^{d-\mathrm{2个}}}-C_d）\right|}^{\frac{\mathrm{1个}}{\mathrm{2个}}}，$ 在哪里 $C_d$ 是将库仑核归一化为均值0的常数。我们用它来证明紧凑流形上离散绿色能量的极小值具有最佳收敛速度 ${\mathrm{w\; ^}}_{\mathrm{2个}}（\frac{\mathrm{1个}}{ñ}{\sum }_{ķ=\mathrm{1个}}^{ñ}{\delta }_{X_{ķ}}，dX）\lesssim {ñ}^{-\mathrm{1个}/d}$。第二个不等式意味着最小化库仑能量的结果相同。${\mathbb{小号}}^d$ Marzo＆Mas最近证明了这一点。