Endpoint capacity at strictly stable minimal boundaries:Gaussian layers and Schwarzschild deficits

Let $(M^n,\mathfrak{g})$, $3\le n\le 7$, be a one-ended asymptotically flat manifold whose boundary $\mathrm{\Sigma }$ is a connected, strictly stable, outermost minimal hypersurface. As $p\downarrow 1$, the $p$-capacity of $\mathrm{\Sigma }$ converges to its area, by a theorem of Fogagnolo and Mazzieri. We establish a higher-order asymptotic expansion governed by a single function on $\mathrm{\Sigma }$: the positive solution $g$ of the reciprocal Jacobi equation $Jg=g^{-1}$. After a universal term, the first correction is determined by the average of $\log g$, and the capacitary potential drops from one to zero across a Gaussian boundary layer of width $g\sqrt{p-1}$; under the normalized energy measure, the logarithm of the gradient has limiting variance $\frac{1}{2}$ plus the variance of $\log g$. In dimension three, assuming in addition that $R_{\mathfrak{g}}+\mathrm{\mid }\mathrm{I}\mathrm{I}{\mathrm{\mid }}^2\ge 0$ along $\mathrm{\Sigma }$, we identify the capacity deficit of $\mathrm{\Sigma }$ relative to the Schwarzschild horizon of the same area at order $p-1$: it is nonnegative, vanishes exactly when the induced metric of $\mathrm{\Sigma }$ is round and $R_{\mathfrak{g}}+\mathrm{\mid }\mathrm{I}\mathrm{I}{\mathrm{\mid }}^2\equiv 0$, and controls the deviation of $g$ from a constant through a relative-entropy identity.