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

Abstract

Let $(M^n,\mathfrak g)$, $3\le n\le7$, be a one-ended asymptotically flat manifold whose boundary $Σ$ is a connected, strictly stable, outermost minimal hypersurface. As $p\downarrow1$, the $p$-capacity of $Σ$ converges to its area, by a theorem of Fogagnolo and Mazzieri. We establish a higher-order asymptotic expansion governed by a single function on $Σ$: 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 $\frac12$ plus the variance of $\log g$. In dimension three, assuming in addition that $R_{\mathfrak g}+|\mathrm{II}|^2\ge0$ along $Σ$, we identify the capacity deficit of $Σ$ relative to the Schwarzschild horizon of the same area at order $p-1$: it is nonnegative, vanishes exactly when the induced metric of $Σ$ is round and $R_{\mathfrak g}+|\mathrm{II}|^2\equiv0$, and controls the deviation of $g$ from a constant through a relative-entropy identity.