Area, Volume, and Capacity in Non-compact 3-Manifolds with Non-negative Scalar Curvature

Let $(M,g)$ be a 3D, complete, one-ended Riemannian manifold, with a minimal, compact, and connected boundary. We assume that M has a simple topology and that the scalar curvature of $(M,g)$ is non-negative. Moreover, we suppose that $(M,g)$ admits a $2$-capacitary potential v with v,\, ∇ v → 0 at infinity. In this note, we provide a gradient integral estimate for the level sets of the function $u=1-v$. This estimate leads to a sharp volume comparison for the sub-level sets of u and a sharp area comparison of the level sets of u. From this last comparison, it follows a sharp area–capacity inequality, originally derived by Bray and Miao in [16], thereby extending its cases of validity. This work is based on the recent paper [23] by Colding and Minicozzi. Finally, for completeness, we also show the same type of area and volume comparison, in the case where $(M,g)$ has no boundary, replacing the function u with one related to the minimal positive Green’s function. This volume comparison leads to a more geometric proof of the positive mass inequality than the one given in [7].