We study the asymptotic behavior of the volume preserving mean curvature and the Mullins–Sekerka flat flow in three dimensional space. Motivated by this, we establish a 3D sharp quantitative version of the Alexandrov inequality for C2-regular sets with a perimeter bound.

We show an improvement of Bray sharp mass-capacity inequality and Bray-Miao sharp upper bound of the capacity of the boundary in terms of its area, for three-dimensional, complete, one-ended asymptotically flat manifolds with compact, connected boundary and with nonnegative scalar curvature, under appropriate assumptions on the topology and on the mean curvature of the boundary. Our arguments relies on two monotonicity formulas holding along level sets of a suitable harmonic potential, associated to the boundary of the manifold. This work is an expansion of the results contained in the Ph.D. thesis [F. Oronzio, ADM mass and linear potential theory, Ph.D. thesis, Universita degli studi di Napoli Federico II (2022)] of the author.