Fine properties of nonlinear potentials and a unified perspective on monotonicity formulas

We rigorously show that a large family of monotone quantities along the weak inverse mean curvature flow is the limit case of the corresponding ones along the level sets of $p$-capacitary potentials. Such monotone quantities include Willmore and Minkowski-type functionals on Riemannian manifolds with nonnegative Ricci curvature. In $3$-dimensional manifolds with nonnegative scalar curvature, we also recover the monotonicity of the Hawking mass and its nonlinear potential theoretic counterparts. This unified view is built on a refined analysis of $p$-capacitary potentials. We prove that they strongly converge in $W^{1,q}_{\mathrm{loc}}$ as $p\to 1^+$ to the inverse mean curvature flow and their level sets are curvature varifolds. Finally, we also deduce a Gauss-Bonnet-type theorem for level sets of $p$-capacitary potentials.

💡 Research Summary

The paper establishes a rigorous bridge between the weak inverse mean curvature flow (IMCF) and a broad family of monotone quantities derived from the level sets of p‑capacitary potentials. The authors consider a complete non‑compact Riemannian manifold (M,g) of dimension n≥3 and a bounded domain Ω⊂M with C^{1,1} boundary. For each p∈