Paper · Nov 10, 2024

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

Luca Benatti, Alessandra Pluda, Marco Pozzetta

As on arXiv

Abstract

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 $W_{loc}^{1, q}$ 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.

Comments: 51 pages. Appendix with formulas in nonlinear potential theory has been added

Mean curvature flow Other curvature flows Inverse mean curvature flow Scalar curvature Weak & singular flows Physics

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Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

12C.9
After PerelmanBook 12C · Extinction and the Proof
Singular flows, the Smale conjecture, and Bamler’s theory.
6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.
9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.

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

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