Paper · Mar 24, 2024

Positive mass and isoperimetry for continuous metrics with nonnegative scalar curvature

Gioacchino Antonelli, Mattia Fogagnolo, Stefano Nardulli, Marco Pozzetta

math.DGWider flowsarXiv:2403.15972v3
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Abstract

This paper deals with quasi-local isoperimetric versions of the positive mass theorem on -manifolds endowed with continuous complete metrics having nonnegative scalar curvature in a suitable weak sense. As a corollary, we derive existence results for isoperimetric sets in such low regularity setting. Our main tool is a new local version of the weak inverse mean curvature flow enjoying -stable quantitative estimates.

Comments: 34 pages, 1 figure. Accepted version. To appear in Annales de l'Institut Henri Poincaré C

Mean curvature flowOther curvature flowsInverse mean curvature flowScalar curvatureGeometric inequalitiesPhysics

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  5. A continuous Positive Mass Theorem for perturbations of Euclidean space

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  6. The asymptotic behaviour of -capacitary potentials in Asymptotically Conical manifolds

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    Topics Inverse mean curvature flow, Other curvature flows · Same author Mattia Fogagnolo · Shared terms “weak inverse”

  7. A Positive Mass Type Theorem for a Singular Toroidal Static Model

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  8. Monotone Quantities for -Harmonic functions and the Sharp -Penrose inequality

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Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

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

  1. 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.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.