Paper · Apr 12, 2024

On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space

Brian Harvie

math.DGWider flowsarXiv:2404.08410v4
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Abstract

We prove that a proper weak solution to inverse mean curvature flow in , , is smooth and star-shaped by the time \beginequation* T= (n-1) \log \left( \frac{sinh \left( r_+ \right)}{ sinh \left( r_- \right)} \right), \endequation* where and are the geodesic out-radius and in-radius of the initial domain . The argument is inspired by the Alexandrov reflection method for extrinsic curvature flows in due to Chow-Gulliver and uses a result of Li-Wei. In addition to this, our methods establish expanding spheres as the only proper weak IMCF on in all dimensions. As applications, we extend the Minkowski inequalities of Brendle-Hung-Wang and De Lima-Girao to outer-minimizing domains in dimensions . From this, we also extend a Penrose-type inequality to balanced asymptotically hyperbolic graphs over the exteriors of outer-minimizing domains of in these dimensions.

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Submitted
Apr 12, 2024
Last revised
Jul 6, 2026 (v4)
Subjects
math.DG, math.AP
Journal
Calculus of Variations and Partial Differential Equations 65, 212 (2026)
DOI
10.1007/s00526-026-03384-4

Comments: Extended versions of the Minkowski inequalities to arbtirary domains in hyperbolic space

Mean curvature flowOther curvature flowsInverse mean curvature flowWeak & singular flowsBoundaries & non-compactGeometric inequalities

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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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 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.