Paper · Apr 21, 2024

A Minkowski type inequality in warped cylinders

Shujing Pan, Bo Yang

math.DGWider flowsarXiv:2404.13670v3
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Abstract

We prove a Minkowski type inequality for weakly mean convex and star-shaped hypersurfaces in warped cylinders which are asymptotically flat or hyperbolic. In particular, we show that this sharp inequality holds for outward minimizing hypersurfaces in the Schwarzschild manifold or the hyperbolic space using the weak solution of the inverse mean curvature flow.

Comments: 26 pages. All comments are welcome

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

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    Brian HarvieWider flows PDF

    Topics Geometric inequalities, Inverse mean curvature flow, Weak & singular flows · Shared terms “Minkowski type”, “type inequality”, “hyperbolic space”

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    Shujing Pan and Bo YangWider flows PDF

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Same authors Shujing Pan, Bo Yang

  3. The weighted geometric inequalities for static convex domains in static rotationally symmetric spaces

    Shujing Pan and Bo YangWider flows PDF

    Topics Geometric inequalities, Other curvature flows · Same authors Shujing Pan, Bo Yang · Shared terms “hyperbolic space”

  4. New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms

    Jie WuWider flows PDF

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “type inequality”, “Minkowski type”, “hyperbolic space”

  5. Sharp Minkowski-Type Inequality in Cartan-Hadamard 3-Spaces

    Fang HongWider flows PDF

    Topics Geometric inequalities · Shared terms “Minkowski type”, “type inequality”

  6. Michael-Simon type inequalities in hyperbolic space via Brendle-Guan-Li's flows

    Jingshi Cui and Peibiao ZhaoWider flows PDF

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “type inequality”, “Minkowski type”, “hyperbolic space”

  7. A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space

    Yingxiang Hu, Yong Wei, Bo Yang et al.Wider flows PDF

    Topics Geometric inequalities, Inverse mean curvature flow, Boundaries & non-compact · Same author Bo Yang

  8. Sharp -Capacity Estimates via Quermassintegrals in Hyperbolic Space

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    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “hyperbolic space”, “star shaped”, “mean convex”

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.