Paper · Sep 19, 2016

An Alexandrov-Fenchel-type inequality for hypersurfaces in the sphere

Frederico Girão, Neilha M. Pinheiro

math.DGWider flowsarXiv:1609.05806
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Abstract

We find a monotone quantity along the inverse mean curvature flow and use it to prove an Alexandrov-Fenchel-type inequality for strictly convex hypersurfaces in the -dimensional sphere, .

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Submitted
Sep 19, 2016
Subjects
math.DG
Mean curvature flowOther curvature flowsInverse mean curvature flowGeometric inequalities

Its neighbourhood

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  1. Harmonic mean curvature flow and geometric inequalities

    Ben Andrews, Yingxiang Hu and Haizhong LiWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Fenchel type”, “Alexandrov Fenchel”, “type inequality”

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

    Jie WuWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Fenchel type”, “Alexandrov Fenchel”, “type inequality”

  3. Locally constrained inverse curvature flow and Alexandrov-Fenchel type inequalities in de Sitter space

    Kuicheng MaWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Fenchel type”, “Alexandrov Fenchel”, “type inequality”

  4. Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space

    Yingxiang Hu and Haizhong LiWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Alexandrov Fenchel”, “type inequality”

  5. Modified support measures and Alexandrov-Fenchel type inequalities for nonsmooth h-convex bodies in hyperbolic space

    Xinxiu ZhangWider flows

    Topics Geometric inequalities, Other curvature flows · Shared terms “Fenchel type”, “Alexandrov Fenchel”, “type inequality”

  6. Capacity inequalities and rigidity of cornered/conical manifolds

    C. Tiarlos CruzWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Alexandrov Fenchel”, “type inequality”

  7. Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space

    Xinqun Mei and Liangjun WengWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Alexandrov Fenchel”

  8. Upper bounds for the Alexandrov-Fenchel deficit via integral formulas

    Kwok-Kun Kwong and Yong WeiWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Alexandrov Fenchel”

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 in the L² sense. 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) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)

In the guidebook

  1. 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.