Paper · Jul 12, 2018

Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space

Yingxiang Hu, Haizhong Li

math.DGWider flowsarXiv:1807.04653
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Abstract

In this article, we will use inverse mean curvature flow to establish an optimal Sobolev-type inequality for hypersurfaces with nonnegative sectional curvature in . As an application, we prove the hyperbolic Alexandrov-Fenchel inequalities for hypersurfaces with nonnegative sectional curvature in : \beginalign* \int_Σ p_2k\geq ω_n-1\left[\left(\frac|Σ|{ω_n-1}\right)^\frac1k+\left(\frac|Σ|{ω_n-1}\right)^{\frac1k\fracn-1-2kn-1}\right]^k, \endalign* where is the normalized -th mean curvature. Equality holds if and only if is a geodesic sphere in . For a domain with having nonnegative sectional curvature, we prove an optimal inequality for quermassintegral in : \beginalign* W_2k+1(Ω)\geq \frac{ω_n-1}n\sum_i=0^k\fracn-1-2kn-1-2iC_k^i\left(\frac|Σ|{ω_n-1}\right)^\fracn-1-2in-1, \endalign* where is the -th quermassintegral in integral geometry. Equality holds if and only if is a geodesic sphere in . All these inequalities was previously proved by Ge, Wang and Wu under the stronger condition that is horospherical convex.

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Submitted
Jul 12, 2018
Subjects
math.DG

Comments: 20 pages, all comments are welcome!

Mean curvature flowOther curvature flowsInverse mean curvature flowGeometric 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 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.