Paper · Aug 30, 2022

Stability of quermassintegral inequalities along inverse curvature flows

Caroline VanBlargan, Yi Wang

math.DGWider flowsarXiv:2208.14341
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Abstract

In this paper, we consider the stability of quermassintegral inequalities along a inverse curvature flow. We choose a special rescaling of the flow such that the -th quermassintegral is decreasing and the -th quermassintegral is preserved. Along this rescaled flow, we prove that the decreasing rate of the -th quermassintegral is faster than the Fraenkel asymmetry of the domain when approaching to the sphere. This leads to the stability inequality of quermassintegral inequalities for nearly spherical sets using the flow method.

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Submitted
Aug 30, 2022
Subjects
math.DG, math.AP

Comments: arXiv admin note: text overlap with arXiv:2201.04256

Other curvature flowsInverse mean curvature flowStability & uniquenessGeometric inequalities

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    Junfang LiWider flows

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

  2. Quermassintegral inequalities for capillary hypersurfaces in a horoball

    Jinyu Guo and Liangjun WengWider flows

    Topics Geometric inequalities, Other curvature flows · Shared terms “quermassintegral inequalities”

  3. Capillary quermassintegral inequalities in the unit ball

    Shujing Pan and Julian ScheuerWider flows

    Topics Geometric inequalities, Other curvature flows · Shared terms “quermassintegral inequalities”

  4. 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

  5. Sharp -Capacity Estimates via Quermassintegrals in Hyperbolic Space

    Xiaoshang Jin, Yao Wan and Jie XiaoWider flows

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

  6. New quermassintegral and Poincaré type inequalities for non-convex domains

    Yingxiang Hu and Mohammad N. IvakiWider flows

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

  7. Inverse curvature flows for capillary hypersurfaces in the unit ball

    Shujing Pan and Bo YangWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows

  8. Inverse Hessian Curvature Flow in Minkowski Space I: Cocompact Hypersurfaces

    Dake Li, Zhizhang Wang and Shiqi YinWider flows

    Topics Geometric inequalities, Other curvature flows · Shared terms “quermassintegral inequalities”

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.

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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)