Paper · Feb 17, 2016

Isoperimetric inequalities for eigenvalues by inverse mean curvature flow

Fangcheng Guo, Guanghan Li, Chuanxi Wu

math.DGWider flowsarXiv:1602.05290
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Abstract

By studying the monotonicity of the first nonzero eigenvalues of Laplace and p-Laplace operators on a closed convex hypersurface which evolves under inverse mean curvature flow in , the isoperimetric lower bounds for both eigenvalues were founded.

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Submitted
Feb 17, 2016
Subjects
math.DG

Comments: 16 pages

Mean curvature flowOther curvature flowsInverse mean curvature flowGeometric inequalitiesEigenvalues & spectrum

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper An Alexandrov-Fenchel-type inequality for hypersurfaces in the sphere (Frederico Girão and Neilha M. Pinheiro, 2016)7Girão 2016 A Curvature Flow and Applications to an Isoperimetric Inequality (Dylan Cant, 2016)6Cant 2016 Willmore inequality on hypersurfaces in hyperbolic space (Yingxiang Hu, 2016)3Hu 2016 Self similar solutions to the inverse mean curvature flow in ℝ² (Jui-En Chang, 2018)1Chang 2018 Harmonic mean curvature flow and geometric inequalities (Ben Andrews, Yingxiang Hu and Haizhong Li, 2019)8Andrews 2019 On the 1/H-flow by p-Laplace approximation: new estimates via fake distances under Ricci lower bounds (Luciano Mari, Marco Rigoli and Alberto Giulio Setti, 2019)4Mari 2019 Isoperimetry and the properness of weak inverse mean curvature flow (Kai Xu, 2023)2Xu 2023 Weinstock inequalities for outward-minimizing domains (Chaoqun Gao, Yong Wei and Rong Zhou, 2026)5Gao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Self similar solutions to the inverse mean curvature flow in

    Jui-En ChangWider flows

    Topics Inverse mean curvature flow, Other curvature flows

  2. Isoperimetry and the properness of weak inverse mean curvature flow

    Kai XuWider flows

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

  3. Willmore inequality on hypersurfaces in hyperbolic space

    Yingxiang HuWider flows

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

  4. On the -flow by -Laplace approximation: new estimates via fake distances under Ricci lower bounds

    Luciano Mari, Marco Rigoli and Alberto Giulio SettiWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Laplace”, “isoperimetric”, “lower bound”

  5. Weinstock inequalities for outward-minimizing domains

    Chaoqun Gao, Yong Wei and Rong ZhouWider flows

    Topics Eigenvalues & spectrum, Inverse mean curvature flow, Other curvature flows

  6. A Curvature Flow and Applications to an Isoperimetric Inequality

    Dylan CantWider flows

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

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

    Frederico Girão and Neilha M. PinheiroWider flows

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

  8. 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 “convex hypersurfaces”

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.