Paper · Oct 19, 2016

A Curvature Flow and Applications to an Isoperimetric Inequality

Dylan Cant

math.DGWider flowsarXiv:1610.05844
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Abstract

Long time existence and convergence to a circle is proved for radial graph solutions to a mean curvature type curve flow in warped product surfaces (under a weak assumption on the warp potential of the surface). This curvature flow preserves the area enclosed by the evolving curve, and this fact is used to prove a general isoperimetric inequality applicable to radial graphs in warped product surfaces under weak assumptions on the warp potential.

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Submitted
Oct 19, 2016
Subjects
math.DG
Other curvature flowsLong-time existence & convergenceRotational symmetry & warped productsGeometric inequalities

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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 Isoperimetric inequalities for eigenvalues by inverse mean curvature flow (Fangcheng Guo, Guanghan Li and Chuanxi Wu, 2016)3Guo 2016 On the inverse mean curvature flow in warped product manifolds (Thomas Mullins, 2016)7Mullins 2016 Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds (Brian Allen, 2017)2Allen 2017 Locally constrained inverse curvature flows (Julian Scheuer and Chao Xia, 2017)6Scheuer 2017 Inverse curvature flows in Riemannian warped products (Julian Scheuer, 2017)5Scheuer 2017 The isoperimetric inequality in steady Ricci solitons (Yuqiao Li, 2019)1Li 2019 Mean curvature flows of graphs sliding off to infinity in warped product manifolds (Naotoshi Fujihara, 2023)8Fujihara 2023 Length bounds and rigidity for ancient curve shortening flow on warped product surfaces (Beomseok Goh and Taehun Lee, 2026)4Goh 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The isoperimetric inequality in steady Ricci solitons

    Yuqiao Li

    Topics Geometric inequalities, Rotational symmetry & warped products · Shared terms “isoperimetric inequality”

  2. Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds

    Brian AllenWider flows

    Topics Geometric inequalities, Rotational symmetry & warped products, Long-time existence & convergence

  3. Isoperimetric inequalities for eigenvalues by inverse mean curvature flow

    Fangcheng Guo, Guanghan Li and Chuanxi WuWider flows

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

  4. Length bounds and rigidity for ancient curve shortening flow on warped product surfaces

    Beomseok Goh and Taehun LeeWider flows

    Topics Rotational symmetry & warped products · Shared terms “product surfaces”

  5. Inverse curvature flows in Riemannian warped products

    Julian ScheuerWider flows

    Topics Rotational symmetry & warped products, Long-time existence & convergence, Other curvature flows

  6. Locally constrained inverse curvature flows

    Julian Scheuer and Chao XiaWider flows

    Topics Geometric inequalities, Rotational symmetry & warped products, Long-time existence & convergence · Shared terms “isoperimetric”

  7. On the inverse mean curvature flow in warped product manifolds

    Thomas MullinsWider flows

    Topics Rotational symmetry & warped products, Long-time existence & convergence, Other curvature flows

  8. Mean curvature flows of graphs sliding off to infinity in warped product manifolds

    Naotoshi FujiharaWider flows

    Topics Rotational symmetry & warped products, Long-time existence & convergence · Shared terms “flow preserves”

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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994) 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)