Paper · Jan 19, 2016

Stability of Geodesic Spheres in under Constrained Curvature Flows

David Hartley

math.DGWider flowsarXiv:1601.04986
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Abstract

In this paper we discuss the stability of geodesic spheres in under constrained curvature flows. We prove that under some standard assumptions on the speed and weight functions, the spheres are stable under perturbations that preserve a volume type quantity. This extends results by Escher and Simonett, 1998, and the author, 2015, to a Riemannian manifold setting.

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Submitted
Jan 19, 2016
Subjects
math.DG, math.AP

Comments: 13 pages

Other curvature flowsVolume-preserving & constrained flowsStability & uniqueness

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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 Volume preserving non homogeneous mean curvature flow in hyperbolic space (Maria Chiara Bertini and Giuseppe Pipoli, 2016)4Bertini 2016 Locally constrained curvature flows and geometric inequalities in hyperbolic space (Yingxiang Hu, Haizhong Li and Yong Wei, 2020)1Hu 2020 Geometric inequalities for static convex domains in hyperbolic space (Yingxiang Hu and Haizhong Li, 2021)7Hu 2021 Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space (Zheng Huang, Longzhi Lin and Zhou Zhang, 2021)8Huang 2021 A fully nonlinear locally constrained curvature flow for capillary hypersurface (Xinqun Mei and Liangjun Weng, 2024)5Mei 2024 Constrained Curvature Flows on Pinched Hadamard Surfaces (Sara Albert-Niclòs and Esther Cabezas-Rivas, 2026)2Albert-Niclò 2026 Alexandrov–Fenchel inequalities for convex domains in the sphere (Tianci Luo, Yong Wei and Rong Zhou, 2026)3Luo 2026 Geometric C¹-stability of geodesic spheres under volume-preserving mean curvature flow (Esther Cabezas-Rivas and Carlo Sinestrari, 2026)6Cabezas-Riva 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Locally constrained curvature flows and geometric inequalities in hyperbolic space

    Yingxiang Hu, Haizhong Li and Yong WeiWider flows

    Topics Volume-preserving & constrained flows, Other curvature flows · Shared terms “geodesic sphere”, “preserves”

  2. Constrained Curvature Flows on Pinched Hadamard Surfaces

    Sara Albert-Niclòs and Esther Cabezas-RivasWider flows

    Topics Volume-preserving & constrained flows, Other curvature flows

  3. Alexandrov–Fenchel inequalities for convex domains in the sphere

    Tianci Luo, Yong Wei and Rong ZhouWider flows

    Topics Volume-preserving & constrained flows, Other curvature flows, Stability & uniqueness · Shared terms “geodesic sphere”, “preserves”

  4. Volume preserving non homogeneous mean curvature flow in hyperbolic space

    Maria Chiara Bertini and Giuseppe PipoliWider flows

    Topics Volume-preserving & constrained flows, Other curvature flows · Shared terms “geodesic sphere”

  5. A fully nonlinear locally constrained curvature flow for capillary hypersurface

    Xinqun Mei and Liangjun WengWider flows

    Topics Volume-preserving & constrained flows, Other curvature flows · Shared terms “preserves”

  6. Geometric -stability of geodesic spheres under volume-preserving mean curvature flow

    Esther Cabezas-Rivas and Carlo SinestrariWider flows

    Topics Volume-preserving & constrained flows, Stability & uniqueness · Shared terms “geodesic sphere”

  7. Geometric inequalities for static convex domains in hyperbolic space

    Yingxiang Hu and Haizhong LiWider flows

    Topics Volume-preserving & constrained flows, Other curvature flows

  8. Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space

    Zheng Huang, Longzhi Lin and Zhou ZhangWider flows

    Topics Volume-preserving & constrained flows, Stability & uniqueness

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) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) 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)