Paper · Nov 26, 2024

Evolution of the Torsional Rigidity under Geometric Flows

Vicent Gimeno i Garcia, Fernán González-Ibáñez

math.DGarXiv:2411.17435
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Abstract

This paper explores the behavior of the torsional rigidity of a precompact domain as the ambient manifold evolves under a geometric flow. Specifically, we derive bounds on torsional rigidity under the Ricci Flow for Heisenberg spaces and homogeneous spheres. Additionally, we establish bounds under the Inverse Mean Curvature Flow for strictly convex, free-boundary, disk-type hypersurfaces within a ball. In this latter case, by extending the analysis to the maximal existence time of the flow, we obtain inequalities of comparison with the flat disk for both volume and torsional rigidity.

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Submitted
Nov 26, 2024
Subjects
math.DG, math-ph
DOI
10.1007/s11118-025-10243-y
Ricci flowMean curvature flowOther curvature flowsInverse mean curvature flowHomogeneous spacesStability & uniquenessFree boundary & capillary

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 2021, 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 Inverse mean curvature flow with a free boundary in hyperbolic space (Xiaoxiang Chai, 2022)1Chai 2022 Flow by Gauss Curvature to the Orlicz Minkowski Problem for q-torsional rigidity (Xia Zhao and Peibiao Zhao, 2024)2Zhao 2024 Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone (Ya Gao and Jing Mao, 2024)8Gao 2024 The dual Minkowski problem for q-torsional rigidity (Xia Zhao and Peibiao Zhao, 2024)4Zhao 2024 Backward Uniqueness of Extrinsic Geometric Flow in general ambient manifolds (Dasong Li and John Man Shun Ma, 2024)7Li 2024 The Minkowski problem for the k-torsional rigidity (Xia Zhao and Peibiao Zhao, 2025)3Zhao 2025 Inverse curvature flows for capillary hypersurfaces in the unit ball (Shujing Pan and Bo Yang, 2025)6Pan 2025 Contracting Transport Maps on Riemannian Manifolds (Shrey Aryan, Bang-Xian Han and Zhuo-Nan Zhu, 2026)5Aryan 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Inverse mean curvature flow with a free boundary in hyperbolic space

    Xiaoxiang ChaiWider flows

    Topics Free boundary & capillary, Inverse mean curvature flow, Other curvature flows

  2. Flow by Gauss Curvature to the Orlicz Minkowski Problem for q-torsional rigidity

    Xia Zhao and Peibiao ZhaoWider flows

    Topics Other curvature flows, Stability & uniqueness · Shared terms “torsional rigidity”

  3. The Minkowski problem for the -torsional rigidity

    Xia Zhao and Peibiao ZhaoWider flows

    Topics Other curvature flows, Stability & uniqueness · Shared terms “torsional rigidity”

  4. The dual Minkowski problem for -torsional rigidity

    Xia Zhao and Peibiao ZhaoWider flows

    Topics Other curvature flows, Stability & uniqueness · Shared terms “torsional rigidity”

  5. Contracting Transport Maps on Riemannian Manifolds

    Shrey Aryan, Bang-Xian Han and Zhuo-Nan ZhuWider flows

    Topics Free boundary & capillary, Inverse mean curvature flow, Other curvature flows · Shared terms “flat disk”, “strictly convex”

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

    Shujing Pan and Bo YangWider flows

    Topics Free boundary & capillary, Inverse mean curvature flow, Other curvature flows · Shared terms “strictly convex”

  7. Backward Uniqueness of Extrinsic Geometric Flow in general ambient manifolds

    Dasong Li and John Man Shun MaWider flows

    Topics Inverse mean curvature flow, Other curvature flows, Stability & uniqueness · Shared terms “geometric flows”, “ambient manifold”

  8. Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone

    Ya Gao and Jing MaoWider flows

    Topics Free boundary & capillary, Inverse mean curvature flow, Other curvature flows

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. 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)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 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.

  3. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.