Paper · Nov 2, 2023

Gromov-Hausdorff stability of tori under Ricci and integral scalar curvature bounds

Shouhei Honda, Christian Ketterer, Ilaria Mondello, Raquel Perales, Chiara Rigoni

math.DGarXiv:2311.01342v3
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Abstract

We establish a nonlinear analogue of a splitting map into a Euclidean space, as a harmonic map into a flat torus. We prove that the existence of such a map implies Gromov-Hausdorff closeness to a flat torus in any dimension. Furthermore, Gromov-Hausdorff closeness to a flat torus and an integral bound {on , the smallest eigenvalue of the Ricci tensor in }, imply the existence of a harmonic splitting map. Combining these results with Stern's inequality, we provide a new Gromov-Hausdorff stability theorem for flat -tori. The main tools we employ include the harmonic map heat flow, Ricci flow, and both Ricci limits and RCD theories.

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Submitted
Nov 2, 2023
Last revised
Jul 29, 2024 (v3)
Subjects
math.DG, math.MG

Comments: 44 pages. We corrected a mistake in Theorem 5.3, present in the previous version, and added Theorem 1.8

Ricci flowScalar curvatureLimit spaces & RCDStability & uniquenessHarmonic maps & DeTurck trickEigenvalues & spectrum

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  1. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick, Scalar curvature, Stability & uniqueness · Shared terms “map heat”, “heat flow”, “Gromov”

  2. Convergence of Ricci flow and long-time existence of Harmonic map heat flow

    Kyeongsu Choi and Yi Lai

    Topics Harmonic maps & DeTurck trick, Eigenvalues & spectrum, Stability & uniqueness · Shared terms “map heat”, “heat flow”

  3. Short time existence for harmonic map heat flow with time-dependent metrics

    Shaochuang Huang and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “map heat”, “heat flow”

  4. Removing scalar curvature assumption for Ricci flow smoothing

    Adam Martens

    Topics Harmonic maps & DeTurck trick, Limit spaces & RCD, Scalar curvature · Shared terms “Gromov Hausdorff”

  5. Smoothing polyhedral spaces via Ricci flow

    Richard H. Bamler and Esther Cabezas-Rivas

    Topics Harmonic maps & DeTurck trick, Limit spaces & RCD, Stability & uniqueness · Shared terms “Gromov Hausdorff”, “curvature bounds”, “Euclidean”

  6. Scalar curvature rigidity of the four-dimensional sphere

    Simone Cecchini, Jinmin Wang, Zhizhang Xie et al.

    Topics Harmonic maps & DeTurck trick, Scalar curvature, Stability & uniqueness · Shared terms “map heat”, “heat flow”, “Gromov”

  7. Splitting maps in Type I Ricci flows

    Panagiotis Gianniotis

    Topics Limit spaces & RCD · Shared terms “splitting map”, “curvature bounds”, “analogue”

  8. Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows

    Marco Flaim, Erik Hupp and Karl-Theodor Sturm

    Topics Limit spaces & RCD, Scalar curvature · Shared terms “integral scalar”

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) 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)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.