Paper · Oct 14, 2021

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

Shaochuang Huang, Luen-Fai Tam

math.DGarXiv:2110.07142
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Abstract

In this work, we obtain a short time existence result for harmonic map heat flow coupled with a smooth family of complete metrics in the domain manifold. Our results generalize short time existence results for harmonic map heat flow by Li-Tam [The heat equation and harmonic maps of complete manifolds, Invent. Math., 1991] and Chen-Zhu [Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geometry, 2006]. In particular, we prove the short time existence of harmonic map heat flow along a complete Ricci flow on into a complete manifold with curvature bounded from above with a smooth initial map of uniformly bounded energy density, under the assumptions that and is uniformly equivalent to .

Comments: 29 pages

Ricci flowStability & uniquenessBoundaries & non-compactHarmonic maps & DeTurck trick

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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 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 Rigidity of Lipschitz map using harmonic map heat flow (Man-Chun Lee and Luen-Fai Tam, 2022)1Lee 2022 Ricci flow on Finsler manifolds (Behroz Bidabad and Maral K. Sedaghat, 2023)7Bidabad 2023 Scalar curvature rigidity of the four-dimensional sphere (Simone Cecchini, Jinmin Wang, Zhizhang Xie et al., 2024)6Cecchini 2024 Prescribed mean curvature flow for noncompact hypersurfaces in Lorentz manifolds (Luen-Fai Tam, 2024)5Tam 2024 Uniqueness of Ricci flow with scaling invariant estimates (Man-Chun Lee, 2025)4Lee 2025 Convergence of Ricci flow and long-time existence of Harmonic map heat flow (Kyeongsu Choi and Yi Lai, 2025)2Choi 2025 Mean curvature flow into an ambient Riemannian manifold evolving by Ricci flow coupled with harmonic map heat flow (José N. V. Gomes, Matheus Hudson and Carlos M. de Sousa, 2025)3Gomes 2025 The Ricci-DeTurck flow on complete manifolds (Jing-Bin Cai and Bing Wang, 2026)8Cai 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam PDF

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Same author Luen-Fai Tam · Shared terms “map heat”, “heat flow”, “flow coupled”

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

    Kyeongsu Choi and Yi Lai PDF

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

  3. Mean curvature flow into an ambient Riemannian manifold evolving by Ricci flow coupled with harmonic map heat flow

    José N. V. Gomes, Matheus Hudson and Carlos M. de Sousa PDF

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

  4. Uniqueness of Ricci flow with scaling invariant estimates

    Man-Chun Lee PDF

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact, Stability & uniqueness · Shared terms “map heat”, “heat flow”, “Chen Zhu”

  5. Prescribed mean curvature flow for noncompact hypersurfaces in Lorentz manifolds

    Luen-Fai TamWider flows PDF

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact · Same author Luen-Fai Tam · Shared terms “time existence”, “short time”, “complete noncompact”

  6. Scalar curvature rigidity of the four-dimensional sphere

    Simone Cecchini, Jinmin Wang, Zhizhang Xie et al. PDF

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

  7. Ricci flow on Finsler manifolds

    Behroz Bidabad and Maral K. Sedaghat PDF

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”

  8. The Ricci-DeTurck flow on complete manifolds

    Jing-Bin Cai and Bing Wang PDF

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact, Stability & uniqueness · Shared terms “complete manifolds”, “short time”, “time existence”

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) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

In the guidebook

  1. 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.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.