Paper · Feb 15, 2017

Gauss maps of the Ricci-mean curvature flow

Naoyuki Koike, Hikaru Yamamoto

math.DGarXiv:1702.04588
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Abstract

In this paper, we investigate the Gauss maps of a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow and a Ricci flow on the ambient manifold. Ruh and Vilms proved that the Gauss map of a minimal submanifold in a Euclidean space is a harmonic map, and Wang extended this result to a mean curvature flow in a Euclidean space by proving its Gauss maps satisfy the harmonic map heat flow equation. In this paper, we deduce the evolution equation for the Gauss maps of a Ricci-mean curvature flow, and as a direct corollary we prove that the Gauss maps of a Ricci-mean curvature flow satisfy the vertically harmonic map heat flow equation when the codimension of submanifolds is 1.

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Submitted
Feb 15, 2017
Subjects
math.DG
Journal
Geometriae Dedicata vol. 194 (2018) 169-185

Comments: 17 pages

Ricci flowMean curvature flowHarmonic maps & DeTurck trickMinimal & CMC surfaces

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 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 Examples of Ricci-mean curvature flows (Hikaru Yamamoto, 2016)3Yamamoto 2016 On the stability of harmonic maps under the homogeneous Ricci flow (Rafaela F. do Prado and Lino Grama, 2017)5Prado 2017 Moving-centre monotonicity formulae for minimal submanifolds and related equations (Jonathan J. Zhu, 2017)1Zhu 2017 On the nonexistence of harmonic and bi-harmonic maps (Ahmed Mohammed Cherif, 2020)4Cherif 2020 Liouville theorems for harmonic map heat flow along ancient super Ricci flow via reduced geometry (Keita Kunikawa and Yohei Sakurai, 2021)8Kunikawa 2021 Short time existence for harmonic map heat flow with time-dependent metrics (Shaochuang Huang and Luen-Fai Tam, 2021)2Huang 2021 Rigidity of Lipschitz map using harmonic map heat flow (Man-Chun Lee and Luen-Fai Tam, 2022)7Lee 2022 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)6Gomes 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Moving-centre monotonicity formulae for minimal submanifolds and related equations

    Jonathan J. ZhuWider flows

    Topics Minimal & CMC surfaces, Harmonic maps & DeTurck trick · Shared terms “minimal submanifolds”, “map heat”, “heat flow”

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

    Shaochuang Huang and Luen-Fai Tam

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

  3. Examples of Ricci-mean curvature flows

    Hikaru Yamamoto

    Topics Mean curvature flow · Same author Hikaru Yamamoto

  4. On the nonexistence of harmonic and bi-harmonic maps

    Ahmed Mohammed Cherif

    Topics Harmonic maps & DeTurck trick

  5. On the stability of harmonic maps under the homogeneous Ricci flow

    Rafaela F. do Prado and Lino Grama

    Topics Harmonic maps & DeTurck trick

  6. 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

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

  7. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam

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

  8. Liouville theorems for harmonic map heat flow along ancient super Ricci flow via reduced geometry

    Keita Kunikawa and Yohei Sakurai

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

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 in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

In the guidebook

  1. 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.

  2. 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.

  3. 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.