Paper · Sep 28, 2023

Mean curvature flows of graphs sliding off to infinity in warped product manifolds

Naotoshi Fujihara

math.DGWider flowsarXiv:2309.16449v3
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Abstract

We study mean curvature flows in a warped product manifold defined by a closed Riemannian manifold and . In such a warped product manifold, we can define the notion of a graph, called a geodesic graph. We prove that the curve shortening flow preserves a geodesic graph for any warping function, and the mean curvature flow of hypersurfaces preserves a geodesic graph for some monotone convex warping functions. In particular, we consider some warping functions that go to zero at infinity, which means that the curves or hypersurfaces go to a point at infinity along the flow. In such a case, we prove the long-time existence of the flow and that the curvature and its higher-order derivatives go to zero along the flow.

Comments: 20 pages, 2 figures; typos corrected, arguments in section 4 revised, appendix and references for appendix added

Mean curvature flowCurve shorteningLong-time existence & convergenceRotational symmetry & warped products

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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 Curve shortening flows on rotational surfaces generated by monotone convex functions (Naotoshi Fujihara, 2022)3Fujihara 2022 Closed self-similar solutions to flows by negative powers of curvature (Shanze Gao, 2022)6Gao 2022 Curve shortening flows on surfaces that are not convex at infinity (Naotoshi Fujihara, 2024)1Fujihara 2024 Isotropy invariant graphical mean curvature flows in warped products (Naotoshi Fujihara and Naoyuki Koike, 2024)2Fujihara 2024 On the geometry of Riemannian warped product maps (Jyoti Yadav, Harmandeep Kaur and Gauree Shanker, 2025)7Yadav 2025 Translating solitons to higher order mean curvature flows in Riemannian products (Jorge Herbert Soares de Lira and Rafael Rocha de Farias, 2026)5de Lira 2026 Convergence of cohomogeneity-one Lagrangian mean curvature flow in positive Kähler-Einstein manifolds (Naotoshi Fujihara, Toru Kajigaya and Albert Wood, 2026)8Fujihara 2026 Length bounds and rigidity for ancient curve shortening flow on warped product surfaces (Beomseok Goh and Taehun Lee, 2026)4Goh 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Curve shortening flows on surfaces that are not convex at infinity

    Naotoshi FujiharaWider flows PDF

    Topics Curve shortening, Rotational symmetry & warped products · Same author Naotoshi Fujihara · Shared terms “warping function”

  2. Isotropy invariant graphical mean curvature flows in warped products

    Naotoshi Fujihara and Naoyuki KoikeWider flows PDF

    Topics Rotational symmetry & warped products · Same author Naotoshi Fujihara · Shared terms “warping function”

  3. Curve shortening flows on rotational surfaces generated by monotone convex functions

    Naotoshi FujiharaWider flows PDF

    Topics Curve shortening, Long-time existence & convergence · Same author Naotoshi Fujihara · Shared terms “monotone convex”

  4. Length bounds and rigidity for ancient curve shortening flow on warped product surfaces

    Beomseok Goh and Taehun LeeWider flows PDF

    Topics Curve shortening, Rotational symmetry & warped products

  5. Translating solitons to higher order mean curvature flows in Riemannian products

    Jorge Herbert Soares de Lira and Rafael Rocha de FariasWider flows PDF

    Topics Rotational symmetry & warped products · Shared terms “higher order”

  6. Closed self-similar solutions to flows by negative powers of curvature

    Shanze GaoWider flows PDF

    Topics Rotational symmetry & warped products

  7. On the geometry of Riemannian warped product maps

    Jyoti Yadav, Harmandeep Kaur and Gauree Shanker PDF

    Topics Rotational symmetry & warped products

  8. Convergence of cohomogeneity-one Lagrangian mean curvature flow in positive Kähler-Einstein manifolds

    Naotoshi Fujihara, Toru Kajigaya and Albert WoodWider flows PDF

    Topics Curve shortening, Long-time existence & convergence · Same author Naotoshi Fujihara

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.

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) Curve shortening Wider flows Plane curves moving by their curvature. Convex curves shrink to round points (Gage–Hamilton), and every embedded curve becomes convex first (Grayson). Start with: The heat equation shrinking convex plane curves (Gage et al., 1986) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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. 12C.4
    Loops, Disks and Curve ShrinkingBook 12C · Extinction and the Proof

    Perelman III §§2–3 and Morgan–Tian §§18.3–18.7: π₃ as spheres of loops, minimal disks and the −2π estimate, curve shrinking, ramps in M × S¹ and the estimates that make the argument rigorous.