Paper · Sep 16, 2026

Rigidity and Non-existence Results of -Translating Solitons

Li Xiang, Sun Jun

math.DGWider flowsarXiv:2609.18064
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Abstract

A -translating soliton is a hypersurface in satisfying ; equivalently, it has constant weighted mean curvature with respect to the log-linear density , and is an eternal solution of the mean curvature flow with a constant forcing term. In this paper, we first prove that every complete properly immersed -translating soliton with and has at least exponential volume growth, in contrast with the linear growth of ordinary translating solitons. We then prove sharp non-existence results for graphic -translating solitons () with bounded gradient, and two rigidity theorems.

Mean curvature flowAncient solutionsSelf-shrinkers & translatorsStability & uniqueness

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Half-space theorems for translating solitons of the r-mean curvature flow

    Hilário Alencar, G. Pacelli Bessa and Gregório Silva NetoWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “translating solitons”, “properly immersed”

  2. On the spectrum and index of expanding and translating solitons of the mean curvature flow in

    Hilário Alencar and Gregório Silva NetoWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “translating solitons”, “volume growth”, “exponential”

  3. Uniqueness of the -catenary cylinders by their asymptotic behaviour

    A. L. Martínez-Triviño and J. P. dos SantosWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “translating solitons”, “properly”

  4. Solitons to Mean Curvature Flow in the hyperbolic 3-space

    R. F. de Lima, A. K. Ramos and J. P. dos SantosWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “translating solitons”, “properly immersed”

  5. Rigidity results for mean curvature flow graphical translators moving in non-graphical direction

    John Man Shun Ma, Yuan Shyong Ooi and Juncheol PyoWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness

  6. A rigidity theorem of ancient solutions to the mean curvature flow in codimension one

    Qun Chen and Hongbing QiuWider flows PDF

    Topics Ancient solutions, Stability & uniqueness · Shared terms “rigidity theorem”

  7. Graphical translating solitons for the mean curvature flow and isoparametric functions

    Tomoki FujiiWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “translating solitons”, “ordinary”

  8. Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space

    Ronaldo F. de Lima and Álvaro K. RamosWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “translating solitons”

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) 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. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  2. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.