Paper · Oct 19, 2023

Poincaré inequality and topological rigidity of translators and self-expanders for the mean curvature flow

Debora Impera, Michele Rimoldi

math.DGWider flowsarXiv:2310.12722v2
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Abstract

We prove an abstract structure theorem for weighted manifolds supporting a weighted -Poincaré inequality and whose ends satisfy a suitable non-integrability condition. We then study how our arguments can be used to obtain full topological control on two important classes of hypersurfaces of the Euclidean space, namely translators and self-expanders for the mean curvature flow, under either stability or curvature asumptions. As an important intermediate step in order to get our results we get the validity of a Poincaré inequality with respect to the natural weighted measure on any translator and we prove that any end of a translator must have infinite weighted volume. Similar tools can be obtained for properly immersed self-expanders permitting to get topological rigidity under curvature assumptions.

Comments: 15 pages. Minor changes in the exposition in v2. All comments are welcome!

Mean curvature flowSelf-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
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    Debora Impera, Michele Rimoldi and Francesco RuattaWider flows PDF

    Topics Self-shrinkers & translators · Same authors Debora Impera, Michele Rimoldi · Shared terms “immersed self”, “properly immersed”

  2. Classification of quadratically pinched translators in higher codimension

    Debora Impera, Michele Rimoldi and Francesco RuattaWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Same authors Debora Impera, Michele Rimoldi · Shared terms “immersed”

  3. The rigidity theorems for complete self-expander of mean curvature flow

    Zhi Li and Guoxin WeiWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “self expanders”, “Euclidean space”

  4. Self-expanders to the mean curvature flow based on the generalized Lawson-Osserman cone

    Chen-Kuan LeeWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “self expanders”

  5. Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow

    Junming XieWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “self expanders”

  6. No breather theorems for the mean curvature flow

    Liang Cheng and Yongjia ZhangWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “self expanders”, “Euclidean space”

  7. 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 “properly immersed”

  8. Translators to Higher Order Mean Curvature Flows in and

    Ronaldo F. de Lima and Giuseppe PipoliWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “Euclidean space”

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) 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. 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.