Paper · Apr 20, 2018

Quantitative index bounds for translators via topology

Debora Impera, Michele Rimoldi

math.DGWider flowsarXiv:1804.07709v4
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Abstract

We obtain a quantitative estimate on the generalised index of translators for the mean curvature flow with bounded norm of the second fundamental form. The estimate involves the dimension of the space of weighted square integrable f-harmonic 1-forms. By the adaptation to the weighted setting of Li-Tam theory developed in previous works, this yields estimates in terms of the number of ends of the hypersurface when this is contained in a upper halfspace with respect to the translating direction. When there exists a point where all principal curvatures are distinct we estimate the nullity of the stability operator. This permits to obtain quantitative estimates on the stability index via the topology of translators with bounded norm of the second fundamental form which are either two-dimensional or (in higher dimension) have finite topological type and are contained in a upper halfspace.

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Submitted
Apr 20, 2018
Last revised
Jan 13, 2019 (v4)
Subjects
math.DG

Comments: 14 pages. Translators seem to support a weighted L^2 Sobolev inequality only in dimension greater than or equal to 3 and when the translator is contained in a upper halfspace with respect to the translating direction; see Appendix A. Statements of Theorem A, Theorem B, Corollary C and Theorem E fixed accordingly. Theorem D still holds unchanged. Final version: to appear on Math. Z

Mean curvature flowSelf-shrinkers & translatorsSurfacesStability & 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. Index and first Betti number of -minimal hypersurfaces and self-shrinkers

    Debora Impera, Michele Rimoldi and Alessandro SavoWider flows

    Topics Self-shrinkers & translators · Same authors Debora Impera, Michele Rimoldi · Shared terms “weighted square”

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

    Debora Impera and Michele RimoldiWider flows

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

  3. Classification of quadratically pinched translators in higher codimension

    Debora Impera, Michele Rimoldi and Francesco RuattaWider flows

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

  4. Classification of quadratically pinched self-shrinkers in higher codimension

    Debora Impera, Michele Rimoldi and Francesco RuattaWider flows

    Topics Self-shrinkers & translators · Same authors Debora Impera, Michele Rimoldi · Shared terms “weighted setting”, “fundamental form”

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

    Hilário Alencar and Gregório Silva NetoWider flows

    Topics Surfaces, Self-shrinkers & translators, Stability & uniqueness · Shared terms “stability operator”

  6. Index and first Betti number of f-minimal hypersurfaces: general ambients

    Debora Impera and Michele Rimoldi

    Same authors Debora Impera, Michele Rimoldi

  7. Dynamical stability of various convex graphical translators

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    Topics Surfaces, Self-shrinkers & translators, Stability & uniqueness

  8. Stability properties of complete self-shrinking surfaces in

    Hilário Alencar, Gregório Silva Neto and Detang ZhouWider flows

    Topics Surfaces, Self-shrinkers & translators, Stability & uniqueness

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 in the L² sense. 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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 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. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 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.