Paper · Mar 22, 2018

Index and first Betti number of -minimal hypersurfaces and self-shrinkers

Debora Impera, Michele Rimoldi, Alessandro Savo

math.DGWider flowsarXiv:1803.08268
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Abstract

We study the Morse index of self-shrinkers for the mean curvature flow and, more generally, of -minimal hypersurfaces in a weighted Euclidean space endowed with a convex weight. When the hypersurface is compact, we show that the index is bounded from below by an affine function of its first Betti number. When the first Betti number is large, this improves index estimates known in literature. In the complete non-compact case, the lower bound is in terms of the dimension of the space of weighted square summable -harmonic -forms; in particular, in dimension , the procedure gives an index estimate in terms of the genus of the surface.

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Submitted
Mar 22, 2018
Subjects
math.DG

Comments: 20 pages

Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compactMinimal & 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 Quantitative index bounds for translators via topology (Debora Impera and Michele Rimoldi, 2018)2Impera 2018 Index and first Betti number of f-minimal hypersurfaces: general ambients (Debora Impera and Michele Rimoldi, 2018)1Impera 2018 Multiplicity one for min-max theory in compact manifolds with boundary and its applications (Ao Sun, Zhichao Wang and Xin Zhou, 2020)8Sun 2020 Poincaré inequality and topological rigidity of translators and self-expanders for the mean curvature flow (Debora Impera and Michele Rimoldi, 2023)5Impera 2023 Ancient mean curvature flows from minimal hypersurfaces (Yongheng Han, 2023)7Han 2023 Classification of quadratically pinched self-shrinkers in higher codimension (Debora Impera, Michele Rimoldi and Francesco Ruatta, 2026)4Impera 2026 Classification of quadratically pinched translators in higher codimension (Debora Impera, Michele Rimoldi and Francesco Ruatta, 2026)3Impera 2026 The global structure of bi-rotational self-shrinkers (Gregório Silva Neto, 2026)6Neto 2026
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 f-minimal hypersurfaces: general ambients

    Debora Impera and Michele Rimoldi

    Topics Minimal & CMC surfaces · Same authors Debora Impera, Michele Rimoldi · Shared terms “Betti number”, “minimal hypersurfaces”, “Morse index”

  2. Quantitative index bounds for translators via topology

    Debora Impera and Michele RimoldiWider flows

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

  3. Classification of quadratically pinched translators in higher codimension

    Debora Impera, Michele Rimoldi and Francesco RuattaWider flows

    Topics Minimal & CMC surfaces, Self-shrinkers & translators · Same authors Debora Impera, Michele Rimoldi

  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

  5. 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 · Same authors Debora Impera, Michele Rimoldi · Shared terms “Euclidean space”

  6. The global structure of bi-rotational self-shrinkers

    Gregório Silva NetoWider flows

    Topics Minimal & CMC surfaces, Self-shrinkers & translators, Boundaries & non-compact

  7. Ancient mean curvature flows from minimal hypersurfaces

    Yongheng HanWider flows

    Topics Minimal & CMC surfaces · Shared terms “minimal hypersurfaces”, “Morse index”

  8. Multiplicity one for min-max theory in compact manifolds with boundary and its applications

    Ao Sun, Zhichao Wang and Xin ZhouWider flows

    Topics Minimal & CMC surfaces, Self-shrinkers & translators, Boundaries & non-compact · Shared terms “minimal hypersurfaces”, “Morse index”, “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 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) 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.