Paper · Nov 9, 2018

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

Debora Impera, Michele Rimoldi

math.DGarXiv:1811.04009
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Abstract

We generalize a method by L. Ambrozio, A. Carlotto, and B. Sharp to study the Morse index of closed f-minimal hypersurfaces isometrically immersed in a general weighted manifold. The technique permits, in particular, to obtain a linear lower bound on the Morse index via the first Betti number for closed f-minimal hypersurfaces in products of some compact rank one symmetric spaces with an Euclidean factor, endowed with the rigid shrinking gradient Ricci soliton structure. These include, as particular cases, all cylindric shrinking gradient Ricci solitons.

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Submitted
Nov 9, 2018
Subjects
math.DG

Comments: 13 pages

Ricci solitonsMinimal & CMC surfaces

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

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    Topics Minimal & CMC surfaces, Ricci solitons · Shared terms “minimal hypersurfaces”, “shrinking gradient”, “gradient Ricci”

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  4. Ancient mean curvature flows from minimal hypersurfaces

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  5. Semi-Continuity of the Morse Index for Ricci Shrinkers

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    Topics Minimal & CMC surfaces, Ricci solitons · Shared terms “Morse index”, “gradient Ricci”

  6. When can minimal hypersurfaces be connected by mean curvature flow?

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

  7. Quantitative index bounds for translators via topology

    Debora Impera and Michele RimoldiWider flows

    Same authors Debora Impera, Michele Rimoldi

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

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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.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 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.