Paper · Jan 2, 2017

The -Stability Index of the Constant Weighted Mean Curvature Hypersurfaces in Gradient Ricci Solitons

Hilário Alencar, Adina Rocha

math.DGarXiv:1701.00373
View PDF View HTML arXiv page

As on arXiv

Abstract

In this paper, we prove that a noncompact complete hypersurface with finite weighted volume, weighted mean curvature vector bounded in norm, and isometrically immersed in a complete weighted manifold is proper. In addition, we obtain an estimate for -stability index of a constant weighted mean curvature hypersurface with finite weighted volume and isometrically immersed in a shrinking gradient Ricci soliton that admits at least one parallel field globally defined. For such hypersurface, we still give a necessary condition for equality to be achieved in the estimate obtained.

View PDF View HTML arXiv page
Submitted
Jan 2, 2017
Subjects
math.DG
Ricci solitonsStability & uniquenessBoundaries & non-compact

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 Index and first Betti number of f-minimal hypersurfaces: general ambients (Debora Impera and Michele Rimoldi, 2018)7Impera 2018 Volume growth of complete submanifolds in gradient Ricci Solitons with bounded weighted mean curvature (Xu Cheng, Matheus Vieira and Detang Zhou, 2019)2Cheng 2019 Stability properties of complete self-shrinking surfaces in ℝ³ (Hilário Alencar, Gregório Silva Neto and Detang Zhou, 2021)4Alencar 2021 Constant weighted mean curvature hypersurfaces in Shrinking Ricci Solitons (Igor Miranda and Matheus Vieira, 2021)1Miranda 2021 On the spectrum and index of expanding and translating solitons of the mean curvature flow in ℝ³ (Hilário Alencar and Gregório Silva Neto, 2022)3Alencar 2022 Liouville theorems for harmonic 1-forms on gradient Ricci solitons (Chenghong He, Di Wu and Xi Zhang, 2024)8He 2024 Rigidity of weighted manifolds via classification results for semilinear equations (Giulio Ciraolo, Alberto Farina and Troy Petitt, 2025)6Ciraolo 2025 Asymptotically conical shrinking ends of the r-mean curvature flow: rotational construction and uniqueness (Hilário Alencar, Márcio Batista and Gregório Silva Neto, 2026)5Alencar 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Constant weighted mean curvature hypersurfaces in Shrinking Ricci Solitons

    Igor Miranda and Matheus Vieira

    Topics Ricci solitons · Shared terms “constant weighted”, “curvature hypersurfaces”, “weighted mean”

  2. Volume growth of complete submanifolds in gradient Ricci Solitons with bounded weighted mean curvature

    Xu Cheng, Matheus Vieira and Detang Zhou

    Topics Boundaries & non-compact, Ricci solitons · Shared terms “weighted mean”, “vector bounded”, “gradient Ricci”

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

    Hilário Alencar and Gregório Silva NetoWider flows

    Topics Stability & uniqueness · Same author Hilário Alencar · Shared terms “weighted volume”

  4. Stability properties of complete self-shrinking surfaces in

    Hilário Alencar, Gregório Silva Neto and Detang ZhouWider flows

    Topics Stability & uniqueness · Same author Hilário Alencar

  5. Asymptotically conical shrinking ends of the -mean curvature flow: rotational construction and uniqueness

    Hilário Alencar, Márcio Batista and Gregório Silva NetoWider flows

    Topics Boundaries & non-compact, Stability & uniqueness · Same author Hilário Alencar

  6. Rigidity of weighted manifolds via classification results for semilinear equations

    Giulio Ciraolo, Alberto Farina and Troy Petitt

    Topics Boundaries & non-compact, Stability & uniqueness, Ricci solitons · Shared terms “weighted manifolds”, “gradient Ricci”, “shrinking gradient”

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

    Debora Impera and Michele Rimoldi

    Topics Ricci solitons · Shared terms “isometrically immersed”, “weighted manifolds”, “gradient Ricci”

  8. Liouville theorems for harmonic 1-forms on gradient Ricci solitons

    Chenghong He, Di Wu and Xi Zhang

    Topics Boundaries & non-compact, Ricci solitons · Shared terms “noncompact complete”, “gradient Ricci”

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) 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) 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)

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.