Paper · Mar 19, 2022

Poincaré type inequality for hypersurfaces and rigidity results

Hilário Alencar, Márcio Batista, Gregório Silva Neto

math.DGWider flowsarXiv:2203.10334v2
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Abstract

In this article, under mild constraints on the sectional curvature, we exploit a divergence formula for symmetric endomorphisms to deduce a general Poincaré type inequality. We apply such inequality to higher-order mean curvature of hypersurfaces of space forms and Einstein manifolds, to obtain several isoperimetric inequalities, as well as rigidity results for complete r-minimal hypersurfaces satisfying a suitable decay of the second fundamental form at infinity. Furthermore, using these techniques, we prove flatness and non-existence results for self-similar solutions to a large class of fully nonlinear curvature flows.

Comments: 26 pages. Final version. To appear in the Journal of Differential Equations

Other curvature flowsEinstein metricsSphere theoremsStability & uniquenessGeometric inequalitiesMinimal & CMC surfaces

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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 2021, 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 On the spectrum and index of expanding and translating solitons of the mean curvature flow in ℝ³ (Hilário Alencar and Gregório Silva Neto, 2022)5Alencar 2022 Hopf type theorems in Riemannian manifolds (Hilário Alencar, Gregório Silva Neto and Detang Zhou, 2022)6Alencar 2022 Gap theorems for complete self-shrinkers of r-mean curvature flows (Hilário Alencar, G. Pacelli Bessa and Gregório Silva Neto, 2024)2Alencar 2024 New quermassintegral and Poincaré type inequalities for non-convex domains (Yingxiang Hu and Mohammad N. Ivaki, 2024)1Hu 2024 New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms (Jie Wu, 2025)7Wu 2025 Half-space theorems for translating solitons of the r-mean curvature flow (Hilário Alencar, G. Pacelli Bessa and Gregório Silva Neto, 2026)4Alencar 2026 Asymptotically conical shrinking ends of the r-mean curvature flow: rotational construction and uniqueness (Hilário Alencar, Márcio Batista and Gregório Silva Neto, 2026)3Alencar 2026 The global structure of bi-rotational self-shrinkers (Gregório Silva Neto, 2026)8Neto 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. New quermassintegral and Poincaré type inequalities for non-convex domains

    Yingxiang Hu and Mohammad N. IvakiWider flows PDF

    Topics Geometric inequalities, Other curvature flows · Shared terms “Poincaré type”, “type inequality”

  2. Gap theorems for complete self-shrinkers of -mean curvature flows

    Hilário Alencar, G. Pacelli Bessa and Gregório Silva NetoWider flows PDF

    Same authors Hilário Alencar, Gregório Silva Neto · Shared terms “fundamental form”

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

    Hilário Alencar, Márcio Batista and Gregório Silva NetoWider flows PDF

    Topics Stability & uniqueness · Same authors Hilário Alencar, Márcio Batista, Gregório Silva Neto

  4. Half-space theorems for translating solitons of the r-mean curvature flow

    Hilário Alencar, G. Pacelli Bessa and Gregório Silva NetoWider flows PDF

    Same authors Hilário Alencar, Gregório Silva Neto · Shared terms “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 PDF

    Topics Stability & uniqueness · Same authors Hilário Alencar, Gregório Silva Neto

  6. Hopf type theorems in Riemannian manifolds

    Hilário Alencar, Gregório Silva Neto and Detang Zhou PDF

    Topics Other curvature flows · Same authors Hilário Alencar, Gregório Silva Neto

  7. New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms

    Jie WuWider flows PDF

    Topics Sphere theorems, Geometric inequalities, Other curvature flows · Shared terms “type inequality”, “space forms”

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

    Gregório Silva NetoWider flows PDF

    Topics Minimal & CMC surfaces · Same author Gregório Silva Neto

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 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. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.