Paper · Feb 5, 2024

Sharp estimates for the drift heat equation on shrinking Ricci solitons

Heather Macbeth

math.DGarXiv:2402.03304
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Abstract

We prove an estimate for the drift heat equation on a complete gradient shrinking Ricci soliton. This estimate has a time-dependent weight which is Gaussian in its spatial asymptotics. When transferred and scaled to an estimate for the heat equation along the Ricci flow of the soliton, this estimate is uniform up to the singular time.

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Submitted
Feb 5, 2024
Subjects
math.DG

Comments: 16 pages

Ricci flowRicci solitons

Its neighbourhood

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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.

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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. Regularizing estimates for positive solutions of the heat equation under geometric flows

    Alessandro Goffi and Francesco Pediconi

    Topics Ricci flow · Shared terms “heat equation”

  2. Vanishing bach-like tensors on complete gradient shrinking ricci solitons

    James Siene

    Topics Ricci solitons · Shared terms “shrinking Ricci”, “gradient shrinking”, “complete gradient”

  3. Hessian estimates for the conjugate heat equation coupled with the Ricci flow

    Hong Huang

    Topics Ricci flow · Shared terms “heat equation”

  4. Drift-harmonic functions with polynomial growth on asymptotically paraboloidal manifolds

    Michael B. Law

    Topics Ricci solitons · Shared terms “asymptotics”

  5. Kählerity of complete almost-Kähler gradient shrinking Ricci solitons

    Junming Xie

    Topics Ricci solitons · Shared terms “shrinking Ricci”, “gradient shrinking”, “complete gradient”

  6. Log gradient estimates for heat type equations on manifolds

    Qi S. Zhang

    Topics Ricci flow · Shared terms “heat equation”

  7. Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow

    Juanling Lu and Yu Zheng

    Topics Ricci flow · Shared terms “heat equation”

  8. Time analyticity for the heat equation under Bakry-Émery Ricci curvature condition

    Ling Wu

    Topics Ricci solitons · Shared terms “heat equation”

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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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)

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.

  3. 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.