Paper · Nov 7, 2019

Time analyticity for heat equation on gradient shrinking Ricci solitons

Jia-Yong Wu

math.DGarXiv:1911.02735v5
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Abstract

On a complete non-compact gradient shrinking Ricci soliton, we prove the analyticity in time for smooth solutions of the heat equation with quadratic exponential growth in the space variable. This growth condition is sharp. As an application, we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with quadratic exponential growth on shrinkers.

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Submitted
Nov 7, 2019
Last revised
Aug 19, 2022 (v5)
Subjects
math.DG
Journal
Acta Math. Sci., 42, 1690-1700, 2022
DOI
10.1007/s10473-022-0424-1

Comments: Finally version

Ricci solitonsBoundaries & non-compact

Its neighbourhood

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

    Ling Wu

    Topics Boundaries & non-compact, Ricci solitons · Shared terms “time analyticity”, “heat equation”, “space variable”

  2. Harmonic and Schrödinger functions of polynomial growth on gradient shrinking Ricci solitons

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    Topics Boundaries & non-compact, Ricci solitons · Same author Jia-Yong Wu · Shared terms “gradient shrinking”, “shrinking Ricci”, “compact gradient”

  3. Geometric inequalities and rigidity of gradient shrinking Ricci solitons

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    Topics Ricci solitons · Same author Jia-Yong Wu · Shared terms “shrinking Ricci”, “gradient shrinking”

  4. Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons

    Jia-Yong Wu

    Topics Ricci solitons · Same author Jia-Yong Wu · Shared terms “gradient shrinking”, “shrinking Ricci”

  5. Elliptic gradient estimates for a nonlinear heat equation and applications

    Jia-Yong Wu

    Topics Ricci solitons · Same author Jia-Yong Wu · Shared terms “heat equation”

  6. Sharp estimates for the drift heat equation on shrinking Ricci solitons

    Heather Macbeth

    Topics Ricci solitons · Shared terms “heat equation”, “shrinking Ricci”, “gradient shrinking”

  7. Sharp upper diameter bounds for compact shrinking Ricci solitons

    Jia-Yong Wu

    Topics Ricci solitons · Same author Jia-Yong Wu · Shared terms “shrinking Ricci”, “gradient shrinking”

  8. New differential Harnack inequalities for nonlinear heat equations

    Jia-Yong Wu

    Same author Jia-Yong Wu · Shared terms “heat equation”, “exponential”

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) 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.