Paper · Jan 11, 2017

Poisson equation on complete manifolds

Ovidiu Munteanu, Chiung-Jue Anna Sung, Jiaping Wang

math.DGarXiv:1701.02865
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Abstract

We develop heat kernel and Green's function estimates for manifolds with positive bottom spectrum. The results are then used to establish existence and sharp estimates of the solution to the Poisson equation on such manifolds with Ricci curvature bounded below. As an application, we show that the curvature of a steady gradient Ricci soliton must decay exponentially if it decays faster than linear and the potential function is bounded above.

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Submitted
Jan 11, 2017
Subjects
math.DG, math.AP

Comments: 44 pages

Ricci solitonsHeat kernelBoundaries & non-compactEigenvalues & spectrum

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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 Structure at infinity for shrinking Ricci solitons (Ovidiu Munteanu and Jiaping Wang, 2016)3Munteanu 2016 Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons (Jia-Yong Wu, 2020)2Wu 2020 Positive solutions to Schrödinger equations and geometric applications (Ovidiu Munteanu, Felix Schulze and Jiaping Wang, 2020)4Munteanu 2020 Gaussian upper bounds for the heat kernel on evolving manifolds (Reto Buzano and Louis Yudowitz, 2020)6Buzano 2020 Heat kernel on Ricci shrinkers (II) (Yu Li and Bing Wang, 2023)7Li 2023 A Minkowski type inequality for manifolds with positive spectrum (Ovidiu Munteanu and Jiaping Wang, 2023)1Munteanu 2023 The Ricci-DeTurck flow on complete manifolds (Jing-Bin Cai and Bing Wang, 2026)5Cai 2026 The Fisher Metric of the Ricci Flow Heat Kernel (Bennett Chow and Robert Koirala, 2026)8Chow 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A Minkowski type inequality for manifolds with positive spectrum

    Ovidiu Munteanu and Jiaping Wang

    Topics Eigenvalues & spectrum, Boundaries & non-compact, Ricci solitons · Same authors Ovidiu Munteanu, Jiaping Wang · Shared terms “complete manifolds”, “bottom spectrum”

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

    Jia-Yong Wu

    Topics Heat kernel, Eigenvalues & spectrum, Ricci solitons · Shared terms “Green function”

  3. Structure at infinity for shrinking Ricci solitons

    Ovidiu Munteanu and Jiaping Wang

    Topics Ricci solitons · Same authors Ovidiu Munteanu, Jiaping Wang · Shared terms “potential function”

  4. Positive solutions to Schrödinger equations and geometric applications

    Ovidiu Munteanu, Felix Schulze and Jiaping Wang

    Topics Ricci solitons · Same authors Ovidiu Munteanu, Jiaping Wang · Shared terms “gradient Ricci”

  5. The Ricci-DeTurck flow on complete manifolds

    Jing-Bin Cai and Bing Wang

    Topics Heat kernel, Boundaries & non-compact · Shared terms “complete manifolds”

  6. Gaussian upper bounds for the heat kernel on evolving manifolds

    Reto Buzano and Louis Yudowitz

    Topics Heat kernel

  7. Heat kernel on Ricci shrinkers (II)

    Yu Li and Bing Wang

    Topics Heat kernel, Ricci solitons

  8. The Fisher Metric of the Ricci Flow Heat Kernel

    Bennett Chow and Robert Koirala

    Topics Heat kernel, Eigenvalues & spectrum

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) Heat kernel The heat kernel of the conjugate heat equation, coupled to Ricci flow, is the probe Perelman and Bamler use to measure geometry. Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986) 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) Eigenvalues & spectrum Perelman's λ-functional is the lowest eigenvalue of −4Δ+ R, and it never decreases along Ricci flow. Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

  1. 6A.3
    The Heat Equation on ℝⁿBook 6A · The Heat Equation and Its Relatives

    The heat kernel, smoothing, uniqueness, Brownian motion, and a Gaussian that runs to Perelman.

  2. 6A.4
    Fourier Methods and Scale SpaceBook 6A · The Heat Equation and Its Relatives

    Fourier's own method on the circle and the line, smoothing as the decay of high frequencies, and the heat equation as the mathematics of blurring images.

  3. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.