Paper · Jun 11, 2020

Long-time estimates for heat flows on ALE manifolds

Klaus Kroencke, Oliver Lindblad Petersen

math.AParXiv:2006.06662v3
View PDF View HTML arXiv page

As on arXiv

Abstract

We consider the heat equation associated to Schrödinger operators acting on vector bundles on asymptotically locally Euclidean (ALE) manifolds. Novel decay estimates are established, allowing the Schrödinger operator to have a non-trivial -kernel. We also prove new decay estimates for spatial derivatives of arbitrary order, in a general geometric setting. Our main motivation is the application to stability of non-linear geometric equations, primarily Ricci flow, which will be presented in a companion paper. The arguments in this paper use that many geometric Schrödinger operators can be written as the square of Dirac type operators. By a remarkable result of Wang, this is even true for the Lichnerowicz Laplacian, under the assumption of a parallel spinor. Our analysis is based on a novel combination of the Fredholm theory for Dirac type operators on ALE manifolds and recent advances in the study of the heat kernel on non-compact manifolds.

View PDF View HTML arXiv page
Submitted
Jun 11, 2020
Last revised
Dec 16, 2021 (v3)
Subjects
math.AP, math.DG
DOI
10.1093/imrn/rnab350

Comments: 38 pages, final version. To appear in Int. Math Res. Not

Ricci flowRicci-flat & Calabi–YauHeat kernelStability & 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 Stability of ALE Ricci-flat manifolds under Ricci flow (Alix Deruelle and Klaus Kroencke, 2017)2Deruelle 2017 Mean curvature flow in asymptotically flat product spacetimes (Klaus Kroencke, Oliver Lindblad Petersen, Felix Lubbe et al., 2019)3Kroencke 2019 Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons (Jia-Yong Wu, 2020)5Wu 2020 Gaussian upper bounds for the heat kernel on evolving manifolds (Reto Buzano and Louis Yudowitz, 2020)4Buzano 2020 Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity (Klaus Kroencke and Oliver Lindblad Petersen, 2020)1Kroencke 2020 Heat kernel on Ricci shrinkers (II) (Yu Li and Bing Wang, 2023)7Li 2023 Ricci flow and the scalar curvature rigidity of Einstein manifolds (Klaus Kroencke, 2025)8Kroencke 2025 Dynamical functionals on ancient ARF Ricci flows (Isaac M. Lopez and Rio Schillmoeller, 2025)6Lopez 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity

    Klaus Kroencke and Oliver Lindblad Petersen

    Topics Ricci-flat & Calabi–Yau, Stability & uniqueness · Same authors Klaus Kroencke, Oliver Lindblad Petersen · Shared terms “ALE manifolds”, “parallel spinor”

  2. Stability of ALE Ricci-flat manifolds under Ricci flow

    Alix Deruelle and Klaus Kroencke

    Topics Ricci-flat & Calabi–Yau, Stability & uniqueness · Same author Klaus Kroencke · Shared terms “ALE”

  3. Mean curvature flow in asymptotically flat product spacetimes

    Klaus Kroencke, Oliver Lindblad Petersen, Felix Lubbe et al.Wider flows

    Topics Boundaries & non-compact · Same authors Klaus Kroencke, Oliver Lindblad Petersen · Shared terms “long time”

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

    Reto Buzano and Louis Yudowitz

    Topics Heat kernel

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

    Jia-Yong Wu

    Topics Heat kernel · Shared terms “Schrödinger operator”, “Euclidean”

  6. Dynamical functionals on ancient ARF Ricci flows

    Isaac M. Lopez and Rio Schillmoeller

    Topics Heat kernel, Ricci-flat & Calabi–Yau, Stability & uniqueness

  7. Heat kernel on Ricci shrinkers (II)

    Yu Li and Bing Wang

    Topics Heat kernel

  8. Ricci flow and the scalar curvature rigidity of Einstein manifolds

    Klaus Kroencke

    Topics Boundaries & non-compact, Stability & uniqueness · Same author Klaus Kroencke · Shared terms “locally Euclidean”, “asymptotically locally”

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-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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) 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. 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.