Paper · Sep 5, 2017

Scattering theory without injectivity radius assumptions and spectral stability for the Ricci flow

Batu Güneysu, Anton Thalmaier

math.DGarXiv:1709.01612
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Abstract

We prove a completely new integral criterion for the existence and completeness of the wave operators corresponding to the (unique self-adjoint realizations of) the Laplace-Beltrami operators , , that are induced by two quasi-isometric complete Riemannian metrics and on an open manifold . In particular, this result provides a criterion for the absolutely continuous spectra of and to coincide. Our proof relies on estimates that are obtained using a probabilistic Bismut type formula for the gradient of a heat semigroup. Unlike all previous results, our integral criterion only requires some lower control on the Ricci curvatures and some upper control on the heat kernels, but no control at all on the injectivity radii. As a consequence, we obtain a stability result for the absolutely continuous spectrum under a Ricci flow.

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Submitted
Sep 5, 2017
Subjects
math.DG, math-ph, math.SP
Ricci flowGeneralized & G₂ flowsHeat kernelStability & uniquenessEigenvalues & spectrum

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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 2016, 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. Scattering theory for the Hodge Laplacian

    Robert Baumgarth

    Topics Heat kernel, Eigenvalues & spectrum, Generalized & G₂ flows · Shared terms “scattering theory”, “absolutely continuous”, “integral criterion”

  2. Scattering Theory and Spectral Stability under a Ricci Flow for Dirac Operators

    Sebastian Boldt and Batu Güneysu

    Topics Eigenvalues & spectrum, Stability & uniqueness · Same author Batu Güneysu · Shared terms “scattering theory”, “spectral stability”, “absolutely continuous”

  3. Dimension-free Harnack inequalities for conjugate heat equations and their applications to geometric flows

    Li-Juan Cheng and Anton Thalmaier

    Topics Heat kernel · Same author Anton Thalmaier · Shared terms “semigroup”, “probabilistic”, “complete Riemannian”

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

    Jia-Yong Wu

    Topics Heat kernel, Eigenvalues & spectrum

  5. The Ricci-DeTurck flow on complete manifolds

    Jing-Bin Cai and Bing Wang

    Topics Heat kernel, Stability & uniqueness · Shared terms “injectivity radius”, “continuous”, “complete Riemannian”

  6. The Fisher Metric of the Ricci Flow Heat Kernel

    Bennett Chow and Robert Koirala

    Topics Heat kernel, Eigenvalues & spectrum, Stability & uniqueness · Shared terms “heat semigroup”, “criterion”

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

    Reto Buzano and Louis Yudowitz

    Topics Heat kernel

  8. Poisson equation on complete manifolds

    Ovidiu Munteanu, Chiung-Jue Anna Sung and Jiaping Wang

    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 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) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) 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.