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

Güneysu, Batu; Thalmaier, Anton

doi:10.5802/aif.3316

Scattering theory without injectivity radius assumptions, and spectral stability for the Ricci flow
[Théorie du scattering sans hypothèses de rayon d’injectivité, et stabilité spectrale du flot de Ricci]

Güneysu, Batu ¹ ; Thalmaier, Anton ²

¹ Universität Bonn 53115 Bonn (Germany)
² Department of Mathematics University of Luxembourg Maison du Nombre 4364 Esch-sur-Alzette (Luxembourg)

Annales de l'Institut Fourier, Tome 70 (2020) no. 1, pp. 437-456.

Résumé
Abstract

Nous démontrons un nouveau critère intégral pour l’existence et la complétude des opérateurs d’onde $W_{\pm} (- Δ_{h}, - Δ_{g}, I_{g, h})$ correspondant aux uniques réalisations auto-adjointes des opérateurs de Laplace–Beltrami $- Δ_{j}$ , $j = g, h$ , induits par deux métriques riemanniennes complètes quasi-isométriques $g$ et $h$ sur une variété non-compacte (sans bord) $M$ . En particulier, ce résultat fournit un critère pour que les spectres absolument continus de $- Δ_{g}$ et $- Δ_{h}$ coïncident. Notre preuve repose sur des estimations obtenues à l’aide d’une formule probabiliste de type Bismut pour le gradient d’un semigroupe de la chaleur. Contrairement aux résultats précédents, notre critère intégral nécessite seulement un certain contrôle des bornes inférieures sur la courbure de Ricci et des bornes supérieures sur les noyaux de la chaleur, mais aucun contrôle sur le rayon d’injectivité. En conséquence, nous obtenons un résultat de stabilité pour le spectre absolument continu sous le flot de Ricci.

We prove a new integral criterion for the existence and completeness of the wave operators $W_{\pm} (- Δ_{h}, - Δ_{g}, I_{g, h})$ corresponding to the unique self-adjoint realizations of the Laplace–Beltrami operators $- Δ_{j}$ , $j = g, h$ , induced by two quasi-isometric complete Riemannian metrics $g$ and $h$ on a noncompact manifold $M$ (without boundary). In particular, this result provides a criterion for the absolutely continuous spectra of $- Δ_{g}$ and $- Δ_{h}$ 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 on the injectivity radii. As a consequence, we obtain a stability result for the absolutely continuous spectrum under the Ricci flow.

Reçu le : 2017-11-06
Révisé le : 2018-12-06
Accepté le : 2019-03-12
Publié le : 2020-05-28

DOI : 10.5802/aif.3316

Classification : 35P25, 58J50, 58J65
Keywords: spectral theory, scattering theory, Riemannian manifold, Bismut type derivative formulas.
Mots-clés : théorie spectrale, théorie du scattering, variété riemannienne, formules de dérivées de type Bismut.

Affiliations des auteurs :

Güneysu, Batu ¹ ; Thalmaier, Anton ²

¹ Universität Bonn 53115 Bonn (Germany)
² Department of Mathematics University of Luxembourg Maison du Nombre 4364 Esch-sur-Alzette (Luxembourg)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Scattering theory without injectivity radius assumptions, and spectral stability for the {Ricci} flow},
     journal = {Annales de l'Institut Fourier},
     pages = {437--456},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Güneysu, Batu; Thalmaier, Anton. Scattering theory without injectivity radius assumptions, and spectral stability for the Ricci flow. Annales de l'Institut Fourier, Tome 70 (2020) no. 1, pp. 437-456. doi : 10.5802/aif.3316. https://aif.centre-mersenne.org/articles/10.5802/aif.3316/

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