Article (Scientific journals)
Scattering theory without injectivity radius assumptions, and spectral stability for the Ricci flow
Güneysu, Batu; THALMAIER, Anton
2020In Annales de l'Institut Fourier, 70 (1), p. 437-456
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Abstract :
[en] We prove a new integral criterion for the existence and completeness of the wave operators W_{\pm}(-\Delta_h,-\Delta_g, I_{g,h}) corresponding to the (unique self-adjoint realizations of) the Laplace-Beltrami operators -\Delta_j, j=g,h, that are induced by two quasi-isometric complete Riemannian metrics g and h on an open manifold M. In particular, this result provides a criterion for the absolutely continuous spectra of -\Delta_g and -\Delta_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 at all on the injectivity radii. As a consequence, we obtain a stability result for the absolutely continuous spectrum under a Ricci flow.
Disciplines :
Mathematics
Author, co-author :
Güneysu, Batu;  Humboldt-Universität zu Berlin > Institut für Mathematik
THALMAIER, Anton ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit
External co-authors :
yes
Language :
English
Title :
Scattering theory without injectivity radius assumptions, and spectral stability for the Ricci flow
Publication date :
28 May 2020
Journal title :
Annales de l'Institut Fourier
ISSN :
0373-0956
eISSN :
1777-5310
Publisher :
Association des Annales de l'Institut Fourier, Grenoble, France
Volume :
70
Issue :
1
Pages :
437-456
Peer reviewed :
Peer Reviewed verified by ORBi
FnR Project :
FNR7628746 - Geometry Of Random Evolutions, 2014 (01/03/2015-28/02/2018) - Anton Thalmaier
Name of the research project :
R-AGR-0517 - IRP15 - AGSDE (20150901-20190630) - THALMAIER Anton
Funders :
University of Luxembourg - UL
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