Browse issues
no. 3
Scattering Theory and Spectral Stability under a Ricci Flow for Dirac Operators
Sebastian Boldt1; Batu Güneysu1
1 Fakultät für Mathematik, Technische Universität Chemnitz, 09126 Chemnitz, Germany
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 3, pp. 785-829.

Given a noncompact spin manifold M with a fixed topological spin structure and two complete Riemannian metrics g and h on M with bounded sectional curvatures, we prove a criterion for the existence and completeness of the wave operators 𝒲 ± (D h ,D g ,I g,h ) and 𝒲 ± (D h 2 ,D g 2 ,I g,h ), where I g,h is the canonically given unitary map between the underlying L 2 -spaces of spinors. This criterion does not involve any injectivity radius assumptions and leads to a criterion for the stability of the absolutely continuous spectrum of a Dirac operator and its square under a Ricci flow.

Étant donné une variété spin non-compacte M avec une structure spinorielle topologique fixée et deux métriques riemanniennes complètes g et h sur M à courbures sectionnelles bornées, nous prouvons un critère d’existence et de complétude des opérateurs d’onde 𝒲 ± (D h ,D g ,I g,h ) et 𝒲 ± (D h 2 ,D g 2 ,I g,h ), où I g,h est l’application unitaire canoniquement donnée entre les espaces L 2 de spineurs sous-jacents. Ce critère ne requiert aucune hypothèse de rayon d’injectivité et amène à un critère de stabilité du spectre absolument continu d’un opérateur de Dirac et de son carré sous un flot de Ricci.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1787
Classification: 35P25, 53C27, 58J65

Sebastian Boldt 1; Batu Güneysu 1

1 Fakultät für Mathematik, Technische Universität Chemnitz, 09126 Chemnitz, Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{AFST_2024_6_33_3_785_0,
     author = {Sebastian Boldt and Batu G\"uneysu},
     title = {Scattering {Theory} and {Spectral} {Stability} under a {Ricci} {Flow} for {Dirac} {Operators}},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {785--829},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 33},
     number = {3},
     year = {2024},
     doi = {10.5802/afst.1787},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1787/}
}
TY  - JOUR
AU  - Sebastian Boldt
AU  - Batu Güneysu
TI  - Scattering Theory and Spectral Stability under a Ricci Flow for Dirac Operators
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2024
SP  - 785
EP  - 829
VL  - 33
IS  - 3
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1787/
DO  - 10.5802/afst.1787
LA  - en
ID  - AFST_2024_6_33_3_785_0
ER  - 
%0 Journal Article
%A Sebastian Boldt
%A Batu Güneysu
%T Scattering Theory and Spectral Stability under a Ricci Flow for Dirac Operators
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2024
%P 785-829
%V 33
%N 3
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1787/
%R 10.5802/afst.1787
%G en
%F AFST_2024_6_33_3_785_0
Sebastian Boldt; Batu Güneysu. Scattering Theory and Spectral Stability under a Ricci Flow for Dirac Operators. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 3, pp. 785-829. doi : 10.5802/afst.1787. https://afst.centre-mersenne.org/articles/10.5802/afst.1787/

[1] Christian Bär The Dirac operator on hyperbolic manifolds of finite volume, J. Differ. Geom., Volume 54 (2000) no. 3, pp. 439-488 | Zbl

[2] Robert Baumgarth Scattering Theory for the Hodge Laplacian, J. Geom. Anal., Volume 32 (2022) no. 5, 150, 52 pages | Zbl

[3] Francesco Bei; Batu Güneysu; Jörn Müller Scattering theory of the Hodge Laplacian under a conformal perturbation, J. Spectr. Theory, Volume 7 (2017) no. 1, pp. 235-267 | Zbl

[4] Jean-Pierre Bourguignon; Paul Gauduchon Spineurs, opérateurs de Dirac et variations de métriques, Commun. Math. Phys., Volume 144 (1992) no. 3, pp. 581-599 | DOI | Zbl

[5] Ulrich Bunke The spectrum of the Dirac operator on the hyperbolic space, Math. Nachr., Volume 153 (1991), pp. 179-190 | DOI | Zbl

[6] Bruce K. Driver; Anton Thalmaier Heat equation derivative formulas for vector bundles, J. Funct. Anal., Volume 183 (2001) no. 1, pp. 42-108 | DOI | Zbl

[7] Jürgen Eichhorn Global Analysis on Open Manifolds, Nova Science Publishers, 2007, x+644 pages

[8] Thomas Friedrich Dirac operators in Riemannian geometry, Graduate Studies in Mathematics, 25, American Mathematical Society, 2000 | Zbl

[9] Batu Güneysu Covariant Schrödinger semigroups on Riemannian manifolds, Operator Theory: Advances and Applications, 264, Birkhäuser/Springer, 2017, xviii+239 pages

[10] Batu Güneysu; Anton Thalmaier Scattering theory without injectivity radius assumptions, and spectral stability for the Ricci flow, Ann. Inst. Fourier, Volume 70 (2020) no. 1, pp. 437-456 | DOI | Numdam | Zbl

[11] Rainer Hempel; Olaf Post; Ricardo Weder On open scattering channels for manifolds with ends, J. Funct. Anal., Volume 266 (2014) no. 9, pp. 5526-5583 | DOI | Zbl

[12] Elton P. Hsu Stochastic analysis on manifolds, Graduate Studies in Mathematics, 38, American Mathematical Society, 2002, xiv+281 pages

[13] Nobuyuki Ikeda; Shinzo Watanabe Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, 24, North-Holland, 1989, xvi+555 pages | Zbl

[14] H. Blaine Lawson; Marie-Louise Michelsohn Spin geometry, Princeton Mathematical Series, 38, Princeton University Press, 1989, xii+427 pages

[15] Niels Martin Møller Extremal metrics for spectral functions of Dirac operators in even and odd dimensions, Adv. Math., Volume 229 (2012) no. 2, pp. 1001-1046 | DOI | Zbl

[16] Yao-Feng Ren On the Burkholder–Davis–Gundy inequalities for continuous martingales, Stat. Probab. Lett., Volume 78 (2008) no. 17, pp. 3034-3039 | Zbl

[17] Wan-Xiong Shi Deforming the metric on complete Riemannian manifolds, J. Differ. Geom., Volume 30 (1989) no. 1, pp. 223-301 | Zbl

[18] Anton Thalmaier; Feng-Yu Wang Gradient estimates for harmonic functions on regular domains in Riemannian manifolds, J. Funct. Anal., Volume 155 (1998) no. 1, pp. 109-124 | DOI | Zbl

[19] Peter Topping Lectures on the Ricci flow, London Mathematical Society Lecture Note Series, 325, London Mathematical Society, 2006, x+113 pages | DOI

Cited by Sources: