Geodesic
On the Spectrum of Certain Hadamard Manifolds
Symmetry, integrability and geometry: methods and applications, Tome 19 (2023) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We show the absolute continuity of the spectrum and determine the spectrum as a set for two classes of Hadamard manifolds and for specific domains and quotients of one of the classes.
Keywords: Laplace operator, absolutely continuous spectrum, point spectrum, Hadamard manifold, asymptotically harmonic manifold. 
@article{SIGMA_2023_19_a49,
     author = {Werner Ballmann and Mayukh Mukherjee and Panagiotis Polymerakis},
     title = {On the {Spectrum} of {Certain} {Hadamard} {Manifolds}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2023},
     volume = {19},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a49/}
}
TY  - JOUR
AU  - Werner Ballmann
AU  - Mayukh Mukherjee
AU  - Panagiotis Polymerakis
TI  - On the Spectrum of Certain Hadamard Manifolds
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2023
VL  - 19
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a49/
LA  - en
ID  - SIGMA_2023_19_a49
ER  - 
%0 Journal Article
%A Werner Ballmann
%A Mayukh Mukherjee
%A Panagiotis Polymerakis
%T On the Spectrum of Certain Hadamard Manifolds
%J Symmetry, integrability and geometry: methods and applications
%D 2023
%V 19
%U http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a49/
%G en
%F SIGMA_2023_19_a49
Werner Ballmann; Mayukh Mukherjee; Panagiotis Polymerakis. On the Spectrum of Certain Hadamard Manifolds. Symmetry, integrability and geometry: methods and applications, Tome 19 (2023). http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a49/

[1] Azencott R., Wilson E.N., “Homogeneous manifolds with negative curvature. I”, Trans. Amer. Math. Soc., 215 (1976), 323–362 | DOI | MR | Zbl

[2] Azencott R., Wilson E.N., Homogeneous manifolds with negative curvature. II, Mem. Amer. Math. Soc., 8, 1976, iii+102 pp. | DOI | MR

[3] Ballmann W., Lectures on spaces of nonpositive curvature, DMV Seminar, 25, Birkhäuser, Basel, 1995 | DOI | MR | Zbl

[4] Ballmann W., Matthiesen H., Mondal S., “Small eigenvalues of surfaces of finite type”, Compos. Math., 153 (2017), 1747–1768, arXiv: 1506.06541 | DOI | MR | Zbl

[5] Ballmann W., Polymerakis P., “On the essential spectrum of differential operators over geometrically finite orbifolds”, J. Differential Geom. (to appear) , arXiv: 2103.13704 | MR

[6] Bessa G.P., Jorge L.P., Montenegro J.F., “The spectrum of the Martin–Morales–Nadirashvili minimal surfaces is discrete”, J. Geom. Anal., 20 (2010), 63–71, arXiv: 0809.1173 | DOI | MR | Zbl

[7] Donnelly H., Garofalo N., “Riemannian manifolds whose Laplacians have purely continuous spectrum”, Math. Ann., 293 (1992), 143–161 | DOI | MR

[8] Donnelly H., Garofalo N., “Schrödinger operators on manifolds, essential self-adjointness, and absence of eigenvalues”, J. Geom. Anal., 7 (1997), 241–257 | DOI | MR | Zbl

[9] Donnelly H., Xavier F., “On the differential form spectrum of negatively curved Riemannian manifolds”, Amer. J. Math., 106 (1984), 169–185 | DOI | MR | Zbl

[10] Heber J., “On the geometric rank of homogeneous spaces of nonpositive curvature”, Invent. Math., 112 (1993), 151–170 | DOI | MR | Zbl

[11] Ichinose T., “Operators on tensor products of Banach spaces”, Trans. Amer. Math. Soc., 170 (1972), 197–219 | DOI | MR | Zbl

[12] Kasue A., “A note on $L^2$ harmonic forms on a complete manifold”, Tokyo J. Math., 17 (1994), 455–465 | DOI | MR | Zbl

[13] Kazdan J.L., Warner F.W., “Curvature functions for compact $2$-manifolds”, Ann. of Math., 99 (1974), 14–47 | DOI | MR | Zbl

[14] Mazzeo R., “Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic manifolds”, Amer. J. Math., 113 (1991), 25–45 | DOI | MR | Zbl

[15] Polymerakis P., “Spectral estimates for Riemannian submersions with fibers of basic mean curvature”, J. Geom. Anal., 31 (2021), 9951–9980, arXiv: 2003.09843 | DOI | MR | Zbl

[16] Reed M., Simon B., Methods of modern mathematical physics, v. IV, Analysis of operators, Academic Press, New York, 1978 | MR | Zbl

[17] Rellich F., “Über das asymptotische Verhalten der Lösungen von $\Delta u+\lambda u=0$ in unendlichen Gebieten”, Jahresber. Dtsch. Math.-Ver., 53 (1943), 57–65 | MR | Zbl

[18] Wolter T.H., “Geometry of homogeneous Hadamard manifolds”, Internat. J. Math., 2 (1991), 223–234 | DOI | MR | Zbl

[19] Xavier F., “Convexity and absolute continuity of the Laplace–Beltrami operator”, Math. Ann., 282 (1988), 579–585 | DOI | MR | Zbl