Analytic and geometric aspects of Laplace operator on Riemannian manifold

Farah Diyab; B. Surender Reddy

doi:10.26637/MJM0804/0038

Abstract

In the past decade there has been a flurry of work at intersection of spectral theory and Riemannian geometry. In this paper we present some of recent results on abstract spectral theory depending on Laplace-Beltrami operator on compact Riemannian manifold. Also, we will emphasize the interplay between spectrum of operator and geometry of manifolds by discussing two main problems (direct and inverse problems) with an eye towards recent developments.

Keywords:

Spectrum, eigenvalue, Laplacian, spectral geometry, isospectral manifolds

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Farah Diyab Department of Mathematics, Osmania University, Hyderabad-500007, India.
B. Surender Reddy Department of Mathematics, Osmania University, Hyderabad-500007, India.

Pages: 1556-1561
Date Published: 01-10-2020
Vol. 8 No. 04 (2020): Malaya Journal of Matematik (MJM)

Canzani, Yaiza, Analysis on manifolds via the Laplacian, Lecture Notes, 2013.

Cheng, S. Y., Eigenvalue comparison theorems and its geometric applications, MathematischeZeitschrift, $143(3)(1975), 289-297$.

Cruz, Martin Vito, The spectrum of the Laplacian in Riemannian geometry, J Comput. Phys., (2003).

Gordon, Carolyn, S., Isospectral manifolds with different local geometry, Journal of the Korean Mathematical Society, 38(5)(2001), 955-969.

Gallot, Sylvestre and Hulin, Dominique and Lafontaine, Jacques, Riemannian Geometry, Springer-Verlag , Berlin, $(1990)$

Hajime and Urakawa, Geometry of Laplace-Beltrami Operator on a Complete Riemannian Manifold, Differential Geometry, (1993), 347-406.

Hajime and Urakawa, Spectral Geometry of the Laplacian: Spectral Analysis and Differential Geometry of the Laplacian, World Scientific, (2017).

Hansmann and Marcel, On the Discrete Spectrum of Linear Operators in Hilbert Spaces, Univ.-Bibliothek, $(2010)$

Jun Ling and ZhiqinLuy, Bounds of Eigenvalues on Riemannian Manifolds, Partial Differential Equations ALM10, (2010), 241-264.

Kac, Mark, Can one hear the shape of a drum? The American Mathematical Monthly, 73(4P2)(1966), 1-23.

Kulkarni, S. H., Nair, M. T., and Ramesh, G., Some properties of unbounded operators with closed range, Proceedings Mathematical Sciences, 118(4)(2008), 613625.

Ling, J., and Lu, Z., Bounds of eigenvalues on Riemannian manifolds, ALM, 10(2010), 241-264.

Milnor, John., Eigenvalues of the Laplace operator on certain manifolds, Proceedings of the National Academy of Sciences of the United States of America, 51(4)(1964), 542.

Ohno, Y., and Urakawa, H., On the first eigenvalue of the combinatorial Laplacian for a graph, Interdisciplinary Information Sciences, 1(1)(1994), 33-46.

Schmidt, F., The Laplace-beltrami-operator on Riemannian manifolds, In Seminar Shape Analysis, (2014).

Sunada, Toshikazu, Riemannian coverings and isospectral manifolds, Annals of Mathematics , 121(1)(1985), $169-186$.

Szegö, Gábor, Inequalities for certain eigenvalues of a membrane of given area, Journal of Rational Mechanics and Analysis, 3(1954), 343-356.

Weinberger, Hans F., An isoperimetric inequality for the $mathrm{N}$-dimensional free membrane problem, Journal of Rational Mechanics and Analysis, 5(4)(1956), 633-636