Abstract
We use the Ricci flow on surfaces to give upper and lower estimates for the eigenvalues of the square of the Dirac operator of a surface (M, g) of negative Euler characteristic. For each of the eigenvalues, the estimate depends on simple geometric data of (M, g) and on the corresponding eigenvalue of the square of the Dirac operator of the unique constant curvature metric on the conformal class of g whose volume equals that of g.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. Ammann, Spin-Strukturen und das Spektrum des Dirac-Operators, Shaker Verlag, 1998. Doktorarbeit – Universität Freiburg (Germany).
Ammann B.: A spin-conformal lower bound of the first positive Dirac eigenvalue. Differential Geom. Appl. 18, 21–32 (2003)
Ammann B.: Dirac eigenvalue estimates on two-tori. J. Geom. Phys. 51, 372–386 (2004)
Ammann B., Bär C.: Dirac eigenvalue estimates on surfaces. Math. Z. 240, 423–449 (2002)
Bär C.: Lower eigenvalue estimates for Dirac operators. Math. Ann. 293, 39–46 (1992)
B. Chow and D. Knopf, The Ricci flow: an introduction, volume 110 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2004.
Di Cerbo L. F.: Eigenvalues of the Laplacian under the Ricci flow. Rend. Mat. Appl. (7) 27, 183–195 (2007)
Hamilton R. S.: Three-manifolds with positive Ricci curvature. J. Differential Geom. 17, 255–306 (1982)
R. S. Hamilton, The Ricci flow on surfaces, In Mathematics and general relativity (Santa Cruz, CA, 1986), volume 71 of Contemp. Math., Amer. Math. Soc., Providence, RI, 1988, 237–262.
Herzlich M.: A Penrose-like inequality for the mass of Riemannian asymptotically flat manifolds. Comm. Math. Phys. 188, 121–133 (1997)
Hijazi O.: A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors. Comm. Math. Phys. 104, 151–162 (1986)
O. Hijazi, Spectral properties of the Dirac operator and geometrical structures, In Geometric methods for quantum field theory (Villa de Leyva, 1999), 116–169, World Sci. Publ., River Edge, NJ, 2001.
Lott J.: Eigenvalue bounds for the Dirac operator. Pacific J. Math. 125, 117–126 (1986)
Ma L.: Eigenvalue monotonicity for the Ricci–Hamilton flow. Ann. Global Anal. Geom. 29, 287–292 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
The author was partially supported by CNPq, Brazil, grant number 483844/2013-6.
Rights and permissions
About this article
Cite this article
Girão, F. An eigenvalue comparison theorem for the Dirac operator on surfaces. Arch. Math. 107, 295–300 (2016). https://doi.org/10.1007/s00013-016-0947-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-016-0947-6