A global curvature pinching result of the first eigenvalue of the Laplacian on Riemannian manifolds. (English) Zbl 1275.53040

Wang, Peihe; Li, Ying

Abstr. Appl. Anal. 2013, Article ID 237418, 5 p. (2013).

Summary: The paper starts with a discussion involving the Sobolev constant on geodesic balls and then follows with a derivation of a lower bound for the first eigenvalue of the Laplacian on manifolds with small negative curvature. The derivation involves Moser iteration.

MSC:

53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J50	Spectral problems; spectral geometry; scattering theory on manifolds

Keywords:

Sobolev constant; geodesic balls; Moser iteration

References:

[1]	Yau, S. T.; Schoen, R., Lectures in Differential Geometry (1988), Scientific Press
[2]	Petersen, P.; Sprouse, C., Integral curvature bounds, distance estimates and applications, Journal of Differential Geometry, 50, 2, 269-298 (1998) · Zbl 0969.53017
[3]	Yang, D., Convergence of Riemannian manifolds with integral bounds on curvature. I, Annales Scientifiques de l’École Normale Supérieure. Quatrième Série, 25, 1, 77-105 (1992) · Zbl 0748.53025
[4]	Chavel, I., Riemannian Geometry: A Mordern Introduction (2000), Cambridge, UK: Cambridge University Press, Cambridge, UK · doi:10.1017/CBO9780511616822
[5]	Sprouse, C., Integral curvature bounds and bounded diameter, Communications in Analysis and Geometry, 8, 3, 531-543 (2000) · Zbl 0984.53018
[6]	Li, P.; Yau, S. T., Estimates of eigenvalues of a compact Riemannian manifold, AMS Proceedings of Symposia in Pure Mathematics, 205-239 (1980), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 0441.58014