Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


On the gradient estimate of Cheng and Yau

Author: Ovidiu Munteanu
Journal: Proc. Amer. Math. Soc. 140 (2012), 1437-1443
MSC (2010): Primary 53C21; Secondary 58J05
Published electronically: September 1, 2011
MathSciNet review: 2869128
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We improve the well-known local gradient estimate of Cheng and Yau in the case when the Ricci curvature has a negative lower bound.

References [Enhancements On Off] (What's this?)

  • 1. E. Calabi, An extension of E. Hopf’s maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1958), 45–56. MR 0092069
  • 2. S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354. MR 0385749
  • 3. Peter Wai-Kwong Li, Harmonic functions and applications to complete manifolds, XIV Escola de Geometria Diferencial. [XIV School of Differential Geometry], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2006. MR 2369440
  • 4. Peter Li and Jiaping Wang, Complete manifolds with positive spectrum. II, J. Differential Geom. 62 (2002), no. 1, 143–162. MR 1987380
  • 5. R. Schoen and S.-T. Yau, Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu; Translated from the Chinese by Ding and S. Y. Cheng; Preface translated from the Chinese by Kaising Tso. MR 1333601
  • 6. Xiaodong Wang, Harmonic functions, entropy, and a characterization of the hyperbolic space, J. Geom. Anal. 18 (2008), no. 1, 272–284. MR 2365675, 10.1007/s12220-007-9001-z
  • 7. Shing Tung Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228. MR 0431040

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C21, 58J05

Retrieve articles in all journals with MSC (2010): 53C21, 58J05

Additional Information

Ovidiu Munteanu
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027

Received by editor(s): December 28, 2010
Published electronically: September 1, 2011
Additional Notes: The author’s research was partially supported by NSF grant No. DMS-1005484
Communicated by: Michael Wolf
Article copyright: © Copyright 2011 American Mathematical Society