Liouville theorems for the ancient solution of heat flows
Author:
Meng Wang
Journal:
Proc. Amer. Math. Soc. 139 (2011), 3491-3496
MSC (2010):
Primary 35K05, 58J35
DOI:
https://doi.org/10.1090/S0002-9939-2011-11170-5
Published electronically:
May 24, 2011
MathSciNet review:
2813381
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Abstract | References | Similar Articles | Additional Information
Abstract: Let be a complete Riemannian manifold with Ricci curvature bounded from below:
. Let
be a simply connected complete Riemannian manifold with nonpositive sectional curvature. Using a gradient estimate, we prove Liouville's theorem for the ancient solution of heat flows.
- 1. Cheng, S., Yau, S., Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28.3 (1975), 333-354. MR 0385749 (52:6608)
- 2. Li, J., Wang, M., Liouville theorems for self-similar solutions of heat flows. J. Eur. Math. Soc. 11.1 (2009), 207-221. MR 2471137 (2009m:58031)
- 3. Li, P., Yau, S.-T., On the parabolic kernel of the Schrödinger operator, Acta Math. 156.3-4 (1986), 153-201. MR 834612 (87f:58156)
- 4. Philippe, S., Zhang, Q., Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds, Bull. London Math. Soc. 38.6 (2006), 1045-1053. MR 2285258 (2008f:35157)
- 5. Li, P., Tam, L., The heat equation and harmonic maps of complete manifolds, Invent. Math. 105.1 (1991), 1-46. MR 1109619 (93e:58039)
- 6. Hamilton, Richard S., A matrix Harnack estimate for the heat equation. Comm. Anal. Geom. 1.1 (1993), 113-126. MR 1230276 (94g:58215)
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Additional Information
Meng Wang
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou, 310027, People’s Republic of China
Email:
mathdreamcn@zju.edu.cn
DOI:
https://doi.org/10.1090/S0002-9939-2011-11170-5
Keywords:
Liouville theorem,
ancient solution,
quasi-harmonic map,
heat flow
Received by editor(s):
August 18, 2010
Published electronically:
May 24, 2011
Additional Notes:
The author’s research was partially supported by NSFC 10701064, 10931001
Communicated by:
Michael T. Lacey
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.