Bounds on the heat kernel under the Ricci flow
HTML articles powered by AMS MathViewer
- by Mihai Băileşteanu
- Proc. Amer. Math. Soc. 140 (2012), 691-700
- DOI: https://doi.org/10.1090/S0002-9939-2011-11057-8
- Published electronically: June 9, 2011
- PDF | Request permission
Abstract:
We establish an estimate for the fundamental solution of the heat equation on a closed Riemannian manifold $M$ of dimension at least $3$, evolving under the Ricci flow. The estimate depends on some constants arising from a Sobolev imbedding theorem. Considering the case when the scalar curvature is positive throughout the manifold, at any time, we will obtain, as a corollary, a bound similar to the one known for the fixed metric case.References
- D. G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc. 73 (1967), 890–896. MR 217444, DOI 10.1090/S0002-9904-1967-11830-5
- Thierry Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry 11 (1976), no. 4, 573–598 (French). MR 448404
- Mihai Bailesteanu, Xiaodong Cao, and Artem Pulemotov, Gradient estimates for the heat equation under the Ricci flow, J. Funct. Anal. 258 (2010), no. 10, 3517–3542. MR 2601627, DOI 10.1016/j.jfa.2009.12.003
- X. Cao and Q. Zhang, The conjugate heat equation and ancient solutions of the Ricci flow, arXiv:math/1006.0540v1 (2010).
- Bennett Chow, Peng Lu, and Lei Ni, Hamilton’s Ricci flow, Graduate Studies in Mathematics, vol. 77, American Mathematical Society, Providence, RI; Science Press Beijing, New York, 2006. MR 2274812, DOI 10.1090/gsm/077
- Christine M. Guenther, The fundamental solution on manifolds with time-dependent metrics, J. Geom. Anal. 12 (2002), no. 3, 425–436. MR 1901749, DOI 10.1007/BF02922048
- Emmanuel Hebey and Michel Vaugon, Meilleures constantes dans le théorème d’inclusion de Sobolev, Ann. Inst. H. Poincaré C Anal. Non Linéaire 13 (1996), no. 1, 57–93 (French, with English and French summaries). MR 1373472, DOI 10.1016/S0294-1449(16)30097-X
- S.-Y. Hsu, Some results for the Perelman lyh-type inequality, arXiv:0801.3506v2 [math.DG] (2008).
- Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201. MR 834612, DOI 10.1007/BF02399203
- G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159v1 (2002).
- Laurent Saloff-Coste, Aspects of Sobolev-type inequalities, London Mathematical Society Lecture Note Series, vol. 289, Cambridge University Press, Cambridge, 2002. MR 1872526
- Jiaping Wang, Global heat kernel estimates, Pacific J. Math. 178 (1997), no. 2, 377–398. MR 1447421, DOI 10.2140/pjm.1997.178.377
- Qi S. Zhang, Some gradient estimates for the heat equation on domains and for an equation by Perelman, Int. Math. Res. Not. , posted on (2006), Art. ID 92314, 39. MR 2250008, DOI 10.1155/IMRN/2006/92314
- —, Sobolev inequalities, heat kernels under Ricci flow and the Poincaré conjecture, Chapman Hall/CRC Press, Taylor and Francis Group, Boca Raton, FL, 2011.
Bibliographic Information
- Mihai Băileşteanu
- Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853-4201
- Email: mbailesteanu@math.cornell.edu
- Received by editor(s): November 23, 2010
- Published electronically: June 9, 2011
- Communicated by: Chuu-Lian Terng
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 691-700
- MSC (2010): Primary 53C44, 35K05, 35K08; Secondary 53B20, 53B21
- DOI: https://doi.org/10.1090/S0002-9939-2011-11057-8
- MathSciNet review: 2846338