Hostname: page-component-f554764f5-fnl2l Total loading time: 0 Render date: 2025-04-10T08:09:01.688Z Has data issue: false hasContentIssue false
  • English
  • Français

Heat Kernel Estimates Under the Ricci–Harmonic Map Flow

Part of: Parabolic equations and systems Global differential geometry

Published online by Cambridge University Press:  22 February 2017

Mihai Băileşteanu  and
Hung Tran
Mihai Băileşteanu
Affiliation:
Department of Mathematics, Central Connecticut State University, 120 Marcus White Hall, New Britain, CT 06052, USA ([email protected])
Hung Tran
Affiliation:
Department of Mathematics, University of California, Irvine, 440G Rowland Hall, Irvine, CA 92612, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper considers the Ricci flow coupled with the harmonic map flow between two manifolds. We derive estimates for the fundamental solution of the corresponding conjugate heat equation and we prove an analogue of Perelman's differential Harnack inequality. As an application, we find a connection between the entropy functional and the best constant in the Sobolev embedding theorem in ℝn .

Keywords

heat equationheat kernelRicci flowharmonic mapSobolev embeddingdifferential Harnack inequality

MSC classification

Secondary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions 53C43: Differential geometric aspects of harmonic maps 35K55: Nonlinear parabolic equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 4 , November 2017 , pp. 831 - 857
Copyright
Copyright © Edinburgh Mathematical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1. Aubin, T., Problemes isoperimetriques et espaces de sobolev, J. Diff. Geom. 11 (1976), 573–598.Google Scholar
2. Băileşteanu, M., Bounds on the heat kernel under the Ricci flow, Proc. Am. Math. Soc. 140(2) (2012), 691–700.CrossRefGoogle Scholar
3. Băileşteanu, M., Cao, X. and Pulemotov, A., Gradient estimates for the heat equation under the Ricci flow, J. Funct. Analysis 258(10) (2010), 3517–3542.Google Scholar
4. Cao, X. and Hamilton, R. S., Differential Harnack estimates for time-dependent heat equations with potentials, Geom. Funct. Analysis 19(4) (2009), 989–1000.Google Scholar
5. Cao, X. and Zhang, Q. S., The conjugate heat equation and ancient solutions of the Ricci flow, Adv. Math. 228(5) (2011), 2891–2919.Google Scholar
6. Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F. and Ni, L., The Ricci flow: techniques and applications, Part II: analytic aspects, Mathematical Surveys and Monographs, Volume 144 (American Mathematical Society, Providence, RI, 2008).Google Scholar
7. Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F. and Ni, L., The Ricci flow: techniques and applications, Part III: geometric-analytic aspects, Mathematical Surveys and Monographs, Volume 163 (American Mathematical Society, Providence, RI, 2010).Google Scholar
8. Guenther, C. M., The fundamental solution on manifolds with time-dependent metrics, J. Geom. Analysis 12(3) (2002), 425–436.Google Scholar
9. Hamilton, R. S., The Harnack estimate for the Ricci flow, J. Diff. Geom. 37(1) (1993), 225–243.Google Scholar
10. Hamilton, R. S., A matrix Harnack estimate for the heat equation, Commun. Analysis Geom. 1(1) (1993), 113–126.Google Scholar
11. Hamilton, R. S., The formation of singularities in the Ricci flow, in Surveys in differential geometry, Volume II, pp. 7–136 (International Press, Cambridge, MA, 1995).Google Scholar
12. Hebey, E. and Vaugon, M., Meilleures constantes dans le théorème d’inclusion de Sobolev, Annales Inst. H. Poincaré Analyse Non Linéaire 13(1) (1996), 57–93.Google Scholar
13. Li, P. and Yau, S. T., On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153–201.Google Scholar
14. List, B., Evolution of an extended Ricci flow system, Commun. Analysis Geom. 16(5) (2008), 1007–1048.CrossRefGoogle Scholar
15. Liu, S., Gradient estimates for solutions of the heat equation under Ricci flow, Pac. J. Math. 243(1) (2009), 165–180.Google Scholar
16. Müller, R., Monotone volume formulas for geometric flows, J. Reine Angew. Math. 643 (2010), 39–57.Google Scholar
17. Müller, R., Ricci flow coupled with harmonic map flow, Annales Scient. Éc. Norm. Sup. 45(1) (2012), 101–142.CrossRefGoogle Scholar
18. Ni, L., Ricci flow and nonnegativity of sectional curvature, Math. Res. Lett. 11 (2004), 883–904.Google Scholar
19. Perelman, G., The entropy formula for the Ricci flow and its geometric applications, Preprint (arXiv:math/0211159 [math.DG]; 2002).Google Scholar
20. Sesum, N., Tian, G. and Wang, X.-D., Notes on perelman's paper on the entropy formula for the Ricci flow and its geometric applications, Preprint (available at http://users.math.msu.edu/users/xwang/perel.pdf; 2004).Google Scholar
21. Sun, J., Gradient estimates for positive solutions of the heat equation under geometric flow, Pac. J. Math. 253(1) (2011), 489–510.CrossRefGoogle Scholar
22. Tran, H., Harnack estimates for Ricci flow on a warped product, J. Geom. Analysis 26(3) (2016), 1838–1862.Google Scholar
23. Wang, J., Global heat kernel estimates, Pac. J. Math. 178 (1997), 377–398.CrossRefGoogle Scholar
24. Williams, M., Stability of solutions of certain extended Ricci flow systems, Preprint (arXiv:1301.3945v2 [math.DG]; 2015).Google Scholar
25. Xian-Gao, L. and Kui, W., A Gaussian upper bound of the conjugate heat equation along an extended ricci flow, Preprint (arXiv:1412.3200 [math.DG]; 2014).Google Scholar
26. Zhang, Q. S., Some gradient estimates for the heat equation on domains and for an equation by Perelman, Int. Math. Res. Not. 2006 (2006), 92314.Google Scholar
27. Zhang, Q. S., Sobolev inequalities, heat kernels under Ricci flow and the Poincaré conjecture (CRC Press, Boca Raton, FL, 2010).CrossRefGoogle Scholar
28. Zhu, A., Differential Harnack inequalities for the backward heat equation with potential under the harmonic–Ricci flow, J. Math. Analysis Applic. 406(2) (2013), 502–510.CrossRefGoogle Scholar