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Heat Kernel Estimates Under the Ricci–Harmonic Map Flow

  • Mihai Băileşteanu (a1) and Hung Tran (a2)

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 ℝ n .

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1. Aubin, T., Problemes isoperimetriques et espaces de sobolev, J. Diff. Geom. 11 (1976), 573598.
2. Băileşteanu, M., Bounds on the heat kernel under the Ricci flow, Proc. Am. Math. Soc. 140(2) (2012), 691700.
3. Băileşteanu, M., Cao, X. and Pulemotov, A., Gradient estimates for the heat equation under the Ricci flow, J. Funct. Analysis 258(10) (2010), 35173542.
4. Cao, X. and Hamilton, R. S., Differential Harnack estimates for time-dependent heat equations with potentials, Geom. Funct. Analysis 19(4) (2009), 9891000.
5. Cao, X. and Zhang, Q. S., The conjugate heat equation and ancient solutions of the Ricci flow, Adv. Math. 228(5) (2011), 28912919.
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).
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).
8. Guenther, C. M., The fundamental solution on manifolds with time-dependent metrics, J. Geom. Analysis 12(3) (2002), 425436.
9. Hamilton, R. S., The Harnack estimate for the Ricci flow, J. Diff. Geom. 37(1) (1993), 225243.
10. Hamilton, R. S., A matrix Harnack estimate for the heat equation, Commun. Analysis Geom. 1(1) (1993), 113126.
11. Hamilton, R. S., The formation of singularities in the Ricci flow, in Surveys in differential geometry, Volume II, pp. 7136 (International Press, Cambridge, MA, 1995).
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), 5793.
13. Li, P. and Yau, S. T., On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153201.
14. List, B., Evolution of an extended Ricci flow system, Commun. Analysis Geom. 16(5) (2008), 10071048.
15. Liu, S., Gradient estimates for solutions of the heat equation under Ricci flow, Pac. J. Math. 243(1) (2009), 165180.
16. Müller, R., Monotone volume formulas for geometric flows, J. Reine Angew. Math. 643 (2010), 3957.
17. Müller, R., Ricci flow coupled with harmonic map flow, Annales Scient. Éc. Norm. Sup. 45(1) (2012), 101142.
18. Ni, L., Ricci flow and nonnegativity of sectional curvature, Math. Res. Lett. 11 (2004), 883904.
19. Perelman, G., The entropy formula for the Ricci flow and its geometric applications, Preprint (arXiv:math/0211159 [math.DG]; 2002).
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).
21. Sun, J., Gradient estimates for positive solutions of the heat equation under geometric flow, Pac. J. Math. 253(1) (2011), 489510.
22. Tran, H., Harnack estimates for Ricci flow on a warped product, J. Geom. Analysis 26(3) (2016), 18381862.
23. Wang, J., Global heat kernel estimates, Pac. J. Math. 178 (1997), 377398.
24. Williams, M., Stability of solutions of certain extended Ricci flow systems, Preprint (arXiv:1301.3945v2 [math.DG]; 2015).
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).
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.
27. Zhang, Q. S., Sobolev inequalities, heat kernels under Ricci flow and the Poincaré conjecture (CRC Press, Boca Raton, FL, 2010).
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), 502510.

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