Abstract
Let M be a closed Riemannian manifold with a family of Riemannian metrics \({g_{\it ij}(t)}\) evolving by geometric flow \({\partial_{t}g_{\it ij} = -2{S}_{\it ij}}\), where \({S_{\it ij}(t)}\) is a family of smooth symmetric two-tensors on M. In this paper we derive differential Harnack estimates for positive solutions to the nonlinear heat equation with potential:
where \({\gamma (t)}\) is a continuous function on t, a is a constant and \({S=g^{\it ij}S_{\it ij}}\) is the trace of \({S_{\it ij}}\). Our Harnack estimates include many known results as special cases, and moreover lead to new Harnack inequalities for a variety geometric flows.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bailesteanu M., Cao X., Pulemotov A.: Gradient estimates for the heat equation under the Ricci flow. J. Funct. Anal. 258, 3517–3542 (2010)
Cao X.: Differential Harnack estimates for backward heat equations with potentials under the Ricci flow. J. Funct. Anal. 255, 1024–1038 (2008)
Cao X., Hamilton R.: Differential Harnack estimates for time-dependent heat equations with potentials. Geom. Funct. Anal. 19, 989–100 (2009)
Cao, X., Zhang, Z.: Differential Harnack estimates for parabolic equations with potentials. In: Proceeding of Complex and Differential Geometry, pp. 87–98 (2011)
Guenther C.M.: The fundamental solution on manifolds with time-dependent metrics. J. Geom. Anal. 12, 425–436 (2002)
Guo H., He T.: Harnack estimates for the geometric flows, applications to Ricci flow coupled with harmonic map flow. Geom. Dedicata 169, 411–418 (2014)
Guo H., Ishida M.: Harnack estimates for nonlinear backward heat equations in geometric flows. J. Funct. Anal. 267, 2638–2662 (2014)
Guo H., Philipowski R., Thalmaier A.: Entropy and lowest eigenvalue on evolving manifolds. Pac. J. Math. 264, 61–81 (2013)
Hamilton R.: Three manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)
Hamilton R.: The Harnack estimate for the Ricci flow. J. Differ. Geom. 37, 225–243 (1993)
Ishida M.: Geometric flows and differential Harnack estimates for heat equations with potentials. Ann. Global Anal. Geom. 45, 287–302 (2014)
Kuang S., Zhang Q.S.: A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow. J. Funct. Anal. 255, 1008–1023 (2008)
Ni, L.: Monotonicity and Li-Yau-Hamilton inequalities. In: Surveys in Differential Geometry, Vol. XII. Geometric Flows, pp. 251–301, Surv. Differ. Geom., 12, Int. Press, Somerville, MA, (2008)
Li P., Yau S.-T.: On the parabolic kernel of the Schrodinger operator. Acta Math. 156, 153–201 (1986)
List, B.: Evolution of an Extended Ricci Flow System. PhD thesis, AEI Potsdam (2005)
Liu S.: Gradient estimates for solutions of the heat equation under Ricci flow. Pac. J. Math. 243, 165–180 (2009)
Müller R.: Monotone volume formulas for geometric flows. J. Reine Angew. Math. 643, 39–57 (2010)
Müller R.: Ricci flow coupled with harmonic map flow. Ann. Sci. Ec. Norm. Super. 45(4), 101–142 (2012)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159 (2002)
Sun J.: Gradient estimates for positive solutions of the heat equation under geometric flow. Pac. J. Math. 253, 489–510 (2011)
Zhang, Q.S.: Some gradient estimates for the heat equation on domains and for an equation by Perelman. Int. Math. Res. Not. Art. ID 92314, (2006) 39 pp.
Wu J.-Y.: Differential Harnack inequalities for nonlinear heat equations with potentials under the Ricci flow. Pac. J. Math. 257, 199–218 (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Guo, H., Ishida, M. Harnack estimates for nonlinear heat equations with potentials in geometric flows. manuscripta math. 148, 471–484 (2015). https://doi.org/10.1007/s00229-015-0757-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-015-0757-3