TY - JOUR

T1 - Geometric flows and differential Harnack estimates for heat equations with potentials

AU - Ishida, Masashi

N1 - Funding Information:
Acknowledgments The author would like to thank the Max-Plank-Institut für Mathematik in Bonn for its hospitality during the inception of this work. This work was partially supported by the Grant-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, No. 20540090.

PY - 2014/4

Y1 - 2014/4

N2 - Let M be a closed Riemannian manifold with a Riemannian metric gij(t) evolving by a geometric flow ∂tgij = -2Sij, where Sij(t) is a symmetric two-tensor on (M, g(t)). Suppose that Sij satisfies the tensor inequality 2H(S, X)+E(S,X) ≥ 0 for all vector fields X on M, where H(S,X) and E(S, X) are introduced in Definition 1 below. Then, we shall prove differential Harnack estimates for positive solutions to time-dependent forward heat equations with potentials. In the case where Sij = Rij, the Ricci tensor of M, our results correspond to the results proved by Cao and Hamilton (Geom Funct Anal 19:983-989, 2009). Moreover, in the case where the Ricci flow coupled with harmonic map heat flow introduced by Müller (Ann Sci Ec Norm Super 45(4):101-142, 2012), our results derive new differential Harnack estimates. We shall also find new entropies which are monotone under the above geometric flow.

AB - Let M be a closed Riemannian manifold with a Riemannian metric gij(t) evolving by a geometric flow ∂tgij = -2Sij, where Sij(t) is a symmetric two-tensor on (M, g(t)). Suppose that Sij satisfies the tensor inequality 2H(S, X)+E(S,X) ≥ 0 for all vector fields X on M, where H(S,X) and E(S, X) are introduced in Definition 1 below. Then, we shall prove differential Harnack estimates for positive solutions to time-dependent forward heat equations with potentials. In the case where Sij = Rij, the Ricci tensor of M, our results correspond to the results proved by Cao and Hamilton (Geom Funct Anal 19:983-989, 2009). Moreover, in the case where the Ricci flow coupled with harmonic map heat flow introduced by Müller (Ann Sci Ec Norm Super 45(4):101-142, 2012), our results derive new differential Harnack estimates. We shall also find new entropies which are monotone under the above geometric flow.

KW - Differential Harnack Estimates

KW - Geometric flows

KW - Heat equations with potentials

KW - Ricci flow

UR - http://www.scopus.com/inward/record.url?scp=84897020852&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84897020852&partnerID=8YFLogxK

U2 - 10.1007/s10455-013-9401-1

DO - 10.1007/s10455-013-9401-1

M3 - Article

AN - SCOPUS:84897020852

SN - 0232-704X

VL - 45

SP - 287

EP - 302

JO - Annals of Global Analysis and Geometry

JF - Annals of Global Analysis and Geometry

IS - 4

ER -