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
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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 -