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May  2015, 14(3): 793-809. doi: 10.3934/cpaa.2015.14.793

Differential Harnack estimates for backward heat equations with potentials under geometric flows

Shouwen Fang 1, and  Peng Zhu 2,

1. 

School of Mathematical Science, Yangzhou University, Yangzhou 225002, China
2. 

School of Mathematics and physics, Jiangsu University of Technology, Changzhou, Jiangsu 213001, China

Received  January 2014 Revised  December 2014 Published  March 2015

In the paper we consider a closed Riemannian manifold $M$ with a time-dependent Riemannian metric $g_{i j}(t)$ evolving by $\partial_{t}g_{i j}=-2S_{i j}$ where $S_{i j}$ is a symmetric two-tensor on $(M,g(t))$. We prove some differential Harnack inequalities for positive solutions of backward heat equations with potentials when the metric satisfies the geometric flow. Some applications of these inequalities will be obtained. In particular, we show that the shrinking breathers of the Lorentzian mean curvature flow are the shrinking gradient solitons when the ambient Lorentzian manifold has nonnegative sectional curvature.
Keywords: Geometric flow, backward heat equation., Harnack estimate.
Mathematics Subject Classification: Primary: 53C21, 53C44; Secondary: 35K0.
Citation: Shouwen Fang, Peng Zhu. Differential Harnack estimates for backward heat equations with potentials under geometric flows. Communications on Pure & Applied Analysis, 2015, 14 (3) : 793-809. doi: 10.3934/cpaa.2015.14.793
References:
[1]

B. Andrews, Harnack inequalities for evolving hypersurfaces,, \emph{Math. Z.}, 217 (1994), 179.  doi: 10.1007/BF02571941.  Google Scholar

[2]

S. Brendle, A generalization of Hamilton's differential Harnack inequality for the Ricci flow,, \emph{J. Diff. Geom.}, 82 (2009), 207.   Google Scholar

[3]

H. D. Cao, On Harnack's inequalities for the Kähler-Ricci flow,, \emph{Invent. Math.}, 109 (1992), 247.  doi: 10.1007/BF01232027.  Google Scholar

[4]

H. D. Cao and L. Ni, Matrix Li-Yau-Hamilton estimates for the heat equation on Kähler manifolds,, \emph{Math. Ann.}, 331 (2005), 795.  doi: 10.1007/s00208-004-0605-3.  Google Scholar

[5]

X. D. Cao, Differential Harnack estimates for backward heat equations with potentials under the Ricci flow,, \emph{J. Funct. Anal.}, 255 (2008), 1024.  doi: 10.1016/j.jfa.2008.05.009.  Google Scholar

[6]

X. D. Cao and R. Hamilton, Differential Harnack estimates for time-dependent heat equations with potentials,, \emph{Geom. Funct. Anal.}, 19 (2009), 989.  doi: 10.1007/s00039-009-0024-4.  Google Scholar

[7]

B. Chow, On Harnack's inequality and entropy for the Gaussian curvature flow,, \emph{Comm. Pure Appl. Math.}, 44 (1991), 469.  doi: 10.1002/cpa.3160440405.  Google Scholar

[8]

B. Chow, The Ricci flow on the 2-sphere,, \emph{J. Diff. Geom.}, 33 (1991), 325.   Google Scholar

[9]

B. Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature,, \emph{Comm. Pure Appl. Math.}, 45 (1992), 1003.  doi: 10.1002/cpa.3160450805.  Google Scholar

[10]

B. Chow and S. C. Chu, A geometric interpretation of Hamilton's Harnack inequality for the Ricci flow,, \emph{Math. Res. Lett.}, 2 (1995), 701.  doi: 10.4310/MRL.1995.v2.n6.a4.  Google Scholar

[11]

B. Chow and S. C. Chu, Space-time formulation of Harnack inequalities for curvature flows of hypersurfaces,, \emph{J. Geom. Anal.}, 11 (2001), 219.  doi: 10.1007/BF02921963.  Google Scholar

[12]

B. Chow and R. Hamilton, Constrained and linear Harnack inequalities for parabolic equations,, \emph{Invent. Math.}, 129 (1997), 213.  doi: 10.1007/s002220050162.  Google Scholar

[13]

K. Ecker, A formula relating entropy monotonicity to Harnack inequalities,, \emph{Comm. Anal. Geom.}, 15 (2007), 1025.  doi: 10.4310/CAG.2007.v15.n5.a5.  Google Scholar

[14]

S. W. Fang, Local Harnack estimate for Yamabe flow on locally conformally flat manifolds,, \emph{Asian J. Math.}, 12 (2008), 545.  doi: 10.4310/AJM.2008.v12.n4.a8.  Google Scholar

[15]

S. W. Fang, Harnack estimates for curvature flows depending on mean curvature,, \emph{Appl. Math. J. Chinese Univ. B}, 24 (2009), 361.  doi: 10.1007/s11766-009-2019-1.  Google Scholar

[16]

S. W. Fang and F. Ye, Differential Harnack inequalities for heat equations with potentials under Kähler-Ricci flow,, \emph{Acta Math. Sincia (Chinese Series)}, 53 (2010), 597.   Google Scholar

[17]

S. W. Fang, Differential Harnack inequalities for heat equations with potentials under the Bernhard List's flow,, \emph{Geom. Dedicata}, 161 (2012), 11.  doi: 10.1007/s10711-011-9690-0.  Google Scholar

[18]

S. W. Fang, Differential Harnack inequalities for backward heat equations with potentials under an extended Ricci flow,, \emph{Adv. Geom.}, 13 (2013), 741.  doi: 10.1515/advgeom-2013-0020.  Google Scholar

[19]

S. W. Fang, Differential Harnack inequalities for heat equations with potentials under geometric flows,, \emph{Arch. Math.}, 100 (2013), 179.  doi: 10.1007/s00013-013-0482-7.  Google Scholar

[20]

C. M. Guenther, The fundamental solution on manifolds with time-dependent metrics,, \emph{J. Geom. Anal.}, 12 (2002), 425.  doi: 10.1007/BF02922048.  Google Scholar

[21]

R. Hamilton, The Ricci flow on surfaces,, in \emph{Mathematics and General Relativity} (ed. James A. Isenberg), (1988), 237.  doi: 10.1090/conm/071/954419.  Google Scholar

[22]

R. Hamilton, A matrix Harnack estimate for the heat equation,, \emph{Comm. Anal. Geom.}, 1 (1993), 113.   Google Scholar

[23]

R. Hamilton, The Harnack estimate for the Ricci flow,, \emph{J. Diff. Geom.}, 37 (1993), 225.   Google Scholar

[24]

R. Hamilton, The Harnack estimate for the mean curvature flow,, \emph{J. Diff. Geom.}, 41 (1995), 215.   Google Scholar

[25]

S. L. Kuang and Q. S. Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow,, \emph{J. Funct. Anal.}, 255 (2008), 1008.  doi: 10.1016/j.jfa.2008.05.014.  Google Scholar

[26]

P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator,, \emph{Acta. Math.}, 156 (1986), 153.  doi: 10.1007/BF02399203.  Google Scholar

[27]

R. Müller, Monotone volume formulas for geometric flows,, \emph{J. Reine Angew. Math.}, 643 (2010), 39.  doi: 10.1515/CRELLE.2010.044.  Google Scholar

[28]

L. Ni, A matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow,, \emph{J. Diff. Geom.}, 75 (2007), 303.   Google Scholar

[29]

G. Perelman, The entropy formula for the Ricci flow and its geometric applications,, preprint, ().   Google Scholar

[30]

O. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators,, \emph{J. Funct. Anal.}, 42 (1981), 110.  doi: 10.1016/0022-1236(81)90050-1.  Google Scholar

[31]

K. Smoczyk, Harnack inequalities for curvature flows depending on mean curvature,, \emph{New York J. Math}, 3 (1997), 103.   Google Scholar

[32]

K. Smoczyk, Harnack inequality for the Lagrangian mean curvature flow,, \emph{Calc. Var. Partial Differ. Equ.}, 8 (1999), 247.  doi: 10.1007/s005260050125.  Google Scholar

[33]

J. Wang, Harnack estimate for $H^k$-flow,, \emph{Science in China Series A: Mathematics}, 50 (2007), 1642.  doi: 10.1007/s11425-007-0095-3.  Google Scholar

[34]

J. Wang, Local Harnack estimate for mean curvature flow in Euclidean space,, \emph{Asian J. Math.}, 13 (2009), 89.  doi: 10.4310/AJM.2009.v13.n1.a5.  Google Scholar

[35]

A. Q. Zhu, Differential Harnack inequalities for the backward heat equation with potential under the harmonic-Ricci flow,, \emph{J. Math. Anal. Appl.}, 406 (2013), 502.  doi: 10.1016/j.jmaa.2013.05.004.  Google Scholar

show all references

References:
[1]

B. Andrews, Harnack inequalities for evolving hypersurfaces,, \emph{Math. Z.}, 217 (1994), 179.  doi: 10.1007/BF02571941.  Google Scholar

[2]

S. Brendle, A generalization of Hamilton's differential Harnack inequality for the Ricci flow,, \emph{J. Diff. Geom.}, 82 (2009), 207.   Google Scholar

[3]

H. D. Cao, On Harnack's inequalities for the Kähler-Ricci flow,, \emph{Invent. Math.}, 109 (1992), 247.  doi: 10.1007/BF01232027.  Google Scholar

[4]

H. D. Cao and L. Ni, Matrix Li-Yau-Hamilton estimates for the heat equation on Kähler manifolds,, \emph{Math. Ann.}, 331 (2005), 795.  doi: 10.1007/s00208-004-0605-3.  Google Scholar

[5]

X. D. Cao, Differential Harnack estimates for backward heat equations with potentials under the Ricci flow,, \emph{J. Funct. Anal.}, 255 (2008), 1024.  doi: 10.1016/j.jfa.2008.05.009.  Google Scholar

[6]

X. D. Cao and R. Hamilton, Differential Harnack estimates for time-dependent heat equations with potentials,, \emph{Geom. Funct. Anal.}, 19 (2009), 989.  doi: 10.1007/s00039-009-0024-4.  Google Scholar

[7]

B. Chow, On Harnack's inequality and entropy for the Gaussian curvature flow,, \emph{Comm. Pure Appl. Math.}, 44 (1991), 469.  doi: 10.1002/cpa.3160440405.  Google Scholar

[8]

B. Chow, The Ricci flow on the 2-sphere,, \emph{J. Diff. Geom.}, 33 (1991), 325.   Google Scholar

[9]

B. Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature,, \emph{Comm. Pure Appl. Math.}, 45 (1992), 1003.  doi: 10.1002/cpa.3160450805.  Google Scholar

[10]

B. Chow and S. C. Chu, A geometric interpretation of Hamilton's Harnack inequality for the Ricci flow,, \emph{Math. Res. Lett.}, 2 (1995), 701.  doi: 10.4310/MRL.1995.v2.n6.a4.  Google Scholar

[11]

B. Chow and S. C. Chu, Space-time formulation of Harnack inequalities for curvature flows of hypersurfaces,, \emph{J. Geom. Anal.}, 11 (2001), 219.  doi: 10.1007/BF02921963.  Google Scholar

[12]

B. Chow and R. Hamilton, Constrained and linear Harnack inequalities for parabolic equations,, \emph{Invent. Math.}, 129 (1997), 213.  doi: 10.1007/s002220050162.  Google Scholar

[13]

K. Ecker, A formula relating entropy monotonicity to Harnack inequalities,, \emph{Comm. Anal. Geom.}, 15 (2007), 1025.  doi: 10.4310/CAG.2007.v15.n5.a5.  Google Scholar

[14]

S. W. Fang, Local Harnack estimate for Yamabe flow on locally conformally flat manifolds,, \emph{Asian J. Math.}, 12 (2008), 545.  doi: 10.4310/AJM.2008.v12.n4.a8.  Google Scholar

[15]

S. W. Fang, Harnack estimates for curvature flows depending on mean curvature,, \emph{Appl. Math. J. Chinese Univ. B}, 24 (2009), 361.  doi: 10.1007/s11766-009-2019-1.  Google Scholar

[16]

S. W. Fang and F. Ye, Differential Harnack inequalities for heat equations with potentials under Kähler-Ricci flow,, \emph{Acta Math. Sincia (Chinese Series)}, 53 (2010), 597.   Google Scholar

[17]

S. W. Fang, Differential Harnack inequalities for heat equations with potentials under the Bernhard List's flow,, \emph{Geom. Dedicata}, 161 (2012), 11.  doi: 10.1007/s10711-011-9690-0.  Google Scholar

[18]

S. W. Fang, Differential Harnack inequalities for backward heat equations with potentials under an extended Ricci flow,, \emph{Adv. Geom.}, 13 (2013), 741.  doi: 10.1515/advgeom-2013-0020.  Google Scholar

[19]

S. W. Fang, Differential Harnack inequalities for heat equations with potentials under geometric flows,, \emph{Arch. Math.}, 100 (2013), 179.  doi: 10.1007/s00013-013-0482-7.  Google Scholar

[20]

C. M. Guenther, The fundamental solution on manifolds with time-dependent metrics,, \emph{J. Geom. Anal.}, 12 (2002), 425.  doi: 10.1007/BF02922048.  Google Scholar

[21]

R. Hamilton, The Ricci flow on surfaces,, in \emph{Mathematics and General Relativity} (ed. James A. Isenberg), (1988), 237.  doi: 10.1090/conm/071/954419.  Google Scholar

[22]

R. Hamilton, A matrix Harnack estimate for the heat equation,, \emph{Comm. Anal. Geom.}, 1 (1993), 113.   Google Scholar

[23]

R. Hamilton, The Harnack estimate for the Ricci flow,, \emph{J. Diff. Geom.}, 37 (1993), 225.   Google Scholar

[24]

R. Hamilton, The Harnack estimate for the mean curvature flow,, \emph{J. Diff. Geom.}, 41 (1995), 215.   Google Scholar

[25]

S. L. Kuang and Q. S. Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow,, \emph{J. Funct. Anal.}, 255 (2008), 1008.  doi: 10.1016/j.jfa.2008.05.014.  Google Scholar

[26]

P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator,, \emph{Acta. Math.}, 156 (1986), 153.  doi: 10.1007/BF02399203.  Google Scholar

[27]

R. Müller, Monotone volume formulas for geometric flows,, \emph{J. Reine Angew. Math.}, 643 (2010), 39.  doi: 10.1515/CRELLE.2010.044.  Google Scholar

[28]

L. Ni, A matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow,, \emph{J. Diff. Geom.}, 75 (2007), 303.   Google Scholar

[29]

G. Perelman, The entropy formula for the Ricci flow and its geometric applications,, preprint, ().   Google Scholar

[30]

O. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators,, \emph{J. Funct. Anal.}, 42 (1981), 110.  doi: 10.1016/0022-1236(81)90050-1.  Google Scholar

[31]

K. Smoczyk, Harnack inequalities for curvature flows depending on mean curvature,, \emph{New York J. Math}, 3 (1997), 103.   Google Scholar

[32]

K. Smoczyk, Harnack inequality for the Lagrangian mean curvature flow,, \emph{Calc. Var. Partial Differ. Equ.}, 8 (1999), 247.  doi: 10.1007/s005260050125.  Google Scholar

[33]

J. Wang, Harnack estimate for $H^k$-flow,, \emph{Science in China Series A: Mathematics}, 50 (2007), 1642.  doi: 10.1007/s11425-007-0095-3.  Google Scholar

[34]

J. Wang, Local Harnack estimate for mean curvature flow in Euclidean space,, \emph{Asian J. Math.}, 13 (2009), 89.  doi: 10.4310/AJM.2009.v13.n1.a5.  Google Scholar

[35]

A. Q. Zhu, Differential Harnack inequalities for the backward heat equation with potential under the harmonic-Ricci flow,, \emph{J. Math. Anal. Appl.}, 406 (2013), 502.  doi: 10.1016/j.jmaa.2013.05.004.  Google Scholar

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