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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

 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.
Citation: Shouwen Fang, Peng Zhu. Differential Harnack estimates for backward heat equations with potentials under geometric flows. Communications on Pure and Applied Analysis, 2015, 14 (3) : 793-809. doi: 10.3934/cpaa.2015.14.793
##### References:
 [1] B. Andrews, Harnack inequalities for evolving hypersurfaces, Math. Z., 217 (1994), 179-197. doi: 10.1007/BF02571941. [2] S. Brendle, A generalization of Hamilton's differential Harnack inequality for the Ricci flow, J. Diff. Geom., 82 (2009), 207-227. [3] H. D. Cao, On Harnack's inequalities for the Kähler-Ricci flow, Invent. Math., 109 (1992), 247-263. doi: 10.1007/BF01232027. [4] H. D. Cao and L. Ni, Matrix Li-Yau-Hamilton estimates for the heat equation on Kähler manifolds, Math. Ann., 331 (2005), 795-807. doi: 10.1007/s00208-004-0605-3. [5] X. D. Cao, Differential Harnack estimates for backward heat equations with potentials under the Ricci flow, J. Funct. Anal., 255 (2008), 1024-1038. doi: 10.1016/j.jfa.2008.05.009. [6] X. D. Cao and R. Hamilton, Differential Harnack estimates for time-dependent heat equations with potentials, Geom. Funct. Anal., 19 (2009), 989-1000. doi: 10.1007/s00039-009-0024-4. [7] B. Chow, On Harnack's inequality and entropy for the Gaussian curvature flow, Comm. Pure Appl. Math., 44 (1991), 469-483. doi: 10.1002/cpa.3160440405. [8] B. Chow, The Ricci flow on the 2-sphere, J. Diff. Geom., 33 (1991), 325-334. [9] B. Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature, Comm. Pure Appl. Math., 45 (1992), 1003-1014. doi: 10.1002/cpa.3160450805. [10] B. Chow and S. C. Chu, A geometric interpretation of Hamilton's Harnack inequality for the Ricci flow, Math. Res. Lett., 2 (1995), 701-718. doi: 10.4310/MRL.1995.v2.n6.a4. [11] B. Chow and S. C. Chu, Space-time formulation of Harnack inequalities for curvature flows of hypersurfaces, J. Geom. Anal., 11 (2001), 219-231. doi: 10.1007/BF02921963. [12] B. Chow and R. Hamilton, Constrained and linear Harnack inequalities for parabolic equations, Invent. Math., 129 (1997), 213-238. doi: 10.1007/s002220050162. [13] K. Ecker, A formula relating entropy monotonicity to Harnack inequalities, Comm. Anal. Geom., 15 (2007), 1025-1061. doi: 10.4310/CAG.2007.v15.n5.a5. [14] S. W. Fang, Local Harnack estimate for Yamabe flow on locally conformally flat manifolds, Asian J. Math., 12 (2008), 545-552. doi: 10.4310/AJM.2008.v12.n4.a8. [15] S. W. Fang, Harnack estimates for curvature flows depending on mean curvature, Appl. Math. J. Chinese Univ. B, 24 (2009), 361-369. doi: 10.1007/s11766-009-2019-1. [16] S. W. Fang and F. Ye, Differential Harnack inequalities for heat equations with potentials under Kähler-Ricci flow, Acta Math. Sincia (Chinese Series), 53 (2010), 597-606. [17] S. W. Fang, Differential Harnack inequalities for heat equations with potentials under the Bernhard List's flow, Geom. Dedicata, 161 (2012), 11-22. doi: 10.1007/s10711-011-9690-0. [18] S. W. Fang, Differential Harnack inequalities for backward heat equations with potentials under an extended Ricci flow, Adv. Geom., 13 (2013), 741-755. doi: 10.1515/advgeom-2013-0020. [19] S. W. Fang, Differential Harnack inequalities for heat equations with potentials under geometric flows, Arch. Math., 100 (2013), 179-189. doi: 10.1007/s00013-013-0482-7. [20] C. M. Guenther, The fundamental solution on manifolds with time-dependent metrics, J. Geom. Anal., 12 (2002), 425-436. doi: 10.1007/BF02922048. [21] R. Hamilton, The Ricci flow on surfaces, in Mathematics and General Relativity (ed. James A. Isenberg), Contemp. Math. 71, Amer. Math. Soc., Providence, RI, (1988), 237-262. doi: 10.1090/conm/071/954419. [22] R. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom., 1 (1993), 113-126. [23] R. Hamilton, The Harnack estimate for the Ricci flow, J. Diff. Geom., 37 (1993), 225-243. [24] R. Hamilton, The Harnack estimate for the mean curvature flow, J. Diff. Geom., 41 (1995), 215-226. [25] S. L. Kuang and Q. S. Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow, J. Funct. Anal., 255 (2008), 1008-1023. doi: 10.1016/j.jfa.2008.05.014. [26] P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta. Math., 156 (1986), 153-201. doi: 10.1007/BF02399203. [27] R. Müller, Monotone volume formulas for geometric flows, J. Reine Angew. Math., 643 (2010), 39-57. doi: 10.1515/CRELLE.2010.044. [28] L. Ni, A matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow, J. Diff. Geom., 75 (2007), 303-358. [29] G. Perelman, The entropy formula for the Ricci flow and its geometric applications,, preprint, (). [30] O. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators, J. Funct. Anal., 42 (1981), 110-120. doi: 10.1016/0022-1236(81)90050-1. [31] K. Smoczyk, Harnack inequalities for curvature flows depending on mean curvature, New York J. Math, 3 (1997), 103-118. [32] K. Smoczyk, Harnack inequality for the Lagrangian mean curvature flow, Calc. Var. Partial Differ. Equ., 8 (1999), 247-258. doi: 10.1007/s005260050125. [33] J. Wang, Harnack estimate for $H^k$-flow, Science in China Series A: Mathematics, 50 (2007), 1642-1650. doi: 10.1007/s11425-007-0095-3. [34] J. Wang, Local Harnack estimate for mean curvature flow in Euclidean space, Asian J. Math., 13 (2009), 89-100. doi: 10.4310/AJM.2009.v13.n1.a5. [35] A. Q. Zhu, Differential Harnack inequalities for the backward heat equation with potential under the harmonic-Ricci flow, J. Math. Anal. Appl., 406 (2013), 502-510. doi: 10.1016/j.jmaa.2013.05.004.

show all references

##### References:
 [1] B. Andrews, Harnack inequalities for evolving hypersurfaces, Math. Z., 217 (1994), 179-197. doi: 10.1007/BF02571941. [2] S. Brendle, A generalization of Hamilton's differential Harnack inequality for the Ricci flow, J. Diff. Geom., 82 (2009), 207-227. [3] H. D. Cao, On Harnack's inequalities for the Kähler-Ricci flow, Invent. Math., 109 (1992), 247-263. doi: 10.1007/BF01232027. [4] H. D. Cao and L. Ni, Matrix Li-Yau-Hamilton estimates for the heat equation on Kähler manifolds, Math. Ann., 331 (2005), 795-807. doi: 10.1007/s00208-004-0605-3. [5] X. D. Cao, Differential Harnack estimates for backward heat equations with potentials under the Ricci flow, J. Funct. Anal., 255 (2008), 1024-1038. doi: 10.1016/j.jfa.2008.05.009. [6] X. D. Cao and R. Hamilton, Differential Harnack estimates for time-dependent heat equations with potentials, Geom. Funct. Anal., 19 (2009), 989-1000. doi: 10.1007/s00039-009-0024-4. [7] B. Chow, On Harnack's inequality and entropy for the Gaussian curvature flow, Comm. Pure Appl. Math., 44 (1991), 469-483. doi: 10.1002/cpa.3160440405. [8] B. Chow, The Ricci flow on the 2-sphere, J. Diff. Geom., 33 (1991), 325-334. [9] B. Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature, Comm. Pure Appl. Math., 45 (1992), 1003-1014. doi: 10.1002/cpa.3160450805. [10] B. Chow and S. C. Chu, A geometric interpretation of Hamilton's Harnack inequality for the Ricci flow, Math. Res. Lett., 2 (1995), 701-718. doi: 10.4310/MRL.1995.v2.n6.a4. [11] B. Chow and S. C. Chu, Space-time formulation of Harnack inequalities for curvature flows of hypersurfaces, J. Geom. Anal., 11 (2001), 219-231. doi: 10.1007/BF02921963. [12] B. Chow and R. Hamilton, Constrained and linear Harnack inequalities for parabolic equations, Invent. Math., 129 (1997), 213-238. doi: 10.1007/s002220050162. [13] K. Ecker, A formula relating entropy monotonicity to Harnack inequalities, Comm. Anal. Geom., 15 (2007), 1025-1061. doi: 10.4310/CAG.2007.v15.n5.a5. [14] S. W. Fang, Local Harnack estimate for Yamabe flow on locally conformally flat manifolds, Asian J. Math., 12 (2008), 545-552. doi: 10.4310/AJM.2008.v12.n4.a8. [15] S. W. Fang, Harnack estimates for curvature flows depending on mean curvature, Appl. Math. J. Chinese Univ. B, 24 (2009), 361-369. doi: 10.1007/s11766-009-2019-1. [16] S. W. Fang and F. Ye, Differential Harnack inequalities for heat equations with potentials under Kähler-Ricci flow, Acta Math. Sincia (Chinese Series), 53 (2010), 597-606. [17] S. W. Fang, Differential Harnack inequalities for heat equations with potentials under the Bernhard List's flow, Geom. Dedicata, 161 (2012), 11-22. doi: 10.1007/s10711-011-9690-0. [18] S. W. Fang, Differential Harnack inequalities for backward heat equations with potentials under an extended Ricci flow, Adv. Geom., 13 (2013), 741-755. doi: 10.1515/advgeom-2013-0020. [19] S. W. Fang, Differential Harnack inequalities for heat equations with potentials under geometric flows, Arch. Math., 100 (2013), 179-189. doi: 10.1007/s00013-013-0482-7. [20] C. M. Guenther, The fundamental solution on manifolds with time-dependent metrics, J. Geom. Anal., 12 (2002), 425-436. doi: 10.1007/BF02922048. [21] R. Hamilton, The Ricci flow on surfaces, in Mathematics and General Relativity (ed. James A. Isenberg), Contemp. Math. 71, Amer. Math. Soc., Providence, RI, (1988), 237-262. doi: 10.1090/conm/071/954419. [22] R. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom., 1 (1993), 113-126. [23] R. Hamilton, The Harnack estimate for the Ricci flow, J. Diff. Geom., 37 (1993), 225-243. [24] R. Hamilton, The Harnack estimate for the mean curvature flow, J. Diff. Geom., 41 (1995), 215-226. [25] S. L. Kuang and Q. S. Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow, J. Funct. Anal., 255 (2008), 1008-1023. doi: 10.1016/j.jfa.2008.05.014. [26] P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta. Math., 156 (1986), 153-201. doi: 10.1007/BF02399203. [27] R. Müller, Monotone volume formulas for geometric flows, J. Reine Angew. Math., 643 (2010), 39-57. doi: 10.1515/CRELLE.2010.044. [28] L. Ni, A matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow, J. Diff. Geom., 75 (2007), 303-358. [29] G. Perelman, The entropy formula for the Ricci flow and its geometric applications,, preprint, (). [30] O. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators, J. Funct. Anal., 42 (1981), 110-120. doi: 10.1016/0022-1236(81)90050-1. [31] K. Smoczyk, Harnack inequalities for curvature flows depending on mean curvature, New York J. Math, 3 (1997), 103-118. [32] K. Smoczyk, Harnack inequality for the Lagrangian mean curvature flow, Calc. Var. Partial Differ. Equ., 8 (1999), 247-258. doi: 10.1007/s005260050125. [33] J. Wang, Harnack estimate for $H^k$-flow, Science in China Series A: Mathematics, 50 (2007), 1642-1650. doi: 10.1007/s11425-007-0095-3. [34] J. Wang, Local Harnack estimate for mean curvature flow in Euclidean space, Asian J. Math., 13 (2009), 89-100. doi: 10.4310/AJM.2009.v13.n1.a5. [35] A. Q. Zhu, Differential Harnack inequalities for the backward heat equation with potential under the harmonic-Ricci flow, J. Math. Anal. Appl., 406 (2013), 502-510. doi: 10.1016/j.jmaa.2013.05.004.
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