Long time existence of Ricci-harmonic flow

Abstract

We give a survey about recent results on Ricci-harmonic flow.

Keywords

Ricci-harmonic flow (RHF) / curvature pinching estimates / bounded scalar curvature

References

[1] Abolarinwa A. Entropy formulas and their applications on time dependent Riemannian metrics. Electron J Math Anal Appl, 2015, 3(1): 77–88

[2] Bailesteanu M. On the heat kernel under the Ricci flow coupled with the harmonic map flow. arXiv: 1309.0138

[3] Bailesteanu M. Gradient estimates for the heat equation under the Ricci-harmonic map flow. Adv Geom, 2015, 15(4): 445–454

[4] Bailesteanu M, Tranh H. Heat kernel estimates under the Ricci-harmonic map flow. arXiv: 1310.1619

[5] Bamler R H, Zhang Q S. Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature. arXiv: 1501.01291

[6] Cao X D. Curvature pinching estimate and singularities of the Ricci flow. Comm Anal Geom, 2011, 19(5): 975–990

[7] Cao X D, Guo H X, Tran H. Harnack estimates for conjugate heat kernel on evolving manifolds. Math Z, 2015, 281(1-2): 201–214

[8] Chen X X, Wang B. On the conditions to extend Ricci flow (III). Int Math Res Not IMRN, 2013, (10): 2349–2367

[9] Cheng L, Zhu A Q. On the extension of the harmonic Ricci flow. Geom Dedicata, 2013, 164: 179–185

[10] Enders J, Muller R, Topping P M. On type-I singularities in Ricci flow. Comm Anal Geom, 2011, 19(5): 905–922

[11] Fang S W. Differential Harnack inequalities for heat equations with potentials under geometric flows. Arch Math (Basel), 2013, 100(2): 179–189

[12] Fang S W. Differential Harnack estimates for backward heat equations with potentials under an extended Ricci flow. Adv Geom, 2013, 13(4): 741–755

[13] Fang S W, Zheng T. An upper bound of the heat kernel along the harmonic-Ricci flow. arXiv: 1501.00639

[14] Fang S W, Zheng T. The (logarithmic) Sobolev inequalities along geometric flow and applications. J Math Anal Appl, 2016, 434(1): 729–764

[15] Fang S W, Zhu P. Differential Harnack estimates for backward heat equations with potentials under geometric flows. Comm Pure Appl Anal, 2015, 14(3): 793–809

[16] Guo B, Huang Z J, Phong D H. Pseudo-locality for a coupled Ricci flow. arXiv: 1510.04332

[17] Guo H X, He T T. Harnack estimates for geometric flow, applications to Ricci flow coupled with harmonic map flow. Geom Dedicata, 2014, 169: 411–418

[18] Guo H X, Ishida M. Harnack estimates for nonlinear backward heat equations in geometric flows. J Funct Anal, 2014, 267(8): 2638–2662

[19] Guo H X, Philipowski R, Thalmairt A. Entropy and lowest eigenvalue on evolving manifolds. Pacific J Math, 2013, 264(1): 61–81

[20] Guo H X, Philipowski R, Thalmairt A. A stochastic approach to the harmonic map heat flow on manifolds with time-dependent Riemannian metric. Stochastic Process Appl, 2014, 124(11): 3535–3552

[21] Guo H X, Philipowski R, Thalmairt A. An entropy formula for the heat equation on manifolds with time-dependent metric, application to ancient solutions. Potential Anal, 2015, 42(2): 483–497

[22] Hamilton R S. Three-manifolds with positive Ricci curvature. J Differential Geom, 1982, 17(2): 255–306

[23] He C-L, Hu S, Kong D-X, Liu K F. Generalized Ricci flow. I. Local existence and uniqueness. In: Topology and Physics. Nankai Tracts Math, Vol 12. Hackensack: World Sci Publ, 2008, 151–171

[24] Li Y. Generalized Ricci flow I: higher derivative estimates for compact manifolds. Anal PDE, 2012, 5(4): 747–775

[25] Li Y. Eigenvalues and entropies under the harmonic-Ricci flow. Pacific J Math, 2014, 267(1): 141–184

[26] Li Y. Long time existence and bounded scalar curvature in the Ricci-harmonic flow. arXiv: 1510.05788v2

[27] List B. Evolution of an Extended Ricci Flow System. Ph D Thesis. Fachbereich Mathematik und Informatik der Freie Universität Berlin, 2006, http://www.diss.fu-berlin.de/2006/180/index.html

[28] List B. Evolution of an extended Ricci flow system. Comm Anal Geom, 2008, 16(5): 1007–1048

[29] Liu X-G, Wang K. A Gaussian upper bound of the conjugate heat equation along an extended Ricci flow. arXiv: 1412.3200

[30] Lott J, Sesum N. Ricci flow on three-dimensional manifolds with symmetry. Comment Math Helv, 2014, 89(1): 1–32

[31] Ma L, Cheng L. On the conditions to control curvature tensors of Ricci flow. Ann Global Anal Geom, 2010, 37(4): 403–411

[32] Müller R. The Ricci Flow Coupled with Harmonic Map Flow. Ph D Thesis. ETH Zürich, 2009, DOI: 10.3929/ethz-a-005842361

[33] Müller R. Monotone volume formulas for geometric flow. J Reine Angew Math, 2010, 643: 39–57

[34] Müller R. Ricci flow coupled with harmonic map flow. Ann Sci ′ Ec Norm Sup′er (4), 2012, 45(1): 101–142

[35] Müller R, Rupflin M. Smooth long-time existence of harmonic Ricci flow on surfaces. arXiv: 1510.03643

[36] Oliynyk T, Suneeta V, Woolgar E. A gradient flow for worldsheet nonlinear sigma models. Nuclear Phys B, 2006, 739(3): 441–458

[37] Perelman G. The entropy formula for the Ricci flow and its geometric applications. arXiv: math/0211159

[38] Perelman G. Ricci flow with surgery on three-manifolds. arXiv: math/0303109

[39] Perelman G. Finite extinction time for the solutions to the Ricci flow on certain threemanifolds. arXiv: math/0307245

[40] Ringström H. The Cauchy Problem in General Relativity. ESI Lect Math Phys. Zürich: Eur Math Soc, 2009

[41] Sesum N. Curvature tensor under the Ricci flow. Amer J Math, 2005, 127(6): 1315–1324

[42] Simon M. 4D Ricci flows with bounded scalar curvature. arXiv: 1504.02623v1

[43] Streets J D. Ricci Yang-Mills Flow. Ph D Thesis. Duke University, 2007, http://www.math.uci.edu/ jstreets/papers/StreetsThesis.pdf

[44] Streets J D. Regularity and expanding entropy for connection Ricci flow. J Geom Phys, 2008, 58(7): 900–912

[45] Streets J D. Singularities of renormalization group flows. J Geom Phys, 2009, 59(1): 8–16

[46] Streets J D. Ricci Yang-Mills flow on surfaces. Adv Math, 2010, 223(2): 454–475

[47] Tadano H. A lower diameter bound for compact domain manifolds of shrinking Ricciharmonic solitons. Kodai Math J, 2015, 38(2): 302–309

[48] Tadano H. Gap theorems for Ricci-harmonic solitons. arXiv: 1505.03194

[49] Tian G, Zhang Z L. Regularity of Kähler-Ricci flows on Fano manifolds. Acta Math, 2016, 216(1): 127–176

[50] Wang B. On the conditions to extend Ricci flow. Int Math Res Not IMRN, 2008, (8): Art ID rnn012

[51] Wang B. On the conditions to extend Ricci flow (II). Int Math Res Not IMRN, 2012, (14): 3192–3223

[52] Wang L F. Differential Harnack inequalities under a coupled Ricci flow. Math Phys Anal Geom, 2012, 15(4): 343–360

[53] Williams M B. Results on coupled Ricci and harmonic map flows. Adv Geom, 2015, 15(1): 7–25

[54] Ye R G. Curvature estimates for the Ricci flow. I. Calc Var Partial Differential Equations, 2008, 31(4): 417–437

[55] Ye R G. Curvature estimates for the Ricci flow. II. Calc Var Partial Differential Equations, 2008, 31(4): 439–466

[56] Young A. Modified Ricci Flow on a Principal Bundle. Ph D Thesis. The University of Texas at Austin, 2008, http://search.proquest.com/docview/193674070

[57] Zhang Z. Scalar curvature behavior for finite-time singularity of Kähler-Ricci flow. Michigan Math J, 2010, 59(2): 419–433

[58] Zhu A Q. Differential Harnack inequalities for the backward heat equation with potential under the harmonic-Ricci flow. J Math Anal Appl, 2013, 406(2): 502–510

RIGHTS & PERMISSIONS

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg