On the extension and smoothing of the Calabi flow on complex tori

Previous Article
$\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^∞$ via the singular value problem
DCDS Home
This Issue
Next Article
Interactions, specifications, DLR probabilities and the Ruelle operator in the one-dimensional lattice

December 2017, 37(12): 6153-6164. doi: 10.3934/dcds.2017265

On the extension and smoothing of the Calabi flow on complex tori

Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, 87131, USA

Received November 2016 Revised July 2017 Published August 2017

In this paper, we continue to study the Calabi flow on complex tori. We develop a new method to obtain an explicit bound of the curvature of the Calabi flow. As an application, we show that when $n=2$, the Calabi flow starting from a weak Kähler metric will become smooth immediately. It implies that in our settings, the weak minimizer of the Mabuchi energy is a smooth one.

Keywords: Calabi flow, extension and smoothing, complex tori.

Mathematics Subject Classification: Primary: 53C44; Secondary: 35K96.

Citation: Hongnian Huang. On the extension and smoothing of the Calabi flow on complex tori. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6153-6164. doi: 10.3934/dcds.2017265

References:

[1]	M. Abreu, Kähler geometry of toric varieties and extremal metrics, International J. Math, 9 (1998), 641-665. doi: 10.1142/S0129167X98000282.
[2]	R. Berman, T. Darvas and C. Lu, Convexity of the extended K-energy and the large time behaviour of the weak Calabi flow, preprint, arXiv: 1510.01260.
[3]	R. Berman, T. Darvas and C. Lu, Regularity of weak minimizers of the K-energy and applications to properness and K-stability, preprint, arXiv: 1602.03114.
[4]	E. Calabi and X. X. Chen, Space of Kähler metrics and Calabi flow, J. Differential Geom, 61 (2002), 173-193. doi: 10.4310/jdg/1090351383.
[5]	X. X. Chen, Calabi flow in Riemann surfaces revisited, Int. Math Res Not., 6 (2001), 275-297. doi: 10.1155/S1073792801000149.
[6]	X. X. Chen, Space of Kähler metrics (Ⅳ) - On the lower bound of the K-energy, J. Differential Geom., 56 (2000), 189-234, arXiv: 0809.4081. doi: 10.4310/jdg/1090347643.
[7]	X. X. Chen and W. Y. He, On the Calabi flow, Amer. J. Math., 130 (2008), 539-570. doi: 10.1353/ajm.2008.0018.
[8]	X. X. Chen and W. Y. He, The Calabi flow on Kähler surface with bounded Sobolev constant-(Ⅰ), Mathematische Annalen, 354 (2012), 227-261, arXiv: 0710.5159. doi: 10.1007/s00208-011-0723-7.
[9]	X. X. Chen and W. Y. He, The Calabi ow on toric Fano surface, Mathematical Research Letters, 17 (2010), 231-241, arXiv: 0807.3984. doi: 10.4310/MRL.2010.v17.n2.a3.
[10]	X. X. Chen, G. Tian and Z. Zhang, On the weak Kähler-Ricci flow, Trans. Amer. Math. Soc., 363 (2011), 2849-2863. doi: 10.1090/S0002-9947-2011-05015-4.
[11]	P. T. Chrusciél, Semi-global existence and convergence of solutions of the Robison-Trautman(2-dimensional Calabi) equation, Comm. Math. Phys., 137 (1991), 289-313. doi: 10.1007/BF02431882.
[12]	T. Darvas and Y. A. Rubinstein, Tian's properness conjecture and Finsler geometry of the space of Kähler metrics, J. Amer. Math. Soc., 30 (2017), 347-387, arXiv: 1506.07129. doi: 10.1090/jams/873.
[13]	S. K. Donaldson, Scalar curvature and stability of toric varieties, Jour. Differential Geometry, 62 (2002), 289-349. doi: 10.4310/jdg/1090950195.
[14]	S. K. Donaldson, Interior estimates for solutions of Abreu's equation, Collectanea Math., 56 (2005), 103-142.
[15]	S. K. Donaldson, Extremal metrics on toric surfaces: A continuity method, J. Differential Geom., 79 (2008), 389-432. doi: 10.4310/jdg/1213798183.
[16]	S. K. Donaldson, Constant scalar curvature metrics on toric surfaces, Geom. Funct. Anal., 19 (2009), 83-136. doi: 10.1007/s00039-009-0714-y.
[17]	R. Feng and G. Székelyhidi, Periodic solutions of Abreu's equation, Math. Res. Lett., 18 (2011), 1271-1279. doi: 10.4310/MRL.2011.v18.n6.a15.
[18]	R. J. Feng and H. N. Huang, The Global Existence and Convergence of the Calabi Flow on $\mathbb{C}^n = \mathbb{Z}^n + i \mathbb{Z}^n$, J. Funct. Anal., 263 (2012), 1129-1146. doi: 10.1016/j.jfa.2012.05.017.
[19]	D. Guan, Extremal-solitons and exponential $C^{∞}$ convergence of modified Calabi flow on certain $\mathbb{CP}^1$ bundles, Pacific J. Math., 233 (2007), 91-124. doi: 10.2140/pjm.2007.233.91.
[20]	H. Huang, On the extension of the Calabi flow on toric varieties, Ann. Global Anal. Geom., 40 (2011), 1-19. doi: 10.1007/s10455-010-9242-0.
[21]	U. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Comm. Anal. Geom., 6 (1998), 199-253. doi: 10.4310/CAG.1998.v6.n2.a1.
[22]	G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint, arXiv: math/0211159.
[23]	G. Perelman, Ricci flow with surgery on three-manifolds, preprint, arXiv: math/0303109.
[24]	G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain threemanifolds, preprint, arXiv: math/0307245.
[25]	W. X. Shi, Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geom, 30 (1989), 303-394. doi: 10.4310/jdg/1214443595.
[26]	J. Streets, The long time behavior of fourth-order curvature flows, Calc. Var. Partial Differential Equations, 46 (2013), 39-54. doi: 10.1007/s00526-011-0472-1.
[27]	J. Streets, Long time existence of minimizing movement solutions of Calabi flow, Adv. Math., 259 (2014), 688-729. doi: 10.1016/j.aim.2014.03.027.
[28]	J. Streets, The consistency and convergence of K-energy minimizing movements, Trans. Amer. Math. Soc, 368 (2016), 5075-5091. doi: 10.1090/tran/6508.
[29]	G. Székelyhidi, The Calabi functional on a ruled surface, Ann. Sci.Éc. Norm. Supér, 42 (2009), 833-856.
[30]	G. Tian, Canonical Metrics in Kähler Geometry, Birkhäuser, 2000. doi: 10.1007/978-3-0348-8389-4.
[31]	G. Tian, Existence of Einstein metrics on Fano manifolds, Metric and differential geometry, 119-159, Progr. Math., 297, Birkhäser/Springer, Basel, 2012. doi: 10.1007/978-3-0348-0257-4_5.
[32]	N. S. Trudinger and X. J. Wang, The affine plateau problem, J. Amer. Math. Soc., 18 (2005), 253-289. doi: 10.1090/S0894-0347-05-00475-3.
[33]	B. Zhou and X. H. Zhu, Minimizing weak solutions for Calabi's extremal metrics on toric manifolds, Calc. Var. Partial Differential Equations, 32 (2008), 191-217. doi: 10.1007/s00526-007-0136-3.
[34]	W. P. Ziemer, Weakly Differentiable Functions, Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

show all references

References:

[1]	M. Abreu, Kähler geometry of toric varieties and extremal metrics, International J. Math, 9 (1998), 641-665. doi: 10.1142/S0129167X98000282.
[2]	R. Berman, T. Darvas and C. Lu, Convexity of the extended K-energy and the large time behaviour of the weak Calabi flow, preprint, arXiv: 1510.01260.
[3]	R. Berman, T. Darvas and C. Lu, Regularity of weak minimizers of the K-energy and applications to properness and K-stability, preprint, arXiv: 1602.03114.
[4]	E. Calabi and X. X. Chen, Space of Kähler metrics and Calabi flow, J. Differential Geom, 61 (2002), 173-193. doi: 10.4310/jdg/1090351383.
[5]	X. X. Chen, Calabi flow in Riemann surfaces revisited, Int. Math Res Not., 6 (2001), 275-297. doi: 10.1155/S1073792801000149.
[6]	X. X. Chen, Space of Kähler metrics (Ⅳ) - On the lower bound of the K-energy, J. Differential Geom., 56 (2000), 189-234, arXiv: 0809.4081. doi: 10.4310/jdg/1090347643.
[7]	X. X. Chen and W. Y. He, On the Calabi flow, Amer. J. Math., 130 (2008), 539-570. doi: 10.1353/ajm.2008.0018.
[8]	X. X. Chen and W. Y. He, The Calabi flow on Kähler surface with bounded Sobolev constant-(Ⅰ), Mathematische Annalen, 354 (2012), 227-261, arXiv: 0710.5159. doi: 10.1007/s00208-011-0723-7.
[9]	X. X. Chen and W. Y. He, The Calabi ow on toric Fano surface, Mathematical Research Letters, 17 (2010), 231-241, arXiv: 0807.3984. doi: 10.4310/MRL.2010.v17.n2.a3.
[10]	X. X. Chen, G. Tian and Z. Zhang, On the weak Kähler-Ricci flow, Trans. Amer. Math. Soc., 363 (2011), 2849-2863. doi: 10.1090/S0002-9947-2011-05015-4.
[11]	P. T. Chrusciél, Semi-global existence and convergence of solutions of the Robison-Trautman(2-dimensional Calabi) equation, Comm. Math. Phys., 137 (1991), 289-313. doi: 10.1007/BF02431882.
[12]	T. Darvas and Y. A. Rubinstein, Tian's properness conjecture and Finsler geometry of the space of Kähler metrics, J. Amer. Math. Soc., 30 (2017), 347-387, arXiv: 1506.07129. doi: 10.1090/jams/873.
[13]	S. K. Donaldson, Scalar curvature and stability of toric varieties, Jour. Differential Geometry, 62 (2002), 289-349. doi: 10.4310/jdg/1090950195.
[14]	S. K. Donaldson, Interior estimates for solutions of Abreu's equation, Collectanea Math., 56 (2005), 103-142.
[15]	S. K. Donaldson, Extremal metrics on toric surfaces: A continuity method, J. Differential Geom., 79 (2008), 389-432. doi: 10.4310/jdg/1213798183.
[16]	S. K. Donaldson, Constant scalar curvature metrics on toric surfaces, Geom. Funct. Anal., 19 (2009), 83-136. doi: 10.1007/s00039-009-0714-y.
[17]	R. Feng and G. Székelyhidi, Periodic solutions of Abreu's equation, Math. Res. Lett., 18 (2011), 1271-1279. doi: 10.4310/MRL.2011.v18.n6.a15.
[18]	R. J. Feng and H. N. Huang, The Global Existence and Convergence of the Calabi Flow on $\mathbb{C}^n = \mathbb{Z}^n + i \mathbb{Z}^n$, J. Funct. Anal., 263 (2012), 1129-1146. doi: 10.1016/j.jfa.2012.05.017.
[19]	D. Guan, Extremal-solitons and exponential $C^{∞}$ convergence of modified Calabi flow on certain $\mathbb{CP}^1$ bundles, Pacific J. Math., 233 (2007), 91-124. doi: 10.2140/pjm.2007.233.91.
[20]	H. Huang, On the extension of the Calabi flow on toric varieties, Ann. Global Anal. Geom., 40 (2011), 1-19. doi: 10.1007/s10455-010-9242-0.
[21]	U. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Comm. Anal. Geom., 6 (1998), 199-253. doi: 10.4310/CAG.1998.v6.n2.a1.
[22]	G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint, arXiv: math/0211159.
[23]	G. Perelman, Ricci flow with surgery on three-manifolds, preprint, arXiv: math/0303109.
[24]	G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain threemanifolds, preprint, arXiv: math/0307245.
[25]	W. X. Shi, Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geom, 30 (1989), 303-394. doi: 10.4310/jdg/1214443595.
[26]	J. Streets, The long time behavior of fourth-order curvature flows, Calc. Var. Partial Differential Equations, 46 (2013), 39-54. doi: 10.1007/s00526-011-0472-1.
[27]	J. Streets, Long time existence of minimizing movement solutions of Calabi flow, Adv. Math., 259 (2014), 688-729. doi: 10.1016/j.aim.2014.03.027.
[28]	J. Streets, The consistency and convergence of K-energy minimizing movements, Trans. Amer. Math. Soc, 368 (2016), 5075-5091. doi: 10.1090/tran/6508.
[29]	G. Székelyhidi, The Calabi functional on a ruled surface, Ann. Sci.Éc. Norm. Supér, 42 (2009), 833-856.
[30]	G. Tian, Canonical Metrics in Kähler Geometry, Birkhäuser, 2000. doi: 10.1007/978-3-0348-8389-4.
[31]	G. Tian, Existence of Einstein metrics on Fano manifolds, Metric and differential geometry, 119-159, Progr. Math., 297, Birkhäser/Springer, Basel, 2012. doi: 10.1007/978-3-0348-0257-4_5.
[32]	N. S. Trudinger and X. J. Wang, The affine plateau problem, J. Amer. Math. Soc., 18 (2005), 253-289. doi: 10.1090/S0894-0347-05-00475-3.
[33]	B. Zhou and X. H. Zhu, Minimizing weak solutions for Calabi's extremal metrics on toric manifolds, Calc. Var. Partial Differential Equations, 32 (2008), 191-217. doi: 10.1007/s00526-007-0136-3.
[34]	W. P. Ziemer, Weakly Differentiable Functions, Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

[1]	Noboru Okazawa, Tomomi Yokota. Smoothing effect for generalized complex Ginzburg-Landau equations in unbounded domains. Conference Publications, 2001, 2001 (Special) : 280-288. doi: 10.3934/proc.2001.2001.280
[2]	Michael Hutchings. Mean action and the Calabi invariant. Journal of Modern Dynamics, 2016, 10: 511-539. doi: 10.3934/jmd.2016.10.511
[3]	Francis C. Motta, Patrick D. Shipman. Informing the structure of complex Hadamard matrix spaces using a flow. Discrete and Continuous Dynamical Systems - S, 2019, 12 (8) : 2349-2364. doi: 10.3934/dcdss.2019147
[4]	Ali Unver, Christian Ringhofer, Dieter Armbruster. A hyperbolic relaxation model for product flow in complex production networks. Conference Publications, 2009, 2009 (Special) : 790-799. doi: 10.3934/proc.2009.2009.790
[5]	Amadeu Delshams, Pere Gutiérrez, Tere M. Seara. Exponentially small splitting for whiskered tori in Hamiltonian systems: flow-box coordinates and upper bounds. Discrete and Continuous Dynamical Systems, 2004, 11 (4) : 785-826. doi: 10.3934/dcds.2004.11.785
[6]	Asaf Kislev. Compactly supported Hamiltonian loops with a non-zero Calabi invariant. Electronic Research Announcements, 2014, 21: 80-88. doi: 10.3934/era.2014.21.80
[7]	Stephen Coughlan, Łukasz Gołębiowski, Grzegorz Kapustka, Michał Kapustka. Arithmetically Gorenstein Calabi--Yau threefolds in $\mathbb{P}^7$. Electronic Research Announcements, 2016, 23: 52-68. doi: 10.3934/era.2016.23.006
[8]	Jacek Serafin. A faithful symbolic extension. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1051-1062. doi: 10.3934/cpaa.2012.11.1051
[9]	Hans F. Weinberger, Xiao-Qiang Zhao. An extension of the formula for spreading speeds. Mathematical Biosciences & Engineering, 2010, 7 (1) : 187-194. doi: 10.3934/mbe.2010.7.187
[10]	Augusto Visintin. An extension of the Fitzpatrick theory. Communications on Pure and Applied Analysis, 2014, 13 (5) : 2039-2058. doi: 10.3934/cpaa.2014.13.2039
[11]	Dmitry Jakobson. On quantum limits on flat tori. Electronic Research Announcements, 1995, 1: 80-86.
[12]	Mikko Kemppainen, Peter Sjögren, José Luis Torrea. Wave extension problem for the fractional Laplacian. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4905-4929. doi: 10.3934/dcds.2015.35.4905
[13]	Yuri Chekanov, Felix Schlenk. Notes on monotone Lagrangian twist tori. Electronic Research Announcements, 2010, 17: 104-121. doi: 10.3934/era.2010.17.104
[14]	Grant Cairns, Barry Jessup, Marcel Nicolau. Topologically transitive homeomorphisms of quotients of tori. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 291-300. doi: 10.3934/dcds.1999.5.291
[15]	Mario Jorge Dias Carneiro, Rafael O. Ruggiero. On the graph theorem for Lagrangian minimizing tori. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 6029-6045. doi: 10.3934/dcds.2018260
[16]	Krzysztof Frączek. Polynomial growth of the derivative for diffeomorphisms on tori. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 489-516. doi: 10.3934/dcds.2004.11.489
[17]	Sergio Zamora. Tori can't collapse to an interval. Electronic Research Archive, 2021, 29 (4) : 2637-2644. doi: 10.3934/era.2021005
[18]	Daniel Alpay, Mihai Putinar, Victor Vinnikov. A Hilbert space approach to bounded analytic extension in the ball. Communications on Pure and Applied Analysis, 2003, 2 (2) : 139-145. doi: 10.3934/cpaa.2003.2.139
[19]	Zengjing Chen, Weihuan Huang, Panyu Wu. Extension of the strong law of large numbers for capacities. Mathematical Control and Related Fields, 2019, 9 (1) : 175-190. doi: 10.3934/mcrf.2019010
[20]	Juan Gabriel Brida, Gaston Cayssials, Oscar Córdoba Rodríguez, Martín Puchet Anyul. A dynamic extension of the classical model of production prices determination. Journal of Dynamics and Games, 2020, 7 (3) : 185-196. doi: 10.3934/jdg.2020013

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Hongnian Huang
on Google Scholar
Hongnian Huang

2021 Impact Factor: 1.588

Tools

Article outline

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Hongnian Huang
on Google Scholar
Hongnian Huang

[Back to Top]