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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.

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. ChenG. 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. ChenG. 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.

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