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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 & Continuous Dynamical Systems - A, 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. Google Scholar

[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.Google Scholar

[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.Google Scholar

[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. Google Scholar

[5]

X. X. Chen, Calabi flow in Riemann surfaces revisited, Int. Math Res Not., 6 (2001), 275-297. doi: 10.1155/S1073792801000149. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

S. K. Donaldson, Scalar curvature and stability of toric varieties, Jour. Differential Geometry, 62 (2002), 289-349. doi: 10.4310/jdg/1090950195. Google Scholar

[14]

S. K. Donaldson, Interior estimates for solutions of Abreu's equation, Collectanea Math., 56 (2005), 103-142. Google Scholar

[15]

S. K. Donaldson, Extremal metrics on toric surfaces: A continuity method, J. Differential Geom., 79 (2008), 389-432. doi: 10.4310/jdg/1213798183. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint, arXiv: math/0211159.Google Scholar

[23]

G. Perelman, Ricci flow with surgery on three-manifolds, preprint, arXiv: math/0303109.Google Scholar

[24]

G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain threemanifolds, preprint, arXiv: math/0307245.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

G. Székelyhidi, The Calabi functional on a ruled surface, Ann. Sci.Éc. Norm. Supér, 42 (2009), 833-856. Google Scholar

[30]

G. Tian, Canonical Metrics in Kähler Geometry, Birkhäuser, 2000. doi: 10.1007/978-3-0348-8389-4. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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.Google Scholar

[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.Google Scholar

[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. Google Scholar

[5]

X. X. Chen, Calabi flow in Riemann surfaces revisited, Int. Math Res Not., 6 (2001), 275-297. doi: 10.1155/S1073792801000149. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

S. K. Donaldson, Scalar curvature and stability of toric varieties, Jour. Differential Geometry, 62 (2002), 289-349. doi: 10.4310/jdg/1090950195. Google Scholar

[14]

S. K. Donaldson, Interior estimates for solutions of Abreu's equation, Collectanea Math., 56 (2005), 103-142. Google Scholar

[15]

S. K. Donaldson, Extremal metrics on toric surfaces: A continuity method, J. Differential Geom., 79 (2008), 389-432. doi: 10.4310/jdg/1213798183. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint, arXiv: math/0211159.Google Scholar

[23]

G. Perelman, Ricci flow with surgery on three-manifolds, preprint, arXiv: math/0303109.Google Scholar

[24]

G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain threemanifolds, preprint, arXiv: math/0307245.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

G. Székelyhidi, The Calabi functional on a ruled surface, Ann. Sci.Éc. Norm. Supér, 42 (2009), 833-856. Google Scholar

[30]

G. Tian, Canonical Metrics in Kähler Geometry, Birkhäuser, 2000. doi: 10.1007/978-3-0348-8389-4. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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