-
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
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 |
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.
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. 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. |
[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. 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. |
[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. Google Scholar |
[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. 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. |
[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. 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. |
[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. Google Scholar |
[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] |
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 |
[4] |
Francis C. Motta, Patrick D. Shipman. Informing the structure of complex Hadamard matrix spaces using a flow. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2349-2364. doi: 10.3934/dcdss.2019147 |
[5] |
Amadeu Delshams, Pere Gutiérrez, Tere M. Seara. Exponentially small splitting for whiskered tori in Hamiltonian systems: flow-box coordinates and upper bounds. Discrete & Continuous Dynamical Systems - A, 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 & 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 & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 1999, 5 (2) : 291-300. doi: 10.3934/dcds.1999.5.291 |
[15] |
Krzysztof Frączek. Polynomial growth of the derivative for diffeomorphisms on tori. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 489-516. doi: 10.3934/dcds.2004.11.489 |
[16] |
Mario Jorge Dias Carneiro, Rafael O. Ruggiero. On the graph theorem for Lagrangian minimizing tori. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6029-6045. doi: 10.3934/dcds.2018260 |
[17] |
Daniel Alpay, Mihai Putinar, Victor Vinnikov. A Hilbert space approach to bounded analytic extension in the ball. Communications on Pure & Applied Analysis, 2003, 2 (2) : 139-145. doi: 10.3934/cpaa.2003.2.139 |
[18] |
Xiaoni Du, Chenhuang Wu, Wanyin Wei. An extension of binary threshold sequences from Fermat quotients. Advances in Mathematics of Communications, 2016, 10 (4) : 743-752. doi: 10.3934/amc.2016038 |
[19] |
Ivo Siekmann, Horst Malchow, Ezio Venturino. An extension of the Beretta-Kuang model of viral diseases. Mathematical Biosciences & Engineering, 2008, 5 (3) : 549-565. doi: 10.3934/mbe.2008.5.549 |
[20] |
Vitali Milman, Liran Rotem. $\alpha$-concave functions and a functional extension of mixed volumes. Electronic Research Announcements, 2013, 20: 1-11. doi: 10.3934/era.2013.20.1 |
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]