Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

The consistency and convergence of K-energy minimizing movements


Author: Jeffrey Streets
Journal: Trans. Amer. Math. Soc. 368 (2016), 5075-5091
MSC (2010): Primary 53C44
DOI: https://doi.org/10.1090/tran/6508
Published electronically: September 15, 2015
MathSciNet review: 3456172
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that $ K$-energy minimizing movements agree with smooth solutions to Calabi flow as long as the latter exist. As corollaries we conclude that in a general Kähler class long time solutions of Calabi flow minimize both $ K$-energy and Calabi energy. Lastly, by applying convergence results from the theory of minimizing movements, these results imply that long time solutions to Calabi flow converge in the weak distance topology to minimizers of the $ K$-energy functional on the metric completion of the space of Kähler metrics, assuming one exists.


References [Enhancements On Off] (What's this?)

  • [1] Miroslav Bačák, The proximal point algorithm in metric spaces, Israel J. Math. 194 (2013), no. 2, 689-701. MR 3047087, https://doi.org/10.1007/s11856-012-0091-3
  • [2] Miroslav Bačák, Ian Searston, and Brailey Sims, Alternating projections in $ \rm CAT(0)$ spaces, J. Math. Anal. Appl. 385 (2012), no. 2, 599-607. MR 2834837 (2012h:47027), https://doi.org/10.1016/j.jmaa.2011.06.079
  • [3] Robert J. Berman, A thermodynamical formalism for Monge-Ampère equations, Moser-Trudinger inequalities and Kähler-Einstein metrics, Adv. Math. 248 (2013), 1254-1297. MR 3107540, https://doi.org/10.1016/j.aim.2013.08.024
  • [4] R. Berman and B. Berndtsson, Convexity of the $ K$-energy on the space of Kähler metrics, arXiv:1405.0401
  • [5] Zbigniew Błocki, On geodesics in the space of Kähler metrics, Advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 21, Int. Press, Somerville, MA, 2012, pp. 3-19. MR 3077245
  • [6] Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486 (2000k:53038)
  • [7] Eugenio Calabi, Extremal Kähler metrics, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 259-290. MR 645743 (83i:53088)
  • [8] E. Calabi, The variation of Kähler metrics (Abstract), Bull. Amer. Math. Soc. 60 (1954), 167-168.
  • [9] E. Calabi and X. X. Chen, The space of Kähler metrics. II, J. Differential Geom. 61 (2002), no. 2, 173-193. MR 1969662 (2004i:32039)
  • [10] Xiuxiong Chen, The space of Kähler metrics, J. Differential Geom. 56 (2000), no. 2, 189-234. MR 1863016 (2003b:32031)
  • [11] Xiuxiong Chen, Space of Kähler metrics. III. On the lower bound of the Calabi energy and geodesic distance, Invent. Math. 175 (2009), no. 3, 453-503. MR 2471594 (2010b:32033), https://doi.org/10.1007/s00222-008-0153-7
  • [12] X. X. Chen and G. Tian, Geometry of Kähler metrics and foliations by holomorphic discs, Publ. Math. Inst. Hautes Études Sci. 107 (2008), 1-107. MR 2434691 (2009g:32048), https://doi.org/10.1007/s10240-008-0013-4
  • [13] Brian Clarke and Yanir A. Rubinstein, Ricci flow and the metric completion of the space of Kähler metrics, Amer. J. Math. 135 (2013), no. 6, 1477-1505. MR 3145001, https://doi.org/10.1353/ajm.2013.0051
  • [14] Simon K. Donaldson, Conjectures in Kähler geometry, Strings and geometry, Clay Math. Proc., vol. 3, Amer. Math. Soc., Providence, RI, 2004, pp. 71-78. MR 2103718 (2005i:32024)
  • [15] S. K. Donaldson, Symmetric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, vol. 196, Amer. Math. Soc., Providence, RI, 1999, pp. 13-33. MR 1736211 (2002b:58008)
  • [16] Rafa Espínola and Aurora Fernández-León, $ {\rm CAT}(k)$-spaces, weak convergence and fixed points, J. Math. Anal. Appl. 353 (2009), no. 1, 410-427. MR 2508878 (2010d:47092), https://doi.org/10.1016/j.jmaa.2008.12.015
  • [17] Jürgen Jost, Equilibrium maps between metric spaces, Calc. Var. Partial Differential Equations 2 (1994), no. 2, 173-204. MR 1385525 (98a:58049), https://doi.org/10.1007/BF01191341
  • [18] Toshiki Mabuchi, Some symplectic geometry on compact Kähler manifolds. I, Osaka J. Math. 24 (1987), no. 2, 227-252. MR 909015 (88m:53126)
  • [19] Uwe F. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Comm. Anal. Geom. 6 (1998), no. 2, 199-253. MR 1651416 (99m:58067)
  • [20] Stephen Semmes, Complex Monge-Ampère and symplectic manifolds, Amer. J. Math. 114 (1992), no. 3, 495-550. MR 1165352 (94h:32022), https://doi.org/10.2307/2374768
  • [21] Stephen Semmes, Long time existence of minimizing movement solutions to Calabi flow, arXiv:1208.2718.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C44

Retrieve articles in all journals with MSC (2010): 53C44


Additional Information

Jeffrey Streets
Affiliation: Department of Mathematics, Rowland Hall, University of California, Irvine, Irvine, California 92617
Email: jstreets@uci.edu

DOI: https://doi.org/10.1090/tran/6508
Received by editor(s): February 17, 2014
Received by editor(s) in revised form: June 11, 2014, and June 22, 2014
Published electronically: September 15, 2015
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society