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Transactions of the American Mathematical Society

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Complexified diffeomorphism groups, totally real submanifolds and Kähler-Einstein geometry


Authors: Jason D. Lotay and Tommaso Pacini
Journal: Trans. Amer. Math. Soc. 371 (2019), 2665-2701
MSC (2010): Primary 53CXX; Secondary 32QXX
DOI: https://doi.org/10.1090/tran/7421
Published electronically: November 16, 2018
MathSciNet review: 3896093
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Abstract: Let $ (M,J)$ be an almost complex manifold. We show that the infinite-dimensional space $ \mathcal {T}$ of totally real submanifolds in $ M$ carries a natural connection. This induces a canonical notion of geodesics in $ \mathcal {T}$ and a corresponding definition of when a functional $ f:\mathcal {T}\rightarrow \mathbb{R}$ is convex.

Geodesics in $ \mathcal {T}$ can be expressed in terms of families of $ J$-holomorphic curves in $ M$; we prove a uniqueness result and study their existence. When $ M$ is Kähler we define a canonical functional on $ \mathcal {T}$; it is convex if $ M$ has non-positive Ricci curvature.

Our construction is formally analogous to the notion of geodesics and the Mabuchi functional on the space of Kähler potentials, as studied by Donaldson, Fujiki, and Semmes. Motivated by this analogy, we discuss possible applications of our theory to the study of minimal Lagrangians in negative Kähler-Einstein manifolds.


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  • [1] Brian Joseph Birgen, A characterization of families of loops that are locally closed under polynomial hulls, ProQuest LLC, Ann Arbor, MI, 1997. Thesis (Ph.D.)–University of Michigan. MR 2695840
  • [2] Vincent Borrelli, Maslov form and 𝐽-volume of totally real immersions, J. Geom. Phys. 25 (1998), no. 3-4, 271–290. MR 1619846, https://doi.org/10.1016/S0393-0440(97)00030-2
  • [3] Bang-yen Chen, Geometry of submanifolds and its applications, Science University of Tokyo, Tokyo, 1981. MR 627323
  • [4] John B. Conway, Functions of one complex variable. II, Graduate Texts in Mathematics, vol. 159, Springer-Verlag, New York, 1995. MR 1344449
  • [5] 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, https://doi.org/10.1090/trans2/196/02
  • [6] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR 1079726
  • [7] Paul Gauduchon, Calabi's extremal Kähler metrics: An elementary introduction, lecture notes.
  • [8] Dominic Joyce, Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow, EMS Surv. Math. Sci. 2 (2015), no. 1, 1–62. MR 3354954, https://doi.org/10.4171/EMSS/8
  • [9] Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, Mathematical Surveys and Monographs, vol. 53, American Mathematical Society, Providence, RI, 1997. MR 1471480
  • [10] László Lempert, The problem of complexifying a Lie group, Multidimensional complex analysis and partial differential equations (São Carlos, 1995) Contemp. Math., vol. 205, Amer. Math. Soc., Providence, RI, 1997, pp. 169–176. MR 1447223, https://doi.org/10.1090/conm/205/02662
  • [11] Jason D. Lotay and Tommaso Pacini, From Lagrangian to totally real geometry: coupled flows and calibrations, to appear in Comm. Analysis and Geometry
  • [12] Jason D. Lotay and Tommaso Pacini, Uniqueness and persistence of minimal Lagrangian submanifolds, Boll Unione Mat Ital (2018). DOI 10.1007/s40574-018-0183-z
  • [13] Roberta Maccheroni, Complex analytic properties of minimal Lagrangian submanifolds, arXiv:1805.09651
  • [14] Dusa McDuff and Dietmar Salamon, 𝐽-holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications, vol. 52, American Mathematical Society, Providence, RI, 2004. MR 2045629
  • [15] Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
  • [16] Michael Spivak, A comprehensive introduction to differential geometry. Vol. I, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR 532830
  • [17] François Trèves, Basic linear partial differential equations, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 62. MR 0447753
  • [18] H. Whitney and F. Bruhat, Quelques propriétés fondamentales des ensembles analytiques-réels, Comment. Math. Helv. 33 (1959), 132–160 (French). MR 102094, https://doi.org/10.1007/BF02565913

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Additional Information

Jason D. Lotay
Affiliation: Department of Mathematics, University College London, 25 Gordon Street, London WC1 H0AY, United Kingdom

Tommaso Pacini
Affiliation: Dipartimento di Matematica, Università di Torino, Via Carlo Alberto, 10, 10123 Torino, Italy

DOI: https://doi.org/10.1090/tran/7421
Received by editor(s): September 9, 2016
Received by editor(s) in revised form: July 8, 2017, and October 2, 2017
Published electronically: November 16, 2018
Article copyright: © Copyright 2018 American Mathematical Society