Advertisement

From minimal Lagrangian to J-minimal submanifolds: persistence and uniqueness

  • Jason D. Lotay
  • Tommaso PaciniEmail author
Article

Abstract

Given a minimal Lagrangian submanifold L in a negative Kähler–Einstein manifold M, we show that any small Kähler–Einstein perturbation of M induces a deformation of L which is minimal Lagrangian with respect to the new structure. This provides a new source of examples of minimal Lagrangians. More generally, the same is true for the larger class of totally real J-minimal submanifolds in Kähler manifolds with negative definite Ricci curvature.

Notes

Acknowledgements

We would like to thank Claude LeBrun for suggesting the problem of persistence of minimal Lagrangians to us, André Neves for informing us of the reference [10], and Simon Donaldson, Nicos Kapouleas and Cristiano Spotti for interesting discussions. JDL was partially supported by EPSRC grant EP/K010980/1. TP thanks the Scuola Normale Superiore, in Pisa, for hospitality and research funds. This paper is dedicated to Paolo de Bartolomeis, who was also TP’s advisor. His excellent lectures conveyed the idea that Geometry is not just a body of results. It is also a point of view, which can provide a guiding light in many other fields of Mathematics and Science.

References

  1. 1.
    Borrelli, V.: Maslov form and \(J\)-volume of totally real immersions. J. Geom. Phys. 25, 271–290 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bryant, R.L.: Minimal lagrangian submanifolds of Kähler-einstein manifolds. In: Gu, C., Berger, M., Bryant, R.L. (eds.) Differential Geometry and Differential Equations. Lecture Notes in Mathematics, vol. 1255. Springer, Berlin, Heidelberg (1987)Google Scholar
  3. 3.
    Chen, B.-Y.: Geometry of Submanifolds and Its Applications. Science University of Tokyo, Tokyo (1981)zbMATHGoogle Scholar
  4. 4.
    Lee, Y.-I.: The deformation of Lagrangian minimal surfaces in Kähler–Einstein surfaces. J. Differ. Geom. 50, 299–330 (1998)CrossRefzbMATHGoogle Scholar
  5. 5.
    Lotay, J.D., Pacini, T.: From Lagrangian to totally real geometry: coupled flows and calibrations. CAG. arXiv:1404.4227 (to appear)
  6. 6.
    Lotay, J.D., Pacini, T.: Complexified diffeomorphism groups, totally real submanifolds and Kähler–Einstein geometry. Trans. AMS. arXiv:1506.04630 (to appear)
  7. 7.
    Oh, Y.-G.: Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds. Invent. Math. 101, 501–519 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Pacini, T.: Maslov, Chern–Weil and Mean Curvature. J. Geom. Phys. 135, 129–134 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Smoczyk, K.: The Lagrangian mean curvature flow. Habilitation thesis, Leipzig (2001)Google Scholar
  10. 10.
    White, B.: The space of minimal submanifolds for varying Riemannian metrics. Indiana Univ. Math. J. 40, 161–200 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Unione Matematica Italiana 2018

Authors and Affiliations

  1. 1.University College LondonLondonUK
  2. 2.University of TorinoTurinItaly

Personalised recommendations