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Geometric and Functional Analysis

, Volume 26, Issue 5, pp 1297–1358 | Cite as

Lagrangian isotopy of tori in \({S^2\times S^2}\) and \({{\mathbb{C}}P^2}\)

  • Georgios Dimitroglou Rizell
  • Elizabeth Goodman
  • Alexander Ivrii
Article
  • 136 Downloads

Abstract

We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space \({{\mathbb{R}}^4}\), the projective plane \({{\mathbb{C}}P^2}\), and the monotone \({S^2 \times S^2}\). The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for \({T^*{\mathbb{T}}^2}\), i.e. it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Georgios Dimitroglou Rizell
    • 1
  • Elizabeth Goodman
    • 2
  • Alexander Ivrii
    • 3
  1. 1.Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK
  2. 2.Department of MathematicsStanford UniversityStanfordUSA
  3. 3.IBM Research-Haifa, IBM R & D Labs in Israel, University of Haifa CampusHaifaIsrael

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