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

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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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Authors and Affiliations

  1. Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

    Georgios Dimitroglou Rizell

  2. Department of Mathematics, Stanford University, 450 Serra Mall, Building 380, Stanford, CA, 94305-2125, USA

    Elizabeth Goodman

  3. IBM Research-Haifa, IBM R & D Labs in Israel, University of Haifa Campus, Mount Carmel, Haifa, 3498825, Israel

    Alexander Ivrii

Authors
  1. Georgios Dimitroglou Rizell
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  2. Elizabeth Goodman
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  3. Alexander Ivrii
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Correspondence to Georgios Dimitroglou Rizell.

Additional information

Georgios Dimitroglou Rizell is supported by the Grant KAW 2013.0321 from the Knut and Alice Wallenberg Foundation.

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Dimitroglou Rizell, G., Goodman, E. & Ivrii, A. Lagrangian isotopy of tori in \({S^2\times S^2}\) and \({{\mathbb{C}}P^2}\) . Geom. Funct. Anal. 26, 1297–1358 (2016). https://doi.org/10.1007/s00039-016-0388-1

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