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

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 $T^{\ast }{\mathbb{T}}^2$, i.e.~it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.

arXiv comment66 pages, 10 figures. Final version. Added Lemmas 4.1 and 6.8, added details in the proof of Corollary 3.7, and altered the statement and proof of Lemma 5.14

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• Rigid subsets of symplectic manifolds
• Parametrized Ring-Spectra and the Nearby Lagrangian Conjecture
• On Exotic Lagrangian Tori in CP^2
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• Exact Lagrangian submanifolds in simply-connected cotangent bundles
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Cited by in corpus (12)

• Homological mirror symmetry for log Calabi-Yau surfaces
• The classification of Lagrangians nearby the Whitney immersion
• Families of monotone Lagrangians in Brieskorn--Pham hypersurfaces
• Hamiltonian unknottedness of certain monotone Lagrangian tori in $S^2\times S^2$
• Lagrangian Cobordisms in Liouville manifolds
• Projective twists and the Hopf correspondence
• Packing Lagrangian tori
• Instability of rational and polynomial convexity
• Uniqueness of real Lagrangians up to cobordism
• Exact Lagrangians in four-dimensional symplectisations
• Exact Lagrangian tori in $T^{\ast }{\mathbb{T}}^n$
• On weakly exact Lagrangians in Liouville bi-fillings