English

Non-isotopic monotone Lagrangian submanifolds of $\mathbb{C}^n$

Symplectic Geometry 2020-03-03 v2

Abstract

Let PP be a Delzant polytope in Rk\mathbb{R}^k with n+kn+k facets. We associate a closed Lagrangian submanifold LL of Cn\mathbb{C}^n to each Delzant polytope. We prove that LL is monotone if and only if and only if the polytope PP is Fano. We pose the "Lagrangian version of Delzant Theorem". Then for even pp and nn we construct p2\frac{p}{2} monotone Lagrangian embeddings of Sp1×Snp1×T2S^{p-1} \times S^{n-p-1} \times T^2 into Cn\mathbb{C}^n, no two of which are related by Hamiltonian isotopies. Some of these embeddings are smoothly isotopic and have equal minimal Maslov numbers, but they are not Hamiltonian isotopic. Also, we construct infinitely many non-monotone Lagrangian embeddings of S2p1×S2p1×T2S^{2p-1} \times S^{2p-1} \times T^2 into C4p\mathbb{C}^{4p}, no two of which are related by Hamiltonian isotopies.

Keywords

Cite

@article{arxiv.1911.11407,
  title  = {Non-isotopic monotone Lagrangian submanifolds of $\mathbb{C}^n$},
  author = {Vardan Oganesyan},
  journal= {arXiv preprint arXiv:1911.11407},
  year   = {2020}
}

Comments

This paper has been withdrawn by the author because of the error in the proof