# Lagrangian Kleinian bottles

I remember some talks some time ago about proofs of nonexistence of Lagrangian Kleinian bottles in C^2 for the standard symplectic structure, mentioning that this were the only compact surface for which the problem were open. Question 1: Where to find references for the other surfaces? Question 2: What is the current status for Kleinian bottles? Do there exist written proofs now?

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Just googling "Lagrangian klein bottles" returns the following :

Stefan Nemirovski, Lagrangian Klein Bottles in R2n Geometric And Functional Analysis Volume 19, Number 3 (2009), 902-909

the freely accessible arxiv version is at http://arxiv.org/abs/0712.1760 abstract : "It is shown that the n-dimensional Klein bottle admits a Lagrangian embedding into R^{2n} if and only if n is odd."

There was a previous result by the same author about the $n=2$ case, but it had a flaw initially, and was completely rewritten. See http://arxiv.org/abs/math/0106122

There is also another proof of this result by V. Schevchishin http://front.math.ucdavis.edu/0707.2085

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May be this is relevant:

http://arxiv.org/abs/0712.1760

Lagrangian Klein bottles in R^{2n}

Stefan Nemirovski

It is shown that the n-dimensional Klein bottle admits a Lagrangian embedding into R^{2n} if and only if n is odd.

Comments: V.2 - explicit formula for the Luttinger-type surgery; 6 pages

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