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edited Jan 17 2012 at 17:15

jc
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I'm betting `yes, sure!', but don't see it. Could someone please point me toward, or construct for me, a Lagrangian submanifold immersed in standard symplectic ${\mathbb R}^{2n}$ for $n > 1$, whose closure is all of ${\mathbb R}^{2n}$?

(For an $n =1$ example, one can use the leaves arising from Panov'sthis modification by Panov of irrational flow on the two-torus.)

Strong preference given to analytic immersions of ${\mathbb R}^n$.

Holomorphically immersed complex lines which are dense in complex 2-space -- i.e. dense ${\mathbb C}$'s in ${\mathbb C}^2$ -- are well-known. Ilyashenko in 1968 showed that the typical solution of the typical polynomial ODE (in complex time) yields such a curve. Following his line of thought, it might be easier to construct an entire singular Lagrangian foliation of ${\mathbb R}^{2n}$ whose typical leaf is dense, rather than the one submanifold.

Motivation: I have a certain unstable manifold related to a Hamiltonian system. It is Lagrangian. I would like to be ``as dense as can be'', so I'd like to know how dense can that be.

Can a Lagrangian submanifold of ${\mathbb R}^2n$ R}^{2n}$ be dense ($n>1$)?

I'm betting `yes, sure!', but don't see it. Could someone please point me toward, or construct for me, a Lagrangian submanifold immersed in standard symplectic ${\mathbb R}^2n$ R}^{2n}$ for $n > 1$, whose closure is all of ${\mathbb R}^{2n}$? (For an $n =1$ example, one can use the leaves arising from Panov's
this modification by Panov of irrational flow on the two-torus.)

Strong preference given to analytic immersions of ${\mathbb R}^n$.

Holomorphically immersed complex lines which are dense in complex 2-space -- i.e. dense ${\mathbb C}$'s in ${\mathbb C}^2$ -- are well-known. Ilyashenko in 1968 showed that the typical solution of the typical polynomial ODE (in complex time) yields such a curve. Following his line of thought, it might be easier to construct an entire singular Lagrangian foliation of ${\mathbb R}^2n$ R}^{2n}$ whose typical leaf is dense, rather than the one submanifold.

Motivation: I have a certain unstable manifold related to a Hamiltonian system. It is Lagrangian. I would like to be ``as dense as can be'', so I'd like to know how dense can that be.