# monge ampere equation along totally real submanifolds

hi,

are there some references when solving the complex monge ampere equation along totally real submanifolds of some compact (with boundary or without) complex manifold. i know that there are a lot of references when solving the dirichlet problem on compact complex manifolds with boundary. but now along some submanifold ? and what about in the inhomogenous case ? i am more interested in that. hope for answers.

william

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It's not clear to me what kind of conditions you are assuming and what sort of problem you want to solve. Could you be more explicit about that? In the meantime, maybe the paper at math.osu.edu/~guan.19/Papers/Guan-Li.aim.pdf would be helpful. – Robert Bryant Feb 22 '12 at 20:58
hi, i am interessted in the following (i give only the idea, there might be also other things one has to assume): if we assume that $M$ is a complex manifold and $X$ is a real analytic manifold. consider some $(n,n)$−form on $M$, say $f$. Is there any posibility to solve the equation $(dd^{\mathbb{C}} \varphi)^{n} = f$ with $dd^{\mathbb{C}} \varphi |_{X} = 0$ (or maybe some other constraint)? – william Feb 23 '12 at 6:07
Not sure, but there some work in the late 80's by Lazlo Lempert and Coifman-Semmes that might be along this direction? – Deane Yang Feb 23 '12 at 17:46