2 Changed title and expanded the question

# ThesolutionSolution of Plateau Problem for a simple, smooth closed curve on a Riemannian Manifold (Kahler) gives a surface that can be parametrized by a closed disk?

Hi,

Perhaps it's a stupid question, in that case i'll delete it. Let M be a compact orientable smooth (Kahler if changes things) manifold of dimension $dim_{\mathbb{R}}(M)=2n$ with $n\geq1$, let $\gamma$ be a simple smooth closed curve that lies in a (holomorphic) coordinate chart and that can be taken as small as necessary (one can choose $\gamma$ such $diam(\gamma)<\epsilon$ with $\epsilon>0$). If i want to solve Plateau problem for a $\gamma$ so small such that the solution is contained in the coordinate chart, do i get something that can be parametrized by closed disc?

Edit:

I'll try to clarify my question, this is what i wanted to know. Let $U$ be a sufficiently small geodesically convex set of a manifold $M$ and $\gamma$ a smooth simple closed curve lying in $U$ (no other assumptions on $\gamma$).

1) Can $\gamma$ be the boundary of an embedded closed disk?
2) If $\gamma$ can be the boundary of a closed disk, then can it be the boundary of a minimal (as a surface, not only among the disks that it bounds) embedded closed disk?

I anticipate that i couldn't see works of Douglas so i don't know if the answer to my question is there.

My suspect was that the answer could be yes for dimension 2 (i think about jordan curve theorem), Professor Thusrston example of the knotted curve suggests me that in dimension 3 i need additional assumptions on the curve not only on the linking number. But what happens for dimension $n\geq4$?

1

# The solution of Plateau Problem for a simple, smooth closed curve on a Riemannian Manifold (Kahler) gives a surface that can be parametrized by a closed disk?

Hi,

Perhaps it's a stupid question, in that case i'll delete it. Let M be a compact orientable smooth (Kahler if changes things) manifold of dimension $dim_{\mathbb{R}}(M)=2n$ with $n\geq1$, let $\gamma$ be a simple smooth closed curve that lies in a (holomorphic) coordinate chart and that can be taken as small as necessary (one can choose $\gamma$ such $diam(\gamma)<\epsilon$ with $\epsilon>0$). If i want to solve Plateau problem for a $\gamma$ so small such that the solution is contained in the coordinate chart, do i get something that can be parametrized by closed disc?