Let $\gamma: S^1 \to \partial B \subset \mathbf{R}^3$ be a smooth, simple closed curve in the boundary of the unit ball. Suppose that $\gamma$ intersects every horizontal plane $\Pi\_t = \{ z = t\}$ at most twice:
\begin{equation}
\# \gamma(S^1) \cap \Pi\_t \leq 2 \quad \text{for all $t$.}
\end{equation}



> 
> Does $\gamma$ bound a unique minimal disk (resp. minimal surface)?
> 
> 
> 



A geometric criterion for uniqueness in the Plateau problem?
The answer is no.


Take two parallel circles of unit radius in $z=\pm \epsilon$ with $\epsilon$ small. Tilt the two circles very slightly toward one another. This satisfies your hypotheses. There are clearly at least three minimal surfaces with this boundary the tilted flat disks and a stable and unstable annulus (obtained by perturbing the appropriate pieces of a catenoid).


If you want non-uniqueness in the class of disks then form a small bridge between the two closest points on the circle. This boundary also satisfies your hypotheses. The [bridge principle](https://link.springer.com/article/10.1007/BF01192091) implies there is a stable minimal surfaces with this boundary that should look like the two flat disks joined by a small bridge. However, it is pretty clear the area minimizing disk is the "ribbon" solution going around the outside. Hence, you have two distinct minimal disks.


