# A geometric criterion for uniqueness in the Plateau problem?

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: $$$$\# \gamma(S^1) \cap \Pi_t \leq 2 \quad \text{for all t.}$$$$

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

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).