Geometric criterion for uniqueness in asymptotic Plateau problems

Determine whether a natural geometric condition on Jordan curves at infinity yields uniqueness for the asymptotic Plateau problem in hyperbolic three-space, asymptotically hyperbolic three-manifolds, or universal covers of closed negatively curved three-manifolds.

Background

The paper proves uniqueness and strict stability for minimal disks spanning boundary curves in negatively curved Riemannian 3-balls when the boundary curve has total curvature at most 4π. The authors then discuss the analogous asymptotic Plateau problem, in which a Jordan curve at infinity of a simply connected negatively curved 3-manifold is prescribed and one seeks a unique properly embedded minimal disk spanning it.

Total curvature is unavailable for curves at infinity because the ideal boundary carries only a conformal structure. Existing uniqueness results instead use conditions involving quantities such as the width of the curve, its quasisymmetry constant, or quantitative control of the asymptotic geometry. The stated open question asks for a natural geometric condition at infinity that would play the role of the total-curvature bound in Nitsche’s theorem and in the paper’s main result.