  We remark on two different free-boundary problems for holomorphic curves in
nearly-K\"{a}hler 6-manifolds. First, we observe that a holomorphic curve in a
geodesic ball $B$ of the round 6-sphere that meets $\partial B$ orthogonally
must be totally geodesic. Consequently, we obtain rigidity results for
reflection-invariant holomorphic curves in $\mathbb{S}^6$ and associative cones
in $\mathbb{R}^7$.
  Second, we consider holomorphic curves with boundary on a Lagrangian
submanifold in a strict nearly-K\"{a}hler 6-manifold. By deriving a suitable
second variation formula for area, we observe a topological lower bound on the
Morse index. In both settings, our methods are complex-geometric, closely
following arguments of Fraser-Schoen and Chen-Fraser.

Differential Geometry