## Abstract

Let *N* be a compact domain with weakly two-convex boundary *∂N* in a Riemannian 4-manifold *M* with nonnegative isotropic curvature. If *D* is a stable minimal disk in *N* with *∂D* ⊂ *∂N* that solves the free boundary problem, then *D* is infinitesimally holomorphic; moreover, it is ± holomorphic if *M* is a Kähler surface with positive scalar curvature, and it is holomorphic for some complex structure if *M* is a hyperkähler surface. We also show that if *N* is a compact domain in *M* of dim *M* ≧ 4 with nonnegative isotropic curvature and *∂N* is two-convex, then *π*
_{1}(*∂N*) → *π*
_{1}(*N*) is injective.

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