Symplectic geometry and the uniqueness of Grauert tubes
  A compact real analytic Riemannian manifold M admits a canonical complexification with plurisubharmonic exhaustion function satisfying the homogeneous complex Monge-Ampere equation, called a Grauert tube. From the point of view of complex analysis, several authors have considered whether a given complex manifold can arise in more than one way from this construction. We show that given a compact M and a finite exhaustion, the underlying Riemannian structure is unique. The proof uses the technique of holomorphic disks spanning two exact Lagrangian submanifolds of the cotangent bundle of M, and Schwarz reflection. 
D. Burns (Ann Arbor) and R. Hind (South Bend)