  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.
