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Symplectic geometry and the uniqueness of Grauert tubes

D. Burns·R. Hind
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Abstract

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.

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D. Burns, R. Hind. 2000-10-30. Symplectic geometry and the uniqueness of Grauert tubes. https://arxiv.org/abs/math/0010299

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