On rigidity of Grauert tubes over Riemannian manifolds of constant curvature

It is well-known that a real analytic manifold X admits a complexification XC , a complex manifold that contains X as the fixed point set of an antiholomorphic involution. This can be seen as follows.The transition functions defining the manifold X are real-analytic local diffeomorphisms of Rn. The Taylor expansions of these transition functions can be considered as local biholomorphisms of Cn, hence they serve as transition functions of a complex manifold. The germ of the complexification XC is unique. Every sufficiently small tubular neighborhood Ω of X in the tangent bundle TX admits a real analytic diffeomorphism into XC that fixes X . Therefore a sufficiently small tubular neighborhood ofX in TX has a complex structure and can be considered as a complexification of X . In general the complex structure on the tubular neighborhood is not unique since there are manyways to embed it intoXC . There have been a lot of interest in finding canonical complex structures on tubular neighborhoods ofX in TX . With additional datum of a real analytic Riemannian metric g on X a canonical complex structure can be specified for sufficiently small tubular neighborhoods Ω of X in TX (see [GS, LS, S1]). There is a unique complex structure on Ω such that the map f(σ + iτ) = (τγ′(σ))γ(σ), σ + iτ ∈ C, is holomorphic, wherever it is defined,