On Rigidity of Grauert Tubes

Given a real-analytic Riemannian manifold M there exists a canonical complex structure on part of its tangent bundle which turns leaves of the Riemannian foliation on TM into holomorphic curves. A Grauert tube over M of radius r, denoted as T rM , is the collection of tangent vectors of M of length less than r equipped with this canonical complex structure. We say the Grauert tube T rM is rigid if Aut(T rM) is coming from Isom(M). In this article, we prove the rigidity for Grauert tubes over quasi-homogeneous Riemannian manifolds. A Riemannian manifold (M, g) is quasi-homogeneous if the quotient space M/Isom0(M) is compact. This category has included compact Riemannian manifolds, homogeneous Riemannian manifolds, co-compact Riemannian manifolds whose isometry groups have dimensions ≥ 1, and products of the above spaces.