On rigidity of Grauert tubes over homogeneous Riemannian manifolds
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Su-Jen Kan
Abstract
Given a real-analytic Riemannian manifold X there is a canonical complex structure, which is compatible with the canonical complex structure on T*X and makes the leaves of the Riemannian foliation on TX into holomorphic curves, on its tangent bundle. A Grauert tube over X of radius r, denoted as TrX, is the collection of tangent vectors of X of length less than r equipped with this canonical complex structure.
In this article, we prove the following two rigidity properties of Grauert tubes. First, for any real-analytic Riemannian manifold such that rmax > 0, we show that the identity component of the automorphism group of TrX is isomorphic to the identity component of the isometry group of X provided that r < rmax. Secondly, let X be a homogeneous Riemannian manifold and let the radius r < rmax, then the automorphism group of TrX is isomorphic to the isometry group of X and there is a unique Grauert tube representation for such a complex manifold TrX.
© Walter de Gruyter
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Articles in the same Issue
- The resolution property for schemes and stacks
- Beyond Endoscopy and special forms on GL(2)
- A characterization of quantum groups
- Local properties of self-dual harmonic 2-forms on a 4-manifold
- Wild Euler systems of elliptic units and the Equivariant Tamagawa Number Conjecture
- Existence of families of Galois representations and new cases of the Fontaine-Mazur conjecture
- The rank of elliptic surfaces in unramified abelian towers
- Crystal dissolution and precipitation in porous media: Pore scale analysis
- On rigidity of Grauert tubes over homogeneous Riemannian manifolds
