The Geometry of Grauert Tubes and Complexification of Symmetric Spaces

2001/09/24 by Burns, D., Halverscheid, S., Hind, R. · 1 citation
#32M15 #53C35 (Primary) 32M15 #53C55 (Secondary) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0109186

Abstract

We study the canonical complexifications of non-compact Riemannian symmetric spaces G/K by the Grauert tube construction. We determine the maximal such complexification, a domain already constructed in another context by Akhiezer and Gindikin (Math. Ann., 1990), and show that this domain is Stein. We show there is an alternative for a G-invariant complexification: it is either "rigid" (its automorphism group is G), or it is a Hermmitian symmetric space. We also determine when invariant complexifications, especially the maximal one, are Hermitian symmetric. This is expressed simply in terms of the ranks of the symmetric spaces involved.

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• Maximal Weinstein neighborhoods of symmetric R-spaces and their symplectic capacities

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