Some Global Properties of Tubes

A Grauert tube, or adapted complex structure, gives a canonical complexification of a real analytic Riemannian manifold defined on a neighborhood of the zero section of the (co-)tangent bundle. Such a tube is called entire if it is defined on the whole tangent bundle of the real analytic Riemannian manifold M, g. Very few examples of such entire tubes are known, but a general statement has proved elusive since Szőke found all circle invariant such metrics g on S. We review known properties of entire tubes and relate them to algebraic geometry, Boutet de Monvel’s Paley-Wiener theorem and the Ricci flow on S. We also compare the tube as a complexification to constructions of LeBrun-Mason and Donaldson. This represents collaborations with several colleagues: Raul Aguilar, Zhou Zhang, Zuoqin Wang and Kin-Kwan Leung.