We construct a complete Ricci-flat K~hler metric on the complexification of a compact rank one symmetric space. Our method is to look for a K~ihler potential of the form r = f(u), where u satisfies the homogeneous Monge-Amp~re equation. We use the high degree of symmetry present to reduce the non-linear partial differential equation governing the Ricci curvature to a simple second-order ordinary differential equation for the function f. To prove that the resulting metric is complete requires some techniques from symplectic geometry.

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We study the Segal Bargmann transform on a symmetric space X of compact type, mapping L 2 (X ) into holomorphic functions on the complexification X C . We invert this transform by integrating against a``dual'' heat kernel measure in the fibers of a natural fibration of X C over X. We prove that the Segal Bargmann transform is an isometry from L 2 (X ) onto the space of holomorphic functions on X C which are square integrable with respect to a natural measure. These results extend those of B. Hall in the compact group case. Academic Press

Intrinsic microlocal analysis and inversion formulae for the heat equation on compact real-analytic riemannian manifolds Annales scientifiques de l'É.N.S. 4 e série, tome 29, n o 6 (1996), p. 669-736 © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1996, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

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