Holomorphic Isometries between irreducible bounded symmetric domains with respect to the Bergman metric

Abstract

The study of holomorphic isometries between K¨ahler manifolds with real-analytic potential functions dated back to Bochner and Calabi. Especially, in his seminal work on holomorphic isometries in 1953 in which the diastasis was introduced, Calabi established results on existence, uniqueness and analytic continuation of holomorphic isometries into PN , 1 ≤ N ≤ ∞, from which one derives analytic continuation of germs of holomorphic isometries f between bounded domains with respect to the Bergman metric, and the question remained as to whether analytic continuation persists across the boundary. In 2012, the author solved the problem of boundary extension in a very general context, proving in particular that Graph(f) extends to an affine algebraic variety provided that Bergman kernels are rational functions, which applies in particular to the case of germs of holomorphic isometries from the complex unit ball Bn into bounded symmetric domains Ω in their standard embeddings. In 2016 the author published examples of holomorphic isometric embeddings of higher dimensional complex unit balls into irreducible bounded symmetric domains Ω. Images of such isometries are intersections of Ω with cones of minimal rational curves passing through a vertex lying on Reg(∂Ω). In the case where Ω is a Lie sphere (i.e., a type-IV domain), Chan-Mok classified all holomorphic isometric embeddings of complex unit balls into Ω (the codimension 1 cases being also classified by Uppmeier-Wang-Zhang and Xiao-Yuan). When Ω is an irreducible bounded symmetric domain of rank 2 other than a Lie sphere, Mok-Yang proved the uniqueness of holomorphic isometric embeddings of complex unit balls of maximal admissible dimenion into Ω modulo reparametrization. The proof relies on the use of a “duality principle” leading to the determination of isomorphism types of tangent spaces of images of holomorphic isometric embeddings, the method of reconstruction of uniruled projective varieties by means of varieties of minimal rational tangents (VMRTs) and the construction of essentially smooth neighborhoods of certain minimal rational curves by means of the “Thickening Lemma” in the recent work of Mok-Zhang on geometric substructures modelled on pairs of VMRTs..