Yuan Yuan, Johns Hopkins University, USA

Rigidity of the local holomorphic isometries between ball and products of balls

Abstract

I will talk about a joint work with Yuan Zhang on local holomorphic isometric embeddings from the unit ball into the product of unit balls with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic map between complex Euclidean spaces by the algebraic extension theorem derived along the lines of Mok and Mok-Ng. Applying holomorphic continuation and analyzing real analytic subvarieties carefully, each component is either a constant map or a proper holomorphic map between balls. The total geodesy of non-constant components then follows from a linearity criterion of Huang.