Institute for Mathematical Physics Rigidity for Local Holomorphic Conformal Embeddings from B Rigidity for Local Holomorphic Conformal Embeddings from B

In this article, we study local holomorphic conformal embeddings from B into B1 × · · · ×BNm with respect to the normalized Bergman metrics. 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, we show that each component is either a constant map or a proper holomorphic map between balls. Applying a linearity criterion of Huang, we conclude the total geodesy of non-constant components.