# Rigidity of CR morphisms between compact strongly pseudoconvex CR manifolds

### Stephen S.-T. Yau

University of Illinois at Chicago, United States

## Abstract

Let $X_{1}$ and $X_{2}$ be two compact strongly pseudoconvex CR manifolds of dimension $2n−1≥5$ which bound complex varieties $V_{1}$ and $V_{2}$ with only isolated normal singularities in $C_{N_{1}}$ and $C_{N_{2}}$ respectively. Let $S_{1}$ and $S_{2}$ be the singular sets of $V_{1}$ and $V_{2}$ respectively and $S_{2}$ is nonempty. If $2n–N_{2}–1≥1$ and the cardinality of $S_{1}$ is less than $2$ times the cardinality of $S_{2}$, then we prove that any non-constant CR morphism from $X_{1}$ to $X_{2}$ is necessarily a CR biholomorphism. On the other hand, let $X$ be a compact strongly pseudoconvex CR manifold of dimension $3$ which bounds a complex variety $V$ with only isolated normal non-quotient singularities. Assume that the singular set of $V$ is nonempty. Then we prove that any non-constant CR morphism from $X$ to $X$ is necessarily a CR biholomorphism.

## Cite this article

Stephen S.-T. Yau, Rigidity of CR morphisms between compact strongly pseudoconvex CR manifolds. J. Eur. Math. Soc. 13 (2011), no. 1, pp. 175–184

DOI 10.4171/JEMS/247