A new Poincar\'e type rigidity phenomenon with applications

TL;DR

This paper establishes a new rigidity phenomenon for local CR mappings over bounded symmetric domains, showing such maps extend to rational biholomorphisms and revealing new geometric structures and classifications.

Contribution

It introduces an optimal rigidity theorem for local CR maps over symmetric domains, generalizing classical results and constructing novel examples of CR hypersurfaces with unique properties.

Findings

Nonconstant local CR maps extend to rational biholomorphisms.

CR diffeomorphisms between anti-canonical circle bundles extend to isomorphisms in higher rank domains.

Existence of infinitely many non-spherical, obstruction flat CR hypersurfaces in higher dimensions.

Abstract

We discover a new Poincar\'e type phenomenon by establishing an optimal rigidity theorem for local CR mappings between circle bundles that are defined in a canonical way over (possibly reducible) bounded symmetric domains. We prove such a local CR map, if nonconstant, must extend to a rational biholomorphism between the corresponding disk bundles. The result includes as a special case the classical Poincar\'e--Tanaka--Alexander theorem. Among other applications, we show, for two irreducible bounded symmetric domains with rank at least two, a local CR diffeomorphism between (open connected pieces of) their anti-canonical circle bundles extends to a norm-preserving holomorphic isomorphism between their anti-canonical bundles. The statement fails in the rank one case. As another application, we construct, for any $n \geq 2,$ a countably infinite family of compact locally homogeneous strongly pseudoconvex CR hypersurfaces (in complex manifolds) of real dimension $2n+1$ with transverse symmetry such that they are all obstruction flat and Bergman logarithmically flat. Moreover, their local CR structures are mutually inequivalent. Such a family cannot exist in dimension three by known results: A Bergman logarithmically flat CR hypersurface must be spherical, and so is a compact obstruction flat CR hypersurface with transverse symmetry.