# Semi-isometric CR immersions of CR manifolds into K\"ahler manifolds and applications

## TL;DR

This paper investigates semi-isometric CR immersions into K"ahler manifolds, providing new conditions for CR umbilicality, extending linearity results, and establishing a gap theorem using eigenvalue analysis of the Kohn Laplacian.

## Contribution

It introduces new criteria for CR umbilicality, extends linearity theorems to three-dimensional cases, and proves a gap theorem for semi-isometric CR immersions, utilizing eigenvalue techniques.

## Key findings

- Extended Webster's theorem on CR umbilicality to real hypersurfaces.
- Generalized the linearity theorem for 3D CR manifolds into spheres.
- Proved a first gap theorem for semi-isometric CR immersions into complex Euclidean space.

## Abstract

We study the second fundamental form of semi-isometric CR immersions from strictly pseudoconvex CR manifolds into K\"ahler manifolds. As an application, we give a precise condition for the CR umbilicality of real hypersurfaces, extending an well-known theorem by Webster on the nonexistence of CR umbilical points on generic real ellipsoids. As other applications, we extend the linearity theorem of Ji-Yuan for CR immersions into spheres with vanishing second fundamental form to the important case of three-dimensional manifolds, and prove the ``first gap'' theorem in the spirit of Webster, Faran, Cima-Suffridge, and Huang for semi-isometric CR immersions into a complex euclidean space of ``low'' codimension. Our new approach to the linearity theorem is based on the study of the first positive eigenvalue of the Kohn Laplacian.