Isometric and CR pluriharmonic immersions of three dimensional CR manifolds in Euclidean spaces

TL;DR

This paper characterizes three-dimensional strongly pseudoconvex CR-manifolds immersed in Euclidean spaces using a differential complex and integral representations, and classifies CR-pluriharmonic immersions in four-dimensional space.

Contribution

It introduces a new bigraded differential complex framework for isometric immersions of CR-manifolds and provides a complete classification of CR-pluriharmonic immersions in -dimensional Euclidean space.

Findings

Characterization of 3D strongly pseudoconvex CR-manifolds in Euclidean space.

Development of an integral representation of Weierstrass type.

Complete classification of CR-pluriharmonic immersions in 4-space.

Abstract

Using a bigraded differential complex depending on the CR and pseudohermitian structure, we give a characterization of three-dimensional strongly pseudoconvex pseudo-hermitian CR-manifolds isometrically immersed in Euclidean space $\mathbb{R}^n$ in terms of an integral representation of Weierstrass type. Restricting to the case of immersions in $\mathbb{R}^4$, we study harmonicity conditions for such immersions and give a complete classification of CR-pluriharmonic immersions.