CR-Analogue of Siu-$\partial\bar{\partial}$-formula and Applications to Rigidity problem for pseudo-Hermitian harmonic maps

TL;DR

This paper extends Siu's $ar{d}$-formula to pseudo-Hermitian manifolds, introduces pseudo-Hermitian harmonic maps, and proves a CR rigidity theorem for such maps into negatively curved K"ahler manifolds.

Contribution

It develops a CR analogue of Siu's $ar{d}$-formula and establishes a rigidity theorem for pseudo-Hermitian harmonic maps.

Findings

Established several versions of Siu's $ar{d}$-formula for pseudo-Hermitian manifolds.

Defined and analyzed pseudo-Hermitian harmonic maps.

Proved a CR version of Siu's Rigidity Theorem under specific curvature conditions.

Abstract

We give several versions of Siu's $\partial\bar{\partial}$-formula for maps from a strictly pseudoconvex pseudo-Hermitian manifold $(M^{2m+1}, \theta)$ into a K\"ahler manifold $(N^n, g)$. We also define and study the notion of pseudo-Hermitian harmonicity for maps from $M$ into $N$. In particular, we prove a CR version of Siu Rigidity Theorem for pseudo-Hermitian harmonic maps from a pseudo-Hermitian manifold with vanishing Webster torsion into a K\"ahler manifold having strongly negative curvature.