# $C_0$-positivity and a classification of closed three-dimensional CR torsion solitons

## TL;DR

This paper classifies three-dimensional CR torsion solitons based on $C_0$-positivity, establishes an obstruction for such positivity, and shows that torsion solitons are standard Sasakian space forms, with implications for CR torsion flow.

## Contribution

It introduces a classification of CR Yamabe solitons using $C_0$-positivity and proves that CR torsion solitons are standard Sasakian space forms, providing new insights into CR geometry.

## Key findings

- Obstruction for $C_0$-positive pseudohermitian curvature.
- Classification of CR Yamabe solitons by $C_0$-positivity and negativity.
- CR torsion solitons are standard Sasakian space forms.

## Abstract

A closed CR 3-manifold is said to have $C_{0}$-positive pseudohermitian curvature if $(W+C_{0}Tor)(X,X)>0$ for any $0\neq X\in T_{1,0}(M)$. We discover an obstruction for a closed CR 3-manifold to possess $C_{0}$-positive pseudohermitian curvature. We classify closed three-dimensional CR Yamabe solitons according to $C_{0}$-positivity and $C_{0}$-negativity whenever $C_{0}=1$ and the potential function lies in the kernel of Paneitz operator. Moreover, we show that any closed three-dimensional CR torsion soliton must be the standard Sasakian space form. At last, we discuss the persistence of $C_{0}$-positivity along the CR torsion flow starting from a pseudo-Einstein contact form.