# Contact structures, CR Yamabe invariant, and connected sum

## TL;DR

This paper introduces a new global invariant for contact manifolds with CR structures, analyzes its behavior under topological operations, and provides bounds in specific cases, advancing understanding of contact geometry and CR invariants.

## Contribution

It defines the contact CR Yamabe invariant $\sigma_c$, studies its monotonicity under handle attaching and connected sum, and establishes a lower bound in certain cases.

## Key findings

- $\sigma_c$ is non-decreasing under handle attaching
- $\sigma_c$ is non-decreasing under connected sum
- Lower bounds for $\sigma_c$ in specific cases

## Abstract

We propose a global invariant $\sigma_c$ for contact manifolds which admit a strictly pseudoconvex CR structure, analogous to the Yamabe invariant $\sigma$. We prove that this invariant is non-decreasing under handle attaching and under connected sum. We then give a lower bound on $\sigma_c$ in a particular case.