Existence and Multiplicity results for the prescribed Webster Scalar Curvature Problem on three $C R $ manifolds

TL;DR

This paper establishes new existence and multiplicity results for contact forms with prescribed Webster scalar curvature on certain 3-dimensional CR manifolds, using Euler-Hopf formulas and Morse index bounds.

Contribution

It introduces a novel approach using Euler-Hopf type formulas to prove existence and multiplicity of solutions, providing bounds on Morse index and number of solutions.

Findings

Existence of contact forms with prescribed Webster scalar curvature under certain conditions.

Upper bounds on the Morse index of solutions.

Lower bounds on the number of conformal contact forms with the same curvature.

Abstract

This paper is devoted to the existence of contact forms of prescribed Webster scalar curvature on a $3-$dimensional CR compact manifold locally conformally CR equivalent to the unit sphere $\mathbb{S}^{3}$ of $\mathbb{C}^{2}$. Due to Kazdan-Warner type obstructions, conditions on the function $H$ to be realized as a Webster scalar curvature have to be given. We prove new existence results based on a new type of Euler-Hopf type formula. Our argument gives an upper bound on the Morse index of the obtained solution. We also give a lower bound on the number of conformal contact forms having the same Webster scalar curvature.