Non-homothetic complete periodic contact forms with constant Tanaka--Webster scalar curvature

TL;DR

This paper investigates the existence of complete contact forms with constant Tanaka-Webster scalar curvature on non-compact CR manifolds, revealing conditions under which infinitely many such forms exist, especially related to the fundamental group's properties.

Contribution

It establishes the existence of infinitely many non-homothetic contact forms with constant scalar curvature on certain non-compact CR manifolds, linking this to the fundamental group's profinite completion.

Findings

Existence of infinitely many non-homothetic contact forms under specific conditions.

Application to complements of spheres and circle bundles over Kähler manifolds.

Connection between fundamental group properties and contact form existence.

Abstract

We study the existence problem for complete contact forms with constant Tanaka--Webster scalar curvature on non-compact strictly pseudoconvex CR manifolds. We prove that, under mild assumptions, the universal cover of a compact strictly pseudoconvex CR manifold admits infinitely many non-homothetic such contact forms whenever its fundamental group has infinite profinite completion. As applications, we treat complements of real or complex spheres in the standard CR sphere, as well as circle bundles over compact K\"{a}hler manifolds and the boundary of a Reinhardt domain.