On the Yamabe Problem on contact Riemannian Manifolds

TL;DR

This paper extends the Yamabe problem to contact Riemannian manifolds with non-integrable complex structures, proving the problem's solvability and establishing bounds on the Yamabe invariant.

Contribution

It introduces a framework for the Yamabe problem on contact Riemannian manifolds with non-integrable structures and proves its solvability in this setting.

Findings

Yamabe invariant is always less than that of the Heisenberg group for non-integrable structures.

The Yamabe problem is always solvable on contact Riemannian manifolds with non-integrable complex structures.

Abstract

Contact Riemannian manifolds, whose complex structures are not necessarily integrable, are generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno connection plays the role of the Tanaka-Webster connection of a pseudohermitian manifold. Conformal transformations and the Yamabe problem are also defined naturally in this setting. By constructing the special frames and the normal coordinates on a contact Riemannian manifold, we prove that if the complex structure is not integrable, its Yamabe invariant on a contact Riemannian manifold is always less than the Yamabe invariant of the Heisenberg group. So the Yamabe problem on a contact Riemannian manifold is always solvable.