The k-Yamabe problem on CR manifolds

TL;DR

This paper introduces the pseudohermitian k-curvature and studies the k-Yamabe problem on CR manifolds, providing variational characterizations, bounds, and classification results for solutions, extending classical Yamabe problem insights to CR geometry.

Contribution

It defines the pseudohermitian k-curvature, formulates the k-Yamabe problem for CR manifolds, and establishes variational and classification results, including bounds and uniqueness for solutions.

Findings

Solutions with null Cotton tensor are critical points of a geometric functional.

The infimum of the functional on the CR sphere is computed.

Upper bounds for the functional are established for all compact CR manifolds.

Abstract

We introduce the notion of pseudohermitian k-curvature, which is a natural extension of the Webster scalar curvature, on an orientable manifold endowed with a strictly pseudoconvex pseudohermitian structure (referred here as a CR manifold) and raise the k-Yamabe problem on a compact CR manifold. When k=1, the problem was proposed and partially solved by Jerison and Lee for CR manifolds non-locally CR-equivalent to the CR sphere. For k > 1, the problem can be translated in terms of the study of a fully nonlinear equation of type complex k-Hessian. We provide some partial answers related to the CR k-Yamabe problem. We establish that its solutions with null Cotton tensor are critical points of a suitable geometric functional constrained to pseudohermitian structures of unit volume. Thanks to this variational property, we establish a Obata type result for the problem and also compute the infimum of the functional on the CR sphere. Furthermore, we show that this value is an upper bound for the corresponding one on any compact CR manifolds and, assuming the CR Yamabe invariant is positive, we prove that such an upper bound is only attained for compact CR manifolds locally CR-equivalent to the CR sphere. In the Riemannian field, recent advances have been produced in a series of outstanding works.