Perturbation of the CR fractional Yamabe problem

Abstract

In this note, we first prove the non‐degeneracy property of extremals for the optimal Hardy–Littlewood–Sobolev inequality on the Heisenberg group, as an application, a perturbation result for the CR fractional Yamabe problem is obtained, this generalizes a classical result of Malchiodi and Uguzzoni .

Existence results for the conformal Dirac-Einstein system

“…The condition on the critical point at the south pole of the sphere is needed since we are going to use the standard stereographic projection π, however this condition can be always satisfied by making a unitary transformation which does not affect the generality of the result. The previous result is the analogous of several ones obtained with this kind of hypothesis of Bahri-Coron type on the function k: for instance, for the standard Riemannian case of prescribing the scalar curvature and its generalization to the Q γ curvature see [2,6,8]; in the case of prescribing the Webster curvature in the CR setting and its fractional generalization see [20] and [7]; for the spinorial Yamabe type equations involving the Dirac operator on the sphere see [12]. The idea of the proof follows the abstract perturbation method introduced in [1].…”

The fractional CR curvature equation on the three-dimensional CR sphere

“…We recall that if the function K is constant, then problem (1.1) is the fractional CR Yamabe problem which was studied in [GMM18]. However, the fractional CR Q-curvature problem has been addressed in [CW17] and in [LW18]. Our aim is to handle such a question using some topological and dynamical tools related to the theory of critical points at infinity (see , Bahri [Bah89]) as well as to generalizations of Morse theory.…”