CR-HARMONIC MAPS

We develop the notion of renormalized energy in Cauchy–Riemann (CR) geometry for maps from a strictly pseudoconvex pseudo-Hermitian manifold to a Riemannian manifold. This energy is a CR invariant functional whose critical points, which we call CR-harmonic maps, satisfy a CR covariant partial differential equation. The corresponding operator coincides on functions with the CR Paneitz operator.

A Perturbation Approach for Paneitz Energy on Standard Three Sphere

We present another proof of the sharp inequality for Paneitz operator on the standard three sphere, in the spirit of subcritical approximation for the classical Yamabe problem. To solve the perturbed problem, we use a symmetrization process which only works for extremal functions. This gives a new example of symmetrization for higher-order variational problems.

Compactness properties for geometric fourth order elliptic equations with application to the Q-curvature flow

We prove the compactness of solutions of general fourth order elliptic equations which areL^{1}-perturbations of theQ-curvature equation on compact Riemannian 4-manifolds. Consequently, we prove the global existence and convergence of theQ-curvature flow on a generic class of Riemannian 4-manifolds. As a by-product, we give a positive answer to an open question by A. Malchiodi [J. reine angew. Math. 594 (2006), 137–174] on the existence of bounded Palais–Smale sequences for theQ-curvature problem when the Paneitz operator is positive with trivial kernel.