Concentration phenomena for the fractional Q-curvature equation in dimension 3 and fractional Poisson formulas

Abstract

We study the compactness properties of metrics of prescribed fractional (Formula presented.) -curvature of order 3 in (Formula presented.). We will use an approach inspired from conformal geometry, seeing a metric on a subset of (Formula presented.) as the restriction of a metric on (Formula presented.) with vanishing fourth-order (Formula presented.) -curvature. We will show that a sequence of such metrics with uniformly bounded fractional (Formula presented.) -curvature can blow up on a large set (roughly, the zero set of the trace of a non-positive bi-harmonic function (Formula presented.) in (Formula presented.)), in analogy with a four-dimensional result of Adimurthi–Robert–Struwe, and construct examples of such behaviour. In doing so, we produce general Poisson-type representation formulas (also for higher dimension), which are of independent interest