Prescribing $Q$-curvature on even-dimensional manifolds with conical singularities

TL;DR

This paper establishes existence and multiplicity results for prescribed $Q$-curvature metrics with conical singularities on even-dimensional manifolds, using blow-up analysis and variational methods, especially in the supercritical regime.

Contribution

It provides the first known existence results for supercritical conic manifolds in dimensions higher than two, extending the theory beyond the sphere case.

Findings

Proves existence of prescribed $Q$-curvature metrics with conical singularities.

Develops a blow-up analysis for a high-order PDE in this setting.

Applies a min-max variational approach to establish multiplicity results.

Abstract

On a $2m$-dimensional closed manifold we investigate the existence of prescribed $Q$-curvature metrics with conical singularities. We present here a general existence and multiplicity result in the supercritical regime. To this end, we first carry out a blow-up analysis of a $2m$th-order PDE associated to the problem and then apply a variational argument of min-max type. For $m>1$, this seems to be the first existence result for supercritical conic manifolds different from the sphere.