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

Aleks Jevnikar, Yannick Sire, Wen Yang

TL;DR

This work addresses prescribing the $Q$-curvature on closed even-dimensional manifolds with conical singularities in the supercritical regime. It combines a detailed blow-up analysis of the associated $2m$-th order PDE with a variational min-max framework and StruweMonotonicity trick, under a nonresonance condition ${kappa}_{g_v} otin {Gamma}$. The authors establish a sharp compactness result and prove existence; when ${kappa}_{g_v}$ lies between consecutive multiples of ${Lambda}_m$, they obtain at least one solution, and Morse theory yields multiplicity via the topology of low sublevels modeled by barycenters away from the singularities. This advances conformal geometry with singularities by providing the first general existence results for supercritical conic manifolds beyond the sphere and highlights the role of singular AdamsTrudingerMoser inequalities in controlling concentration phenomena, with potential implications for geometric analysis and related PDEs.

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.

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

TL;DR

This work addresses prescribing the -curvature on closed even-dimensional manifolds with conical singularities in the supercritical regime. It combines a detailed blow-up analysis of the associated -th order PDE with a variational min-max framework and StruweMonotonicity trick, under a nonresonance condition . The authors establish a sharp compactness result and prove existence; when lies between consecutive multiples of , they obtain at least one solution, and Morse theory yields multiplicity via the topology of low sublevels modeled by barycenters away from the singularities. This advances conformal geometry with singularities by providing the first general existence results for supercritical conic manifolds beyond the sphere and highlights the role of singular AdamsTrudingerMoser inequalities in controlling concentration phenomena, with potential implications for geometric analysis and related PDEs.

Abstract

On a -dimensional closed manifold we investigate the existence of prescribed -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 th-order PDE associated to the problem and then apply a variational argument of min-max type. For , this seems to be the first existence result for supercritical conic manifolds different from the sphere.
Paper Structure (5 sections, 17 theorems, 134 equations)

This paper contains 5 sections, 17 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Preliminary facts
  3. Compactness property
  4. Existence result
  5. Appendix: Pohozaev identity

Key Result

Theorem 1.1

Let $(M,D)$ be a supercritical singular $2m$-dimensional closed manifold with $\alpha_j>0$ for $j=1,\dots,N$ and let $Q$ be a smooth positive function on $M$. Suppose that there exists a retraction $R:M\to M^{\hbox{\tiny{R}}}$, with $M^{\hbox{\tiny{R}}}\subset M$ as above. If moreover then there exists a conformal metric on $(M,D)$ with $Q^{2m}$-curvature equal to $Q$.

Theorems & Definitions (33)

  • Definition 1.1
  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Remark 2.1
  • ...and 23 more