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Moduli space theory for complete, constant Q-curvature metrics on finitely punctured spheres

Rayssa Caju, Jesse Ratzkin, Almir Silva Santos

TL;DR

This work studies complete metrics of constant $Q$-curvature conformal to the round sphere with finitely many punctures, characterizing the local structure of the marked moduli space as a real analytic variety of formal dimension $k$ and, under nondegeneracy, as a $k$-dimensional real-analytic manifold. The authors develop a detailed linear analysis around Delaunay ends and singular Yamabe metrics, introducing a deficiency space and using Lyapunov–Schmidt reduction to control nearby solutions. They construct a natural $2k$-dimensional parameter space with a symplectic form and show that, in the smooth case, a small neighborhood of the moduli space embeds as a Lagrangian submanifold in this parameter space. Collectively, the results advance the understanding of singular constant $Q$-curvature metrics, their moduli, and the geometric–analytic mechanisms governing their deformation and stability.

Abstract

We study constant Q-curvature metrics conformal to the round metric on the sphere with finitely many point singularities. We show that the moduli space of solutions with finitely many punctures in fixed positions, equipped with the Gromov-Hausdorff topology, has the local structure of a real analytic variety with formal dimension equal to the number of the punctures. If a nondegeneracy hypothesis holds, we show that a neighborhood in the moduli space is actually a real-analytic manifold of the expected dimension. We also construct a geometrically natural set of parameters, construct a symplectic structure on this parameter space and show that in the smooth case a small neighborhood of the moduli space embeds as a Lagrangian submanifold in the parameter space.

Moduli space theory for complete, constant Q-curvature metrics on finitely punctured spheres

TL;DR

This work studies complete metrics of constant -curvature conformal to the round sphere with finitely many punctures, characterizing the local structure of the marked moduli space as a real analytic variety of formal dimension and, under nondegeneracy, as a -dimensional real-analytic manifold. The authors develop a detailed linear analysis around Delaunay ends and singular Yamabe metrics, introducing a deficiency space and using Lyapunov–Schmidt reduction to control nearby solutions. They construct a natural -dimensional parameter space with a symplectic form and show that, in the smooth case, a small neighborhood of the moduli space embeds as a Lagrangian submanifold in this parameter space. Collectively, the results advance the understanding of singular constant -curvature metrics, their moduli, and the geometric–analytic mechanisms governing their deformation and stability.

Abstract

We study constant Q-curvature metrics conformal to the round metric on the sphere with finitely many point singularities. We show that the moduli space of solutions with finitely many punctures in fixed positions, equipped with the Gromov-Hausdorff topology, has the local structure of a real analytic variety with formal dimension equal to the number of the punctures. If a nondegeneracy hypothesis holds, we show that a neighborhood in the moduli space is actually a real-analytic manifold of the expected dimension. We also construct a geometrically natural set of parameters, construct a symplectic structure on this parameter space and show that in the smooth case a small neighborhood of the moduli space embeds as a Lagrangian submanifold in the parameter space.
Paper Structure (12 sections, 12 theorems, 111 equations)

This paper contains 12 sections, 12 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The choice of a gauge
  4. Delaunay metrics
  5. Local asymptotics
  6. Weighted function spaces
  7. Linear analysis
  8. The linearization about a Delaunay solution
  9. The linearization about a singular Yamabe metric
  10. Local structure in the nondegenerate case
  11. Local structure in the degenerate case
  12. Symplectic structure

Key Result

Theorem 1

For each finite subset $\Lambda \subset \mathbf{S}^n$ with $\# \Lambda =k\geq 3$ the moduli space $\mathcal{M}_\Lambda$ is locally a real analytic variety of formal dimension $k$.

Theorems & Definitions (22)

  • Theorem 1
  • Definition 2
  • Theorem 3
  • Remark 4
  • Theorem 5: Frank and König FK
  • Definition 6
  • Definition 7
  • Lemma 8
  • Proposition 9
  • Proposition 10
  • ...and 12 more