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An end to end gluing construction for metrics of constant Q-curvature

A. Sophie Aiken, Rayssa Caju, Jesse Ratzkin, Almir Silva Santos

TL;DR

This paper develops an end-to-end gluing method to produce new complete metrics of constant $Q$-curvature on finitely punctured spheres by joining known Delaunay-type and singular Yamabe pieces. By a careful linear theory around Delaunay ends and a contraction mapping, the authors perturb approximate glued metrics to exact solutions of $Q_g=\frac{n(n^2-4)}{8}$, establishing unmarked nondegeneracy. They then show that for $k\ge4$ punctures the unmarked moduli space is not contractible by constructing a nontrivial loop via end-to-end gluing and $SO(n-1)$-rotations, together with Liouville-type classification of conformal structures. The results extend gluing techniques to a fourth-order Paneitz-type problem and have implications for the global geometry and topology of singular Yamabe-type moduli spaces.

Abstract

We produce many new complete, constant Q-curvature metrics on finitely punctured spheres by gluing together known examples. In our construction we truncate one end of each summand and glue the two summands together "end-to-end," where we've truncated them. We use this construction to show that the unmarked moduli space of solutions with a fixed number of punctures is topologically nontrivial provided the number of punctures is at least four.

An end to end gluing construction for metrics of constant Q-curvature

TL;DR

This paper develops an end-to-end gluing method to produce new complete metrics of constant -curvature on finitely punctured spheres by joining known Delaunay-type and singular Yamabe pieces. By a careful linear theory around Delaunay ends and a contraction mapping, the authors perturb approximate glued metrics to exact solutions of , establishing unmarked nondegeneracy. They then show that for punctures the unmarked moduli space is not contractible by constructing a nontrivial loop via end-to-end gluing and -rotations, together with Liouville-type classification of conformal structures. The results extend gluing techniques to a fourth-order Paneitz-type problem and have implications for the global geometry and topology of singular Yamabe-type moduli spaces.

Abstract

We produce many new complete, constant Q-curvature metrics on finitely punctured spheres by gluing together known examples. In our construction we truncate one end of each summand and glue the two summands together "end-to-end," where we've truncated them. We use this construction to show that the unmarked moduli space of solutions with a fixed number of punctures is topologically nontrivial provided the number of punctures is at least four.

Paper Structure

This paper contains 17 sections, 13 theorems, 136 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Delaunay metrics
  4. Deformations of Delaunay metrics
  5. Asymptotics
  6. The Approximate Solution
  7. Linear Analysis
  8. Function spaces
  9. The Jacobi operator on a Delaunay solution
  10. Linear analysis on the approximate solution
  11. Nonlinear Analysis
  12. The geometric deformations
  13. Solving the gluing problem with a contraction
  14. Nondegeneracy
  15. Auxiliary Lemmas
  16. ...and 2 more sections

Key Result

Theorem 1.2

For $i=1,2$ consider $M_i=\mathbf{S}^n\backslash\{q_0^i,\ldots,q_{k_i}^i\}$, with $k_i\geq 2$. Let $(M_1, g_1)$ and $(M_2, g_2)$ be complete, constant $Q$-curvature metrics. We also assume the summands $(M_i, g_i)$ are both unmarked nondegenerate, that the asymptotic necksizes at $q_0^1$ and $q_0^2$

Figures (4)

  • Figure 1: The two annuli we identify in $M_1$ and $M_2$.
  • Figure 2: The approximate solution $(M,g)$ with $k_1+k_2$ punctures.
  • Figure 3: The punctured ball $B_{r_0}(q_0^i)\backslash\{q_0^i\}$ in $\mathbf{S}^n$ is identified with the half cylinder $(0,+\infty)\times\mathbf{S}^{n-1}$.
  • Figure 4: Deforming a metric by an element of the deficiency space.

Theorems & Definitions (27)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 2.1: Frank and König MR3869387
  • Lemma 3.1
  • proof
  • Definition 4.1
  • Definition 4.2
  • ...and 17 more