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Singular metrics of constant negative $Q$-curvature in Euclidean spaces

Tobias König, Yamin Wang

TL;DR

This work analyzes singular metrics with constant negative $Q$-curvature in $\mathbb{R}^n$ by studying solutions to $(-\Delta)^{n/2}u=-e^{nu}$ on $\mathbb{R}^n\setminus\{0\}$ with finite volume $\Lambda$. It develops a precise decomposition $u=v+p+q(\cdot/|\cdot|^2)+\beta\ln|x|$ linking the regular part $v$ to polynomial and Kelvin-transformed components, and proves sharp bounds and asymptotics for $v$, enabling a complete classification of singular solutions. For $n=1,2$ no singular solutions exist, while for $n\ge 3$ the authors establish existence results with prescribed asymptotics via a variational framework tied to the Paneitz operator, yielding metrics of constant negative $Q$-curvature that may be smooth or possess logarithmic or polynomial singularities at the origin. The results extend the theory of prescribed $Q$-curvature to negative curvature in all dimensions and refine prior work on nonsingular negative curvature as well as the positive-curvature regime.

Abstract

We study singular metrics of constant negative $Q$-curvature in the Euclidean space $\mathbb{R}^n$ for every $n \geq 1$. Precisely, we consider solutions to the problem \[ (-Δ)^{n/2}u=-e^{nu}\quad \text{on}\quad\mathbb{R}^{n}\backslash \{0\}, \] under a finite volume condition $Λ:=\int_{\mathbb{R}^n}e^{nu}dx$. We classify all singular solutions of the above equation based on their behavior at infinity and zero. As a consequence of this, when $n=1,2$, we show that there is actually no singular solution. Then adapting a variational technique, we obtain that for any $n\geq 3$ and $Λ>0$, the equation admits solutions with prescribed asymptotic behavior. These solutions correspond to metrics of constant negative $Q$-curvature, which are either smooth or have a singularity at the origin of logarithmic or polynomial type. The present paper complements previous works on the case of positive $Q$-curvature, and also sharpens previous results in the nonsingular negative $Q$-curvature case.

Singular metrics of constant negative $Q$-curvature in Euclidean spaces

TL;DR

This work analyzes singular metrics with constant negative -curvature in by studying solutions to on with finite volume . It develops a precise decomposition linking the regular part to polynomial and Kelvin-transformed components, and proves sharp bounds and asymptotics for , enabling a complete classification of singular solutions. For no singular solutions exist, while for the authors establish existence results with prescribed asymptotics via a variational framework tied to the Paneitz operator, yielding metrics of constant negative -curvature that may be smooth or possess logarithmic or polynomial singularities at the origin. The results extend the theory of prescribed -curvature to negative curvature in all dimensions and refine prior work on nonsingular negative curvature as well as the positive-curvature regime.

Abstract

We study singular metrics of constant negative -curvature in the Euclidean space for every . Precisely, we consider solutions to the problem under a finite volume condition . We classify all singular solutions of the above equation based on their behavior at infinity and zero. As a consequence of this, when , we show that there is actually no singular solution. Then adapting a variational technique, we obtain that for any and , the equation admits solutions with prescribed asymptotic behavior. These solutions correspond to metrics of constant negative -curvature, which are either smooth or have a singularity at the origin of logarithmic or polynomial type. The present paper complements previous works on the case of positive -curvature, and also sharpens previous results in the nonsingular negative -curvature case.
Paper Structure (8 sections, 19 theorems, 140 equations)

This paper contains 8 sections, 19 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Preliminary steps
  3. Bounds on $v$
  4. A decomposition of $u$
  5. Proof of Theorems \ref{['tr2']} and \ref{['th2x']}
  6. Proof of Theorem \ref{['th2']}
  7. Proof of Theorem \ref{['tr3s']}
  8. Appendix: Some auxiliary lemmas

Key Result

Theorem 1.1

Let $n\geq 1$ and $u$ be a solution to (az2). Let $v(x)$ be defined as in (aaz1). Then there exist $\beta\in\mathbb{R}$ and polynomials $p$, $q$ of even degree at most $n-1$ bounded from above such that where $v$ satisfies Moreover, assume either (i) $\beta>-1$, $p(x)\rightarrow-\infty$ as $|x|\rightarrow\infty$, or (ii) $\beta\leq-1$, $q(x)\rightarrow-\infty$ as $|x|\rightarrow\infty.$ Then for

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 3.1
  • ...and 11 more