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Conformal metrics with finite total Q-curvature revisited

Mingxiang Li

TL;DR

This work extends the Chern–Gauss–Bonnet program to higher dimensions by studying complete conformal metrics with finite total $Q$-curvature and elucidating how scalar curvature controls the $Q$-curvature integral. The authors introduce the conformal mass $m_c(g)$ and prove that normality is equivalent to having $m_c(g)>- frac{1}{ ext{}}$, with an explicit formula for $m_c(g)$ in terms of the total $Q$-curvature when the metric is normal. They establish sharp bounds on $ rac{1}{(n-1)!|S^n|}\,ig|S^nig| igotimes Q e^{nu}$ under various decay assumptions on $R_g$ near infinity, derive asymptotics for normal solutions, and develop a framework for complete manifolds with finitely many simple ends. These results yield volume-entropy-type controls, a positive-mass-type theorem for Q-curvature, and rigidity phenomena, providing powerful tools for understanding the global geometry of conformal manifolds with finite total Q-curvature.

Abstract

Given a conformal metric with finite total Q-curvature, we show that the assumptions on scalar curvature sensitively govern the Q-curvature integral. Additionally, we introduce a conformal mass for such manifolds. Using such mass, we provides a necessary and sufficient condition for the metric to be normal without assuming metric completeness. As applications, we derive volume comparison theorems and prove a positive mass type theorem related to Q-curvature.

Conformal metrics with finite total Q-curvature revisited

TL;DR

This work extends the Chern–Gauss–Bonnet program to higher dimensions by studying complete conformal metrics with finite total -curvature and elucidating how scalar curvature controls the -curvature integral. The authors introduce the conformal mass and prove that normality is equivalent to having , with an explicit formula for in terms of the total -curvature when the metric is normal. They establish sharp bounds on under various decay assumptions on near infinity, derive asymptotics for normal solutions, and develop a framework for complete manifolds with finitely many simple ends. These results yield volume-entropy-type controls, a positive-mass-type theorem for Q-curvature, and rigidity phenomena, providing powerful tools for understanding the global geometry of conformal manifolds with finite total Q-curvature.

Abstract

Given a conformal metric with finite total Q-curvature, we show that the assumptions on scalar curvature sensitively govern the Q-curvature integral. Additionally, we introduce a conformal mass for such manifolds. Using such mass, we provides a necessary and sufficient condition for the metric to be normal without assuming metric completeness. As applications, we derive volume comparison theorems and prove a positive mass type theorem related to Q-curvature.
Paper Structure (5 sections, 31 theorems, 196 equations)

This paper contains 5 sections, 31 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Some estimates for normal solutions
  3. Scalar curvature and Q-curvature integral
  4. Complete manifolds with finitely many simple ends
  5. Conformal mass and normal metric

Key Result

Theorem 1.1

(See CQY, Fa, NX) Given a complete and conformal metric $g=e^{2u}|dx|^2$ on $\mathbb{R}^n$ where $n\geq 4$ is an even integer with finite total Q-curvature. Suppose that the scalar curvature $R_g\geq 0$ near infinity. Then there holds where $|\mathbb{S}^n|$ denotes the volume of standard n-sphere.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 37 more