A∞-weights and compactness of conformal metrics under Ln∕2 curvature bounds

Aldana, Clara L.; Carron, Gilles; Tapie, Samuel

doi:10.2140/apde.2021.14.2163

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$A_\infty$-weights and compactness of conformal metrics under $L^{n/2}$ curvature bounds

Clara L. Aldana, Gilles Carron and Samuel Tapie

Vol. 14 (2021), No. 7, 2163–2205

DOI: 10.2140/apde.2021.14.2163

Abstract

We study sequences of conformal deformations of a smooth closed Riemannian manifold of dimension $n$ , assuming uniform volume bounds and $L^{n ∕ 2}$ bounds on their scalar curvatures. Singularities may appear in the limit. Nevertheless, we show that under such bounds the underlying metric spaces are precompact in the Gromov–Hausdorff topology. Our study is based on the use of $A_{\infty}$ -weights from harmonic analysis and the geometric controls that this property induces on the limit spaces thus obtained. Our techniques also show that any conformal deformation of the Euclidean metric on $ℝ^{n}$ with infinite volume and finite $L^{n ∕ 2}$ norm of the scalar curvature satisfies the Euclidean isoperimetric inequality.

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Keywords

compactness of conformal metrics, Muckenhoupt weights, Yamabe equation

Mathematical Subject Classification 2010

Primary: 53C20, 53C23, 58J60

Milestones

Received: 3 August 2019

Accepted: 22 April 2020

Published: 10 November 2021

Authors

Clara L. Aldana
Departamento de Matemáticas y Estadística Universidad del Norte Barranquilla Colombia

Gilles Carron
Université de Nantes Laboratoire de Mathématiques Jean Leray, UMR 6629 Nantes France

Samuel Tapie
Université de Lorraine Institut Elie Cartan de Lorraine, UMR 7502 Nancy France	Laboratoire de Mathématiques Jean Leray, UMR 6629 Nantes France

Vol. 14, No. 7, 2021