Vol. 14, No. 7, 2021

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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
Abstract

We study sequences of conformal deformations of a smooth closed Riemannian manifold of dimension n, assuming uniform volume bounds and Ln2 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-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 n with infinite volume and finite Ln2 norm of the scalar curvature satisfies the Euclidean isoperimetric inequality.

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