#### 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 ${L}^{n∕2}$ 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.

##### Keywords
compactness of conformal metrics, Muckenhoupt weights, Yamabe equation
##### Mathematical Subject Classification 2010
Primary: 53C20, 53C23, 58J60