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

Clara L. Aldana (1) , Gilles Carron (2) , Samuel Tapie (2, 3)

1 Universidad del Norte, Barranquilla
2 LMJL - Laboratoire de Mathématiques Jean Leray
3 IECL - Institut Élie Cartan de Lorraine

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 pre-compact in the Gromov-Hausdorff topology. Our study is based on the use of $A_\infty$-weights from harmonic analysis, and provides geometric controls 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.

Domaines: Géométrie différentielle [math.DG]

Clara L. Aldana, Gilles Carron, Samuel Tapie. $A_\infty$ weights and compactness of conformal metrics under $L^{n/2}$ curvature bounds. Analysis & PDE, 2021, 14 (7), pp.2163-2205. ⟨10.2140/apde.2021.14.2163⟩. ⟨hal-01893599⟩