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$A\_\infty$ weights and compactness of conformal metrics under $L^{n/2}$ curvature bounds. (arXiv:1810.05387v1 [math.DG])
来源于:arXiv
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. 查看全文>>