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