consider Ricci flow invariant cones 𝒞 in the space of curvature
operators lying between the cones “nonnegative Ricci curvature” and
“nonnegative curvature operator”. Assuming some mild control on the scalar curvature
of the Ricci flow, we show that if a solution to the Ricci flow has its
curvature operator which satisfies at the
then it satisfies on some time interval
depending only on
the scalar curvature control.
This allows us to link Gromov–Hausdorff
convergence and Ricci flow convergence when the limit is smooth and
along the sequence of initial
conditions. Another application is a stability result for
manifolds whose curvature operator is almost in 𝒞.
Finally, we study the case where 𝒞 is contained in the
cone of operators whose sectional curvature is nonnegative. This allows
us to weaken the assumptions of the previously mentioned
applications. In particular, we construct a Ricci flow for a class of
(not too) singular Alexandrov spaces.
Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of
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