Lower bounds on Ricci flow invariant curvatures and geometric applications

Thomas Richard 1
  • 1 Mathematics Department, Indian Institute of Science, Bangalore, India

Abstract

We 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 R+εI𝒞 at the initial time, then it satisfies R+KεI𝒞 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 R+I𝒞 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.

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The 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 Crelle’s Journal. In the 190 years of its existence, Crelle’s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals.

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