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Lower bounds on Ricci flow invariant curvatures and geometric applications

Richard, Thomas (2015) Lower bounds on Ricci flow invariant curvatures and geometric applications. In: JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, 703 . pp. 27-41.

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Official URL: http://dx.doi.org/ 10.1515/crelle-2013-0042

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 + epsilon I is an element of C at the initial time, then it satisfies R + epsilon I is an element of C 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 + I is an element of C 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.

Item Type: Journal Article
Additional Information: Copy right for this article belongs to the WALTER DE GRUYTER GMBH, GENTHINER STRASSE 13, D-10785 BERLIN, GERMANY
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Depositing User: Id for Latest eprints
Date Deposited: 19 Jul 2015 06:12
Last Modified: 19 Jul 2015 06:12
URI: http://eprints.iisc.ac.in/id/eprint/51817

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