Lower bounds on Ricci flow invariant curvatures and geometric applications.

Thomas Richard (1, 2)

1 IF - Institut Fourier
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

We consider Ricci flow invariant cones C in the space of curvature operators lying between 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 Ricci flow has its curvature operator which satsisfies R+εI ∈ C at the initial time, then it satisfies R +Kε I ∈ 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 ∈ 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 allow 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.

Mots clés: Riemannian geometry; Ricci flow; Curvature bounded from below.; Curvature bounded from below

Domaines: Géométrie différentielle [math.DG]

Thomas Richard. Lower bounds on Ricci flow invariant curvatures and geometric applications.. Journal für die reine und angewandte Mathematik, 2015, 2015 (703), pp.27-41. ⟨10.1515/crelle-2013-0042⟩. ⟨hal-00638017⟩