2011/11/03 by Tom L. Richard, Thomas Richard, Richard, Thomas
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG
paper · pdf · doi:10.48550/arxiv.1111.0859
arxiv created 2011/11/03 · arxiv updated 2011/11/04
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 +KεI ∈ 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.