2007/05/25 by Tobias Colding, Tobias H. Colding, Colding, Tobias H. +3
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometry and complex manifolds #Point processes and geometric inequalities #math.AP #math.DG #math.GT
paper · pdf · doi:10.48550/arxiv.0705.3827
Includes applications to flows by powers of mean curvature
openalex publication_date 2007/05/25 · arxiv created 2007/06/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a Riemannian metric on a homotopy n-sphere, sweep it out by a continuous one-parameter family of closed curves starting and ending at point curves. Pull the sweepout tight by, in a continuous way, pulling each curve as tight as possible yet preserving the sweepout. We show: Each curve in the tightened sweepout whose length is close to the length of the longest curve in the sweepout must itself be close to a closed geodesic. In particular, there are curves in the sweepout that are close to closed geodesics. Finding closed geodesics on the 2-sphere by using sweepouts goes back to Birkhoff in 1917. As an application, we bound from above, by a negative constant, the rate of change of the width for a one-parameter family of convex hypersurfaces that flows by mean curvature. The width is loosely speaking up to a constant the square of the length of the shortest closed curve needed to ``pull over'' M. This estimate is sharp and leads to a sharp estimate for the extinction time; cf. [CM1], [CM2] where a similar bound for the rate of change for the two dimensional width is shown for homotopy 3-spheres evolving by the Ricci flow (see also Perelman).