2003/12/26 by Li Ma, Ma, Li, Dezhong Chen +1
Mathematics · Medicine · #53C44 #Advanced Neuroimaging Techniques and Applications #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:53C44
paper · pdf · doi:10.48550/arxiv.math/0312463
23 pages
arxiv created 2003/12/26 · openalex publication_date 2003/12/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we systemally study the long time behavior of the curve shortening flow in a closed or non-compact complete locally Riemannian symmetric manifold. Assume that we have a global flow. Then we can exhibit a a limit for the global behavior of the flow. In particular, we show the following results. 1). Let M be a compact locally symmetric space. If the curve shortening flow exists for infinite time, and limt→∞L(γt)gt;0, then for every n>0, limt→ ∞sup(|\fracDnT∂ sn|)=0. In particular, the limiting curve exists and is a closed geodesic in M. 2). For γ0 is a ramp, we have a global flow and the flow converges to a geodesic in C∞ norm.