 

Knut Smoczyk
Closed Legendre geodesics in Sasaki manifolds


Published: 
January 28, 2003 
Keywords: 
Legrendrian, mean curvature flow, geodesic, minimal, Lagrangian, Lagrangian cone, Hamiltonian minimal, volume decreasing 
Subject: 
Primary 53C44; Secondary 53D99, 58E10 


Abstract
If L⊂ M is a Legendre submanifold in a Sasaki manifold, then
the mean curvature flow does not preserve the Legendre condition. We define
a kind of mean curvature flow for Legendre submanifolds which slightly differs
from the standard one and then we prove that closed Legendre curves L
in a Sasaki space form M converge to closed Legendre geodesics, if
k^{2}+σ+3≦ 0 and rot(L)=0, where σ denotes the
sectional curvature of the contact plane ξ and k and rot(L)
are the curvature respectively the rotation number of L. If
rot(L)≠ 0, we obtain convergence of a subsequence to Legendre
curves with constant curvature. In case σ+3≦ 0 and if the
Legendre angle α of the initial curve satisfies osc(α)
≦ \pi, then we also prove convergence to a closed Legendre geodesic.


Author information
Max Planck Institute for Mathematics in the Sciences, Inselstr. 2226, 04103 Leipzig, Germany
Knut.Smoczyk@mis.mpg.de
http://personalhomepages.mis.mpg.de/smoczyk/index.html

