Preprint 80/2002

Closed Legendre geodesics in Sasaki manifolds

Knut Smoczyk

Contact the author: Please use for correspondence this email.
Submission date: 05. Sep. 2002
published in: New York journal of mathematics, 9 (2003), p. 23-47 (electronic) 
MSC-Numbers: 53C44, 53C42
Keywords and phrases: legendre, curve shortening, geodesic, sasaki

If formula11 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 formula17 and formula19, where formula21 denotes the sectional curvature of the contact plane formula23 and k, formula27 are the curvature respectively the rotation number of L. If formula31, we obtain convergence of a subsequence to Legendre curves with constant curvature. In case formula33 and if the Legendre angle formula35 of the initial curve satisfies formula37, then we also prove convergence to a closed Legendre geodesic.

03.07.2017, 01:40