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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
80/2002

Closed Legendre geodesics in Sasaki manifolds

Knut Smoczyk

Abstract

If $L\subset M$ 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, $k^2+\sigma+3\le 0$ and $\text{rot}(L)=0$, where $\sigma$ denotes the sectional curvature of the contact plane $\xi$ and $k$, $\text{rot}(L)$ are the curvature respectively the rotation number of $L$. If $\text{rot}(L)\neq 0$, we obtain convergence of a subsequence to Legendre curves with constant curvature. In case $\sigma+3\le 0$ and if the Legendre angle $\alpha$ of the initial curve satisfies $\text{osc}\,(\alpha) \le \pi$, then we also prove convergence to a closed Legendre geodesic.

Received:
Sep 5, 2002
Published:
Sep 5, 2002
MSC Codes:
53C44, 53C42
Keywords:
legendre, curve shortening, geodesic, sasaki

Related publications

inJournal
2003 Journal Open Access
Knut Smoczyk

Closed Legendre geodesics in Sasaki manifolds

In: New York journal of mathematics, 9 (2003), 23-47 (electronic)