MiS Preprint Repository

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 $\text{rot}(L)\ne 0$, 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.