New geometric flows on Riemannian manifolds and applications to Schr?dinger-Airy flows

In this paper,a class of new geometric flows on a complete Riemannian manifold is defined. The new flow is related to the generalized(third order) Landau-Lifshitz equation. On the other hand it could be thought of as a special case of the Schr¨odinger-Airy flow when the target manifold is a K¨ahler manifold with constant holomorphic sectional curvature. We show the local existence of the new flow on a complete Riemannian manifold with some assumptions on Ricci tensor. Moreover,if the target manifolds are Einstein or some certain type of locally symmetric spaces,the global results are obtained.

In this paper,a class of new geometric flows on a complete Riemannian manifold is defined. The new flow is related to the generalized（third order） Landau-Lifshitz equation. On the other hand it could be thought of as a special case of the Schroodinger-Airy flow when the target manifold is a Koahler manifold with constant holomorphic sectional curvature. We show the local existence of the new flow on a complete Riemannian manifold with some assumptions on Ricci tensor. Moreover,if the target manifolds are Einstein or some certain type of locally symmetric spaces,the global results are obtained.