New Geometric Flows on Riemannian Manifolds and Applications to Schr\"odinger-Airy Flows

Abstract

In this paper, we define a class of new geometric flows on a complete Riemannian manifold. The new flow is related to the generalized (third order) Landau-Lifishitz equation. On the other hand it could be thought of 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, we obtain the global results.