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

TL;DR

This paper introduces a new class of geometric flows on Riemannian manifolds, related to the Landau-Lifshitz and Schr"odinger-Airy flows, with results on local and global existence under specific geometric conditions.

Contribution

It defines novel geometric flows linked to well-known equations and establishes local and global existence results under certain curvature assumptions.

Findings

Established local existence of the flow on complete manifolds.

Proved global existence when target manifolds are Einstein or locally symmetric.

Connected the new flow to the generalized Landau-Lifshitz and 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.