Arnold stability and Misio{\l}ek curvature

TL;DR

This paper investigates the stability of solutions to the incompressible Euler equation on 2D manifolds, showing that most Misio{ extlangle}ek curvature values are nonpositive, which relates to the absence of conjugate points.

Contribution

It provides a partial positive answer to whether Arnold stable solutions always lack conjugate points on the diffeomorphism group for general manifolds.

Findings

Almost all Misio{ extlangle}ek curvature of Arnold stable solutions is nonpositive.

Positivity of Misio{ extlangle}ek curvature indicates potential conjugate points.

Results extend understanding of stability and conjugate points beyond specific domains.

Abstract

Let $M$ be a compact 2-dimensional Riemannian manifold with smooth boundary and consider the incompressible Euler equation on $M$. In the case that $M$ is the straight periodic channel, the annulus or the disc with the Euclidean metric, it was proved by T. D. Drivas, G. Misio{\l}ek, B. Shi, and the second author that all Arnold stable solutions have no conjugate point on the volume-preserving diffeomorphism group ${\mathcal D}_{\mu}^{s}(M)$. They also proposed a question which asks whether this is true or not for any $M$. In this article, we give a partial positive answer. More precisely, we show that almost all the Misio{\l}ek curvature of any Arnold stable solution is nonpositive. The positivity of the Misio{\l}ek curvature is a sufficient condition for the existence of a conjugate point.