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.
Categories: math.DG math.AP