On stability of Euler flows on closed surfaces with positive genus

Incompressible flows of an ideal two-dimensional fluid on a closed, orientable surface with positive genus are studied. Linear stability of the harmonic, i.e. irrotational and incompressible, solutions to the Euler equations is shown using the Hodge-Helmholtz decomposition. We also demonstrate that any surface Euler flow is stable with respect to small harmonic velocity perturbations.