Orbital Stability of First Laplacian Eigenstates for the Incompressible Euler Equation on a Flat 2-Torus

On a two-dimensional flat torus, Laplacian eigenfunctions admit explicit trigonometric representations.

It is known that every first eigenstate on a rectangular or square torus is stable under the incompressible Euler dynamics modulo translations.

We extend this result to flat tori of arbitrary shape and thereby obtain, to the best of our knowledge, the first family of orbitally stable sinusoidal Euler flows on a hexagonal torus.

The proof uses a Burton-type stability criterion and has two main ingredients: (i) a variational characterization of each equimeasurable class in the first eigenspace and (ii) the finiteness of the number of translational orbits contained in each such class.

The second ingredient is particularly delicate in the hexagonal case, where it reduces to the analysis of a polynomial system reflecting both the symmetry of the torus and the structure of its first eigenspace.