On the universality of the incompressible Euler equation on compact manifolds, II. Non-rigidity of Euler flows

The incompressible Euler equations on a compact Riemannian manifold $(M,g)$ take the form \begin{align*} \partial_t u + \nabla_u u &= - $\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}$_g p \ $\mathrm{d}\mathrm{i}\mathrm{v}$_g u &= 0, \end{align*} where $u:[0,T]\to \mathrm{\Gamma }(TM)$ is the velocity field and $p:[0,T]\to C^{\mathrm{\infty }}(M)$ is the pressure field. In this paper we show that if one is permitted to extend the base manifold $M$ by taking an arbitrary warped product with a torus, then the space of solutions to this equation becomes "non-rigid’"in the sense that a non-empty open set of smooth incompressible flows $u:[0,T]\to \mathrm{\Gamma }(TM)$ can be approximated in the smooth topology by (the horizontal component of) a solution to these equations. We view this as further evidence towards the “universal” nature of Euler flows.