Nonuniqueness of solutions of the Navier-Stokes equations on negatively curved Riemannian manifolds

In a well-known work, M. Anderson constructed a Hadamard manifold $(M^n,g)$ which carries non-zero $L^2$ harmonic $p$-forms when $p\mathrm{\ne }n\mathrm{/}2$, thus disproving the Dodziuk-Singer conjecture. In this paper, we use the manifold $(M^3,g)$ in order to solve another problem in geometric analysis, namely the nonuniqueness of solutions of Leray-Hopf type of the Navier-Stokes equations.