  We show that the low-density strongly interacting electron liquid,
interacting via the long-range Coulomb interaction, could develop a dispersion
instability at a critical density associated with the approximate flattening of
the quasiparticle energy dispersion. At the critical density the quasiparticle
effective mass diverges at the Fermi surface, but the signature of this Fermi
surface instability manifests itself away from the Fermi momentum at higher
densities. For densities below the critical density the system is unstable
since the quasiparticle velocity becomes negative. We show that one physical
mechanism underlying the dispersion instability is the emission of soft
plasmons by the quasiparticles. The dispersion instability occurs both in two
and three dimensional electron liquids. We discuss the implications of the
dispersion instability for experiments at low electron densities.
