  In his lecture notes on mean curvature flow, Ilmanen conjectured the
existence of noncompact self-shrinkers with arbitrary genus. Here, we employ
min-max techniques to give a rigorous existence proof for these surfaces.
Conjecturally, the self-shrinkers that we obtain have precisely one
(asymptotically conical) end. We confirm this for large genus via a precise
analysis of the limiting object of sequences of such self-shrinkers for which
the genus tends to infinity. Finally, we provide numerical evidence for a
further family of noncompact self-shrinkers with odd genus and two
asymptotically conical ends.
