  We show compactness in the locally smooth topology for certain natural
families of asymptotically conical self-expanding solutions of mean curvature
flow. Specifically, we show such compactness for the set of all two-dimensional
self-expanders of a fixed topological type and, in all dimensions, for the set
of self-expanders of low entropy and for the set of mean convex self-expanders
with strictly mean convex asymptotic cones. From this we deduce that the
natural projection map from the space of parameterizations of asymptotically
conical self-expanders to the space of parameterizations of the asymptotic
cones is proper for these classes.
