The global structure of bi-rotational self-shrinkers

Abstract

We describe the global structure of the self-shrinkers of the mean curvature flow in $\mathbb{R}^{p+q+2}$ invariant under $O(p+1)\times O(q+1)$, without assuming properness or embeddedness. The main tool is a pair of piecewise monotone quantities along the profile curves, in the spirit of Hsiang's analysis of constant mean curvature hypersurfaces. We prove that every such self-shrinker other than the two invariant cylinders intersects both of them transversely, and all of them except the Clifford minimal cone are complete. Apart from those of the cylinders, each end is either asymptotic to a cone or meets both cylinders infinitely many times. Consequently, a complete noncompact example is proper if and only if all its ends are asymptotically conical, and this is equivalent to polynomial volume growth, to finite entropy, and to finitely many intersections with the cylinders. The reduced flow preserves a finite measure, and for almost every initial condition the profile curve is either closed or generates a complete nonproper self-shrinker of infinite entropy. We also bound the winding of the profile curve by its number of cylinder crossings and localize its self-intersections.