Hearing the shape of ancient noncollapsed flows in R4

We consider ancient noncollapsed mean curvature flows in R-4 whose tangent flow at -infinity is a bubble-sheet. We carry out a fine spectral analysis for the bubble-sheet function.. that measures the deviation of the renormalized flow from the round cylinder R-2 x S-1(root 2) and prove that for tau -> -infinity we have the fine asymptotics u(y, theta, tau) = (y(inverted perpendicular)Qy- 2tr(Q))/|tau| +o(|tau|(-1)), where Q =Q(tau) is a symmetric 2 x 2-matrix whose eigenvalues are quantized to be either 0 or -1/root 8. This naturally breaks up the classification problem for general ancient noncollapsed flows in R4 into three cases depending on the rank of Q. In the case rk(Q) = 0, generalizing a prior result of Choi, Hershkovits and the second author, we prove that the flow is either a round shrinking cylinder or Rx2d-bowl. In the case rk(Q) = 1, under the additional assumption that the flow either splits off a line or is self-similarly translating, as a consequence of recent work by Angenent, Brendle, Choi, Daskalopoulos, Hershkovits, Sesum and the second authorwe show that the flow must be Rx2d-oval or belongs to the one-parameter family of 3d oval-bowls constructed by Hoffman-Ilmanen-Martin-White, respectively. Finally, in the case rk(Q) = 2 we show that the flow is compact and SO(2)-symmetric and for tau -> -infinity has the same sharp asymptotics as the O(2) x O(2)-symmetric ancient ovals constructed by Hershkovits and the second author. The full classification problem will be addressed in subsequent papers based on the results of the present paper.

Automated summary

This paper explores ancient noncollapsed mean curvature flows and provides insights into their behavior and properties through spectral analysis and precise asymptotic analysis in various cases.