Hearing the shape of ancient noncollapsed flows in \(\mathbb{R}^{4}\)

Abstract

We consider ancient noncollapsed mean curvature flows in \(\mathbb{R}^4\) whose tangent flow at \(-\infty\) is a bubble-sheet. We carry out a fine spectral analysis for the bubble-sheet function \(u\) that measures the deviation of the renormalized flow from the round cylinder \(\mathbb{R}^2 \times S^1(\sqrt{2})\) and prove that for \(\tau\to -\infty\) we have the fine asymptotics \(u(y,\theta,\tau)= (y^\top Qy -2\textrm{tr}(Q))/|\tau| + o(|\tau|^{-1})\), where \(Q=Q(\tau)\) is a symmetric \(2\times 2\)-matrix whose eigenvalues are quantized to be either 0 or \(-1/\sqrt{8}\). This naturally breaks up the classification problem for general ancient noncollapsed flows in \(\mathbb{R}^4\) into three cases depending on the rank of \(Q\). In the case \(\mathrm{rk}(Q)=0\), generalizing a prior result of Choi, Hershkovits and the second author, we prove that the flow is either a round shrinking cylinder or \(\mathbb{R}\times\)2d-bowl. In the case \(\mathrm{rk}(Q)=1\), under the additional assumption that the flow either splits off a line or is selfsimilarly translating, as a consequence of recent work by Angenent, Brendle, Choi, Daskalopoulos, Hershkovits, Sesum and the second author we show that the flow must be \(\mathbb{R}\times\)2d-oval or belongs to the one-parameter family of 3d oval-bowls constructed by Hoffman-Ilmanen-Martin-White, respectively. Finally, in the case \(\mathrm{rk}(Q)=2\) we show that the flow is compact and \(\mathrm{SO}(2)\)-symmetric and for \(\tau\to-\infty\) has the same sharp asymptotics as the \(\mathrm{O}(2)\times\mathrm{O}(2)\)-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.