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.