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

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 $u(y,\theta ,\tau )=(y^{\top }Qy-2\text{tr}(Q))/|\tau |+o(|\tau {|}^{-1})$, 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.