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

Abstract

We consider ancient noncollapsed mean curvature flows in R4$\mathbb {R}^4$ 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 R2×S1(2)$\mathbb {R}^2 \times S^1(\sqrt {2})$ and prove that for τ→−∞$\tau \rightarrow -\infty$ we have the fine asymptotics u(y,θ,τ)=(y⊤Qy−2tr(Q))/|τ|+o(|τ|−1)$u(y,\theta ,\tau )= (y^\top Qy -2\textrm {tr}(Q))/|\tau | + o(|\tau |^{-1})$ , where Q=Q(τ)$Q=Q(\tau )$ is a symmetric 2 × 2‐matrix whose eigenvalues are quantized to be either 0 or −1/8$-1/\sqrt {8}$ . This naturally breaks up the classification problem for general ancient noncollapsed flows in R4$\mathbb {R}^4$ into three cases depending on the rank of Q. In the case rk(Q)=0$\mathrm{rk}(Q)=0$ , generalizing a prior result of Choi, Hershkovits and the second author, we prove that the flow is either a round shrinking cylinder or R×$\mathbb {R}\times$ 2d‐bowl. In the case rk(Q)=1$\mathrm{rk}(Q)=1$ , 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 author we show that the flow must be R×$\mathbb {R}\times$ 2d‐oval or belongs to the one‐parameter family of 3d oval‐bowls constructed by Hoffman‐Ilmanen‐Martin‐White, respectively. Finally, in the case rk(Q)=2$\mathrm{rk}(Q)=2$ we show that the flow is compact and SO(2)‐symmetric and for τ→−∞$\tau \rightarrow -\infty$ has the same sharp asymptotics as the O(2) × 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.