The blowdown of ancient noncollapsed mean curvature flows

In the analysis of mean curvature flow it is crucial to understand ancient noncollapsed flows. We recall that a mean curvature flow $M_{t}$ is called ancient if it is defined for all $t\ll 0$, and noncollapsed if it is mean-convex and there is an $\alpha>0$ so that every point $p\in M_{t}$ admits interior and exterior balls of radius at least $\alpha/H(p)$, c.f. [SW09, And12, HK17] (in fact, by [Bre15, HK15] one can always take $\alpha=1$). By the work of White [Whi00, Whi03, Whi15] and by [HH18] it is known that all blowup limits of mean-convex mean curvature flow are ancient noncollapsed flows. More generally, by Ilmanen’s mean-convex neighborhood conjecture [Ilm03], which has been proved recently in the case of neck-singularities in [CHH18, CHHW19], it is expected even without mean-convexity assumption that all blowup limits near any cylindrical singularity are ancient noncollapsed flows.

In this paper, we consider the asymptotic structure of ancient noncollapsed flows, specifically the blowdown of any time-slice. To describe this, denote by $M_{t}=\partial K_{t}\subset\mathbb{R}^{n+1}$ any ancient noncollapsed mean curvature flow. Recall that such flows are always convex thanks to [HK17, Theorem 1.10]. We can assume without essential loss of generality that the solution does not split off a line (otherwise we can reduce the problem to a lower dimensional one). Equivalently, this means that $M_{t}$ is strictly convex.

It is easy to see that the blowdown exists (in fact it is given as a decreasing intersection of convex sets) and is a cone, i.e. satsifies $\lambda\check{K}_{t_{0}}=\check{K}_{t_{0}}$ for all $\lambda>0$. The blowdown captures important information about the asymptotic structure of the time slices. By general theory the blowdown is at most $n-1$ dimensional. Indeed, the blowdown cannot be bigger than the singular set in a mean convex flow, which is at most $n-1$ dimensional by the deep theory of White [Whi00, Whi03] (see also [HK17]). Here, we improve this to:

This generalizes the main result from [CHH21a], where Choi, Hershkovits and the second author ruled out the potential scenario ancient wing-like noncollapsed flows in $\mathbb{R}^{4}$. Our theorem shows that the blowdown is always at least $1$ dimension smaller than what one gets from the general bound for the dimension of the singular set. In particular, it implies:

For comparison, we recall the basic result that there does not exist any nontrivial ancient noncollapsed flow in $\mathbb{R}^{n+1}$ that is asymptotic to an $n$-dimensional cone, as follows easily from the fact that the asymptotic area ratio vanishes [HK17]. Our corollary rules out solutions that are asymptotic to $n-1$ dimension cones, i.e. solutions asymptotic to cones whose dimension is smaller by $1$ than the dimension of the evolving hypersurfaces, which is much more subtle. It even rules out flows asymptotic to singular cones.

The dimension bound can be improved further in the uniformly $k$-convex case. Recall that an ancient noncollapsed flow is uniformly $k$-convex if there is a constant $\beta>0$ such that $\lambda_{1}+\ldots+\lambda_{k}\geq\beta H$, where $\lambda_{1}\leq\ldots\leq\lambda_{n}$ denote the principal curvatures. We prove:

Setting $k=3$ and relaxing the strict convexity assumption we obtain:

Let us explain how the corollary follows: If the flow is strictly convex, then its blowdown is a halfline by our theorem (unless the solution is compact in which case the blowdown is simply a point). If the flow is not strictly convex, then after splitting off a line we get an ancient uniformly $2$-convex noncollapsed flow, and such flows have been classified in the recent breakthrough by Angenent-Daskalopoulos-Sesum [ADS19, ADS20] and Brendle-Choi [BC19, BC]. Finally, the ‘moreover statement’ follows as consequence of our proofs as explained in Section 5 and Section 7.

As another corollary, for self-similarly translating solutions we obtain:

Indeed, any such translator is either an $(n-1)$-dimensional rotationally symmetric bowl soliton times $\mathbb{R}$, or its blowdown is a halfline. The latter in particular implies that the translator has a unique tip point, and hence is $\mathrm{SO}(n-1)$-symmetric by a recent result of Zhu [Zhu21].

Finally, inspecting all the possibilities in Corollary 1.6 (blowdown of ancient 3-convex flows), as yet another corollary we obtain:

To prove our main theorems, we generalize the arguments from [CHH21a], by Choi, Hershkovits and the second author, to higher dimensions as follows:

In Section 2, we collect some basic properties of ancient noncollapsed flows in $\mathbb{R}^{n+1}$. In particular, we show that it is enough to analyze the case where the tangent flow at $-\infty$ is given by

with $\ell=n-1$ (respectively $k-1$).

In Section 3, to facilitate barrier and calibration arguments in later sections, we construct an $\mathrm{O}(\ell)\times\mathrm{O}(n+1-\ell)$ symmetric foliation by rotating and shifting the $d=n+1-\ell$ dimensional shrinker foliation from [ADS19].

In Section 4, we set up a fine cylindrical analysis, which generalizes the fine neck analysis from [ADS19, ADS20, BC19, BC, CHH18, CHHW19] and the fine bubble sheet analysis from [CHH21a]. Given any space-time point $X_{0}=(x_{0},t_{0})$, we consider the renormalized flow

Then, the hypersurfaces $\bar{M}_{\tau}$ converge for $\tau\to-\infty$ to the cylinder

The analysis over $\Gamma$ is governed by the Ornstein-Uhlenbeck operator

Applying the Merle-Zaag lemma, we see that either the unstable eigenfunctions, namely

are dominant, or the neutral eigenfunctions, namely

are dominant. Further, considering $\tilde{M}_{\tau}=S(\tau)\bar{M}_{\tau}$, where $S(\tau)\in\mathrm{SO}(n+1)$ is a suitable rotation, we can kill the contribution from $\{x_{i}x_{J}\}$.

In Section 5, we analyze the case where the neutral eigenfunctions are dominant. We show that along a suitable sequence $\tau_{i}\to-\infty$, the truncated graphical function $\hat{u}$ of $\tilde{M}_{\tau}$ satisfies

where $x^{\prime}=(x_{1},\ldots,x_{\ell})$, and $Q=\{q_{ij}\}$ is a nontrivial semi-negative definite $\ell\times\ell$-matrix. We then prove the crucial inclusion

Roughly speaking, directions that are not in the kernel of the quadratic form are directions of inwards quadratic bending, and thus must be compact directions. We make this precise using the Brunn-Minkowski inequality. Once (1.10) is established, we conclude that $\dim\check{K}_{t_{0}}\leq\ell-1\leq k-2$ for uniformly $k$-convex flows.

In Section 6, we consider ancient noncollapsed flows whose unstable mode is dominant. We prove the fine cylindrical theorem, which says that there exists a nonvanishing vector $(a_{1},\ldots,a_{\ell})$ with the following significance: For every space-time center $X$, after suitable recentering in the $\mathbb{R}^{n+1-\ell}$-plane, the truncated graph function $\check{u}^{X}(\cdot,\tau)$ of the renormalized flow $\bar{M}_{\tau}^{X}$ satisfies

for $\tau\ll 0$ depending only on an upper bound on the cylindrical scale of $X$. To show this, we generalize the proofs of the fine neck theorem from [CHH18, CHHW19] and the fine bubble sheet theorem from [CHH21a].

In Section 7, we rule out the case where the unstable mode is dominant. We suppose towards a contradiction that there is an ancient noncollapsed uniformly $k$-convex flow $M_{t}=\partial K_{t}$ that does not split off a line, which has a $(k-1)$-dimensional blowdown $\check{K}_{t_{0}}$. We select two distinct rays of differentiability $R^{\pm}\subset\partial\check{K}_{t_{0}}$. Considering suitable corresponding sequences of points in $\partial K_{t}$, we can pass to limits $K^{\pm}_{t}$ that split off $k-2$ lines. Using the classification from Brendle-Choi [BC], we argue that $K^{\pm}_{t}$ must be $(n+2-k)$-dimensional bowls times $(k-2)$-dimensional planes. However, since these bowls translate in different directions, we obtain a contradiction with the fact that the fine cylindrical vector $(a_{1},\ldots,a_{\ell})$ is independent of the space-time center point. This concludes the outline of the proof.

Let $M_{t}=\partial K_{t}$ be a noncollapsed mean curvature flow in $\mathbb{R}^{n+1}$, where $n\geq 3$, that is noncompact and strictly convex. Assume further that it is uniformly $k$-convex, where $k\geq 3$ (in case $k=n$, there is no further assumption). It is not hard to see that the blowdown

is in fact independent of the choice of $t_{0}$ prior to the extinction time (indeed, as one increases $t_{0}$, by mean-convexity the blowdown cannot increase, and using suitable spheres as barriers, similarly as in [CHH21a, Section 2], one sees that it cannot decrease either). By [HK17, CM15] the tangent flow of $\{K_{t}\}$ at $-\infty$ in suitable coordinates, is

for some $\ell\in\{1,\ldots,k-1\}$. Together with mean-convexity it follows that

If $\ell\leq k-2$, then $\dim\check{K}\leq k-2$ and we are done. Hence, we can assume from now on that $\ell=k-1$. Moreover, since our solution is strictly convex, $\check{K}$ does not contain any line.

Given any ancient noncollapsed mean curvature flow $\mathcal{M}$, that is not a round shrinking cylinder, any point $X=(x,t)\in\mathcal{M}$ and scale $r>0$, we consider the flow

that is obtained from $\mathcal{M}$ by translating $X$ to the space-time origin and parabolically rescaling by $1/r$.

We fix a small enough parameter $\varepsilon>0$ quantifying the quality of the cylinders for the rest of the paper. Given $X=(x,t)\in\mathcal{M}$, we analyze the solution around $X$ at the diadic scales $r_{j}=2^{j}$, where $j\in\mathbb{Z}$. Using Huisken’s monotonicity formula [Hui90] and quantitative differentiation (see e.g. [CHN13]), for every $X\in\mathcal{M}$, we can find an integer $J(X)\in\mathbb{Z}$ such that

and

We recall from Angenent-Daskalopoulos-Sesum [ADS19] that there is some $L_{0}>1$ such that for every $a\geq L_{0}$ and $b>0$, there are $d$-dimensional shrinkers in $\mathbb{R}^{d+1}$,

as illustrated in [ADS19, Fig. 1]. Here, the parameter $a$ captures where the concave functions $u_{a}$ meet the $x_{1}$-axis, namely $u_{a}(a)=0$, and the parameter $b$ is the asymptotic slope of the convex functions $\tilde{u}_{b}$, namely $\lim_{x_{1}\to\infty}\tilde{u}_{b}^{\prime}(x_{1})=b$. We recall:

We now write points in $\mathbb{R}^{n+1}$ in the form $x=(x^{\prime},x^{\prime\prime})$, where $x^{\prime}=(x_{1},\dots,x_{\ell})$ and $x^{\prime\prime}=(x_{\ell+1},\dots,x_{n+1})$. We choose $d=n+1-\ell$ and shift and rotate the above foliation to construct a suitable foliation in $\mathbb{R}^{n+1}$:

In this section, we set up the fine cylindrical analysis. This is similar to setting up fine bubble sheet analysis in [CHH21a, Section 4], so we will discuss this briefly. Throughout this section, let $\mathcal{M}$ be an ancient noncollapsed flow in $\mathbb{R}^{n+1}$ whose tangent flow at $-\infty$ is a round shrinking cylinder $\mathbb{R}^{\ell}\times S^{n-\ell}(\sqrt{2(n-\ell)t})$. Assume further that $\mathcal{M}$ is not self-similarly shrinking. Given any space-time point $X_{0}=(x_{0},t_{0})$, we consider the renormalized flow

Then, as $\tau\rightarrow-\infty$, the hypersurfaces $\bar{M}^{X_{0}}_{\tau}$ converges to the cylinder

After rescaling, we can assume that $Z(X_{0})\leq 1$, and we can find a universal function $\rho(\tau)>0$ with

so that for every $S\in\mathrm{SO}(n+1)$ with $\sphericalangle(S(\Gamma),\Gamma)<\rho(\tau)^{-3}$, the rotated surface $S(M^{X_{0}}_{\tau})$ is the graph of a function $u=u_{S}(\cdot,\tau)$ over $\Gamma\cap B_{2\rho(\tau)}(0)$ with

We fix a nonnegative smooth cutoff function $\chi$ satisfying $\chi(s)=1$ for $|s|\leq\frac{1}{2}$ and $\chi(s)=0$ for $|s|\geq 1$. We consider the truncated function

where

Using the implicit function theorem, similarly as in [CHH21a, Proposition 4.1], we can find a differentiable function $S^{X_{0}}(\tau)\in\mathrm{SO}(n+1)$, defined for $\tau$ sufficiently negative, such that for $u:=u_{S^{X_{0}}(\tau)}$ we have

Moreover, we can arrange that for all $\tau\ll 0$, the matrix

satisfies $A_{ij}(\tau)=0$ for $1\leq i,j\leq\ell$, and $A_{IJ}(\tau)=0$ for $\ell+1\leq I,J\leq n+1$.

We now set

where $S^{X_{0}}(\tau)\in SO(n+1)$ is the fine tuning rotation from above, and let

so $u$ describes $\tilde{M}_{\tau}^{X_{0}}$ as a graph over the cylinder $\Gamma$.

Let $\mathcal{H}$ be the Hilbert space of all functions $f$ on $\Gamma$ such that

The cylindrical representation of the operator $\mathcal{L}$ is explicitly given by

Analysing the spectrum of $\mathcal{L}$, the Hilbert space $\mathcal{H}$ from (4.13) can be decomposed as

where

and

and where $x_{1},\dots,x_{n+1}$ denotes the restriction of the Euclidean coordinate functions to the cylinder $\Gamma$. Moreover, we have

Consider the functions

where $P_{+},P_{0},P_{-}$ denote the orthogonal projections to $\mathcal{H}_{+},\mathcal{H}_{0},\mathcal{H}_{-}$, respectively.

In this section, we prove our main theorems in the case where the neutral mode is dominant. Namely, throughout this section we assume that our truncated graph function $\hat{u}(\cdot,\tau)$ satisfies

Using this together with the evolution inequality (4.24), given any $\delta>0$, we see that $|\tfrac{d}{d\tau}\log U_{0}|\leq\tfrac{1}{2}\delta$ for sufficiently large $-\tau$, and thus

for sufficiently large $-\tau$.