Precise asymptotics near a generic $\mathbb S^1\times\mathbb R^3$ singularity of mean curvature flow

TL;DR

This paper analyzes a specific generic singularity in mean curvature flow modeled on the bubble-sheet $\

Contribution

It provides a detailed asymptotic profile near a $\

Findings

Derived precise asymptotics for the singularity

Characterized the local geometric structure

Enhanced understanding of bubble-sheet singularities

Abstract

In the present paper we study a type of generic singularity of mean curvature flow modelled on the bubble-sheet $\mathbb S^1\times\mathbb R^3$ , and we derive an asymptotic profile for a neighborhood of singularity.

Equations

$${\frac{d{\mathbf{X}}_M}{dt}}^{\perp }={\mathbf{H}}_M.$$

$$C_1=\{\left[\begin{array}{c}y\\ \sqrt{2}\cos \theta \\ \sqrt{2}\sin \theta \end{array}\right]\in {\mathbb{R}}^5|y\in {\mathbb{R}}^3\}.$$

$$M_t=\left[\begin{array}{c}x\\ u(x,\theta ,t)\cos \theta \\ u(x,\theta ,t)\sin \theta \end{array}\right].$$

$${\tilde{M}}_{\tau }=\sqrt{-t}{M}_t,\tau =-\log (-t).$$

$${\tilde{M}}_{\tau }=\left[\begin{array}{c}y\\ v(y,\theta ,\tau )\cos \theta \\ v(y,\theta ,\tau )\sin \theta \end{array}\right]$$

$$u(x,\theta ,t)=\sqrt{-t}v(y,\theta ,\tau ),$$

$$y:=\frac{x}{\sqrt{-t}},\tau :=-ln|t|.$$

$${\tau }_0>>1,C=O(1)$$

$$v(y,\theta ,\tau )=\sqrt{\frac{2+y^T\cdot B\cdot y}{2a(\tau )}}+\eta (y,\theta ,\tau ),\text{for}|y|\le \Sigma (\tau ),$$

$$\displaystyle B=\tau^{-1}\left[\begin{array}[]{ccc}b_{1}&0&0\\ 0&b_{2}&0\\ 0&0&b_{3}\end{array}\right]+O(\tau^{-2}),\ b_{k}=0\ \text{or}\ 1,\ \text{and}\ k=1,2,3,$$

$$a(\tau )=\frac{1}{2}+O({\tau }^{-1})$$

$$\begin{array}{rl}\Vert ⟨y{⟩}^{-3}{1}_{\Omega }{\partial }_{\theta }^l\eta (\cdot ,\tau ){\Vert }_{\infty }\le & C{\tau }^{-2},l=0,1,2,\\ \Vert ⟨y{⟩}^{-2}{1}_{\Omega }{\partial }_{\theta }^l{\nabla }_y\eta (\cdot ,\tau ){\Vert }_{\infty }\le & C{\Omega }^{-3},l=0,1,\\ \Vert ⟨y{⟩}^{-1}{1}_{\Omega }{\nabla }_y^k\eta (\cdot ,\tau ){\Vert }_{\infty }\le & C{\Omega }^{-2},|k|=2.\end{array}$$

$$\displaystyle 1_{\leq\Omega}(y)=\left[\begin{array}[]{ll}1&\ \text{if}\ |y|\leq\Omega,\\ 0&\ \text{otherwise.}\end{array}\right.$$

$$\Omega (\tau ):=\sqrt{100\ln \tau +9(\tau -{\xi }_0{)}^{\frac{11}{10}}}.$$

$$b_1=b_1=b_3=1,$$

$$\displaystyle\mathbb{R}^{3}\times\mathbb{S}^{1}_{\sqrt{2}}=\left(\begin{array}[]{cc}x\\ \sqrt{2}\cos\theta\\ \sqrt{2}\sin\theta\end{array}\right),\ \text{for}\ x\in\mathbb{R}^{3}.$$

$$\displaystyle\left(\begin{array}[]{cc}x\\ u_{0}(x,\theta)\cos\theta\\ u_{0}(x,\theta)\sin\theta)\end{array}\right)\in\mathbb{R}^{5},\ x\in\mathbb{R}^{3},$$

$$u_0(x,\theta )=\frac{|x|}{\sqrt{-\log |x|}}(1+o(1)),\text{where}o(1)\to 0\text{as}|x|\to 0.$$

$$e^{\frac{{\tau }_1}{2}}|x_1|={\tau }_1^{\frac{1}{2}+\frac{1}{20}}.$$

$$t_1=t_1({\tau }_1)=-e^{-{\tau }_1}.$$

$$|\frac{u(x_1,\theta ,t)}{u(x_1,\theta ,0)}-1|<ϵ.$$

$$u(x_1,\theta ,0)=$$

$$u(x_1,\theta ,t_1)=\sqrt{-t_1}v(\frac{1}{\sqrt{-t_1}}{x}_1,\theta ,{\tau }_1)=\sqrt{-t_1}v(y_1,\theta ,{\tau }_1).$$

$$v(y,\theta ,\tau )=\sqrt{\frac{2+y^T\cdot B\cdot y}{2a(\tau )}}+\eta (y,\theta ,\tau ).$$

$$\displaystyle v(y,\theta,\tau)=\sqrt{2+\tau^{-1}|y|^{2}}\big{(}1+o(1)\big{)},\ \text{for}\ |y|\leq 2\tau^{\frac{1}{2}+\frac{1}{20}},$$

$$|\eta (y,\theta ,\tau )|\le ⟨y{⟩}^3{\tau }^{-2}\le {\tau }^{-\frac{1}{10}},$$

$$u(x_1,\theta ,t_1)=$$

$$=$$

$$-\log |x_1|=\frac{{\tau }_1}{2}(1+o(1)).$$

$$\begin{array}{rl}u(x_1,\theta ,t_1)=& \frac{|x_1|}{{\tau }_1^{\frac{1}{2}+\frac{1}{20}}}{\tau }_1^{\frac{1}{20}}(1+o(1))=\frac{|x_1|}{{\tau }_1^{\frac{1}{2}}}(1+o(1))\\ =& \frac{|x_1|}{\sqrt{-\log |x_1|}}(1+o(1)).\end{array}$$

Full text

1. Introduction

In the present paper we study a family of hypersurfaces $\{M^{n}_{t}\subset\mathbb{R}^{n+1}\}$ evolving by mean curvature flow (MCF), which makes it satisfy the equation

[TABLE]

where $\mathbf{X}$ is the position vector and $\mathbf{H}$ is the mean curvature vector of the hypersurfaces $M_{t}$.

For a rich family of initial hypersurfaces some singularities will form in finite time. A main theme in the study of mean curvature flow or more general geometric flows is to understand the formation of singularities, and find a way to extend the flows through singularities with controlled geometry and topology. In the present paper we would like to understand a class of generic singularities, at the singular time.

In this aspect some understanding is achieved in the setting the flows of 2-convex hypersurfaces. In [7, 8], Gerhard and Sinestrari, Haslhofer and Kleiner developed the existence of mean curvature flow with surgery starting from arbitrary 2-convex hypersurfaces, with a detailed description of a neighborhood of the singularities and surgery procedure with controlled geometry and topology. For the results on non 2-convex flows, see [14, 2, 15].

In a separate direction Colding and Minicozzi proved, in [4], that the generic singularities models are of the type $\mathbb{S}^{k}\times\mathbb{R}^{n-k}$. But in general, there is no canonical neighborhood theorem known for these singularities of generalized cylinder with $k>1$ and no theory of performing surgery through these singularities. Recently, the first author [9] gives a description in a certain regime of a neighborhood of singularities modeled on $\mathbb{S}^{1}\times\mathbb{R}^{3}$. And we will use that result to derive a precise asymptotic of the a pinched disk singularities modeled on $\mathbb{S}^{1}\times\mathbb{R}^{3}$. Hopefully this can be used to give information of the level-set flow through such kind of singularities.

In the situation of cylindrical singularities, the phenomena that the mean curvature evolution collapse to the singular set in the first and only singular time is highly unstable and can only appear in very rare cases: evolution of exact round cylinders collapsing to a line and the evolution of rotational symmetric marriage rings collapsing to a circle for example. Generically, the flow encountering cylindrical singularities will develop pinched-necks (if the axis of cylinder is one-dimensional) or pinched-disks (if the axis of cylinder is higher-dimensional). Our result concerns a pinched-disk singularity modeled on $\mathbb{S}^{1}\times\mathbb{R}^{3}$.

The paper is organized as following: the Main Theorem will be stated in Section 2, and it will be proved in Section 3. In Section 4 we will prove a key technical lemma.

2. Main Theorem

Up to parabolic rescaling and translations, we can assume without loss of generality that a singularity in space-time is at spatial origin and at time [math]. Colding and Minicozzi proved in [4] that the generic singularities are modeled on self-shrinking spheres or generalized cylinders $\mathbb{S}^{n},\ \mathbb{S}^{k}_{\sqrt{2k}}\times\mathbb{R}^{n-k}$.

In this paper, we consider MCF of four dimensional hypersurfaces $M^{4}_{t}$ in $\mathbb{R}^{5}$ that develops a generic singularity modeled on the self-shrinking cylinder $\mathbb{S}^{1}_{\sqrt{2}}\times\mathbb{R}^{3}$ at the space-time point $(0,0)$. The result of Colding and Minicozzi [5] on the uniqueness of the blow-up of cylindrical type singularities implies the uniqueness of a fixed axis. Up to rotations, we can assume the unique singularity model is the cylinder

[TABLE]

In a neighborhood of the blowup point the unscaled flow can also be parametrized by

[TABLE]

Define a rescaled flow about this singularity by

[TABLE]

It has the effect of zooming in the singularity. Hence the time $\tau$ slice of rescaled flow $\tilde{M}_{\tau}$ can be parametrized by

[TABLE]

where the function $v$ is obtained after rescaling $u$ by

[TABLE]

where $y\in\mathbb{R}^{3}$ and $\tau$ are defined as

[TABLE]

Under our assumption, this rescaled flow will converge (without passing to a subsequence) to the unique cylinder $C_{1}$ as measures on any compact subset, see [5].

By a recent result of the first author [9], we have a more detailed estimate of this convergence of the rescaled flow in neighborhoods of size that is increasing with $\tau$. The precise statement is as follows:

Theorem 2.1**.**

[Theorem 2.1 of [9]] Let $M^{4}_{t}$ be a mean curvature flow in $\mathbb{R}^{5}$ that develops a singularity of type at space-time $(0,0)\subset\mathbb{R}^{5}\times\mathbb{R}$ and is modeled on the self-shrinker $C_{1}$ defined above.

Then there exist two independent positive constants $\tau_{0}$ and $C$ satisfying

[TABLE]

such that for $\tau\geq\tau_{0}$, $v$ has the following asymptotic

[TABLE]

where $B$ is a $3\times 3$ symmetric real matrix satisfying the estimate

[TABLE]

$a$* is a scalar function satisfying*

[TABLE]

and $\eta$ satisfies the estimate

[TABLE]

Here $1_{\leq\Omega}$ is the Heaviside function defined as

[TABLE]

and the scalar function $\Omega$ is defined as

[TABLE]

Under a further generic condition, specific in (2.10),

[TABLE]

it is shown in [9] that the flow will develop an isolated singularity of pinched-disk type and verify the Ilmanen’s mean-convex neighborhood conjecture in this case. We also noticed recently that the mean-convex neighborhood conjecture has been verified by [3] for flows of hypersurfaces in $\mathbb{R}^{3}$.

The main theorem is

Theorem 2.2**.**

Let $\{M^{4}_{t}\subset\mathbb{R}^{5}\}$ be an evolution of mean curvature flow. Assume that it develops a singularity at space-time $(0,0)$, modeled on a cylindrical self-shrinker

[TABLE]

Under the generic assumption (2.17), in a neighborhood $B_{\epsilon}(0)\subset\mathbb{R}^{5}$ of the origin, and at the singular time $t=0$, $M_{0}$ can be parametrized by

[TABLE]

where $u_{0}$ is some nonnegative function , and satisfies the asymptotics

[TABLE]

In [1], Angenent and Knopf has got a similar asymptotic for rotationally symmetric Ricci flow neck-pinch singularities.

3. Proof of Main Theorem 2.2

To prepare for the proof of the main theorem we need a technical lemma. Before stating it we define some constants. For any $x_{1}\in\mathbb{R}^{3}$ with $|x_{1}|\not=0$ small enough, we let $\tau_{1}$ to be the large number such that

[TABLE]

It is not hard to see that such a number exists since $e^{\frac{\tau_{1}}{2}}$ grows much faster than $\tau_{1}^{\frac{1}{2}+\frac{1}{20}}$ as $\tau_{1}$ grows. We denote, by $t_{1}$, the time (of unscaled MCF) corresponding to the rescaled time $\tau_{1}$, which makes

[TABLE]

We have the following results:

Lemma 3.1**.**

Suppose that all the conditions in Theorem 2.2 hold. Then for any $\epsilon>0$, there exits a small constant $\delta>0$ such that if $0\not=|x_{1}|<\delta$, then in the time interval $t\in[t_{1},0)$, with $t_{1}$ defined in (3.2), the unscaled flow can be parametrized by (2.2), and $u(x,\theta,t)$ satisfies the estimate

[TABLE]

This lemma will be proved in Section 4.

The intuitive idea behind Lemma 3.1 is not difficult. Recall that $x=0$ is the only blowup point, and the blowup time is $t=0$. Thus, at least intuitively, for any $x_{1}\not=0$, there exists a time $t_{1}<0$ such that for any $t\in[t_{1},0)$, $u(x_{1},\theta,t)$ will stay away from [math] and vary very little, and hence $\frac{u(x_{1},\theta,t)}{u(x_{1},\theta,t_{1})}\approx 1.$ Moreover $\frac{u(x_{1},\theta,t)}{u(x_{1},\theta,t_{1})}\rightarrow 1$ as $t\geq t_{1}\rightarrow 0.$

3.1. Proof of the Main Theorem 2.2

Proof.

The basic idea is simple. Lemma 3.1 shows that, for each $x_{1}\not=0$ with $|x_{1}|$ sufficiently small, there exists a time $t_{1}$ such that for $t\in[t_{1},0)$,

[TABLE]

Hence it suffices to estimate $u(x_{1},\theta,t_{1}).$ In what follows, we compare the time $t_{1}$ for different $|x_{1}|$ to find information for $u(x_{1},\theta,t_{1}).$ Recall the definitions of $x_{1}$, $\tau_{1}$ and $t_{1}$ before Lemma 3.1.

To estimate $u(x_{1},\theta,t_{1})$ we study the rescaled MCF, which is parametrized by the function $v$. The facts $\tau_{1}=-\ln|t_{1}|$ and $y_{1}=\frac{x_{1}}{\sqrt{-t_{1}}}$ and the definition relating $u$ and $v$ in (2.4) make

[TABLE]

To control $v$ we apply Theorem 2.1, where $v$ is decomposed into two parts

[TABLE]

We claim that

[TABLE]

where $o(1)$ is in the $L_{y,\theta}^{\infty}$-norm in the considered region. To see this we need to control the remainder $\eta$. For that purpose, the first estimate in (2.12), implies that, when $|y|\leq 3\tau^{\frac{1}{2}+\frac{1}{20}}$ and $\tau$ is sufficiently large, then

[TABLE]

thus together with the estimates $B=\tau^{-1}I+O(\tau^{-2})$, $a=\frac{1}{2}+O(\tau^{-1})$ in (2.10) and (2.11), we have the desired (3.7), together

Since the length of $y_{1}$ is $|y_{1}|=\tau_{1}^{\frac{1}{2}+\frac{1}{20}}$, we apply the estimate in (3.7) to obtain,

[TABLE]

The bound in (3.9) is in $\tau_{1}$ while the desired estimate in Main Theorem 2.2 is in terms of $|x_{1}|$. Next we convert it to the desired form.

Since by (3.1), $|x_{1}|=\tau_{1}^{\frac{1}{2}+\frac{1}{20}}e^{\frac{-\tau_{1}}{2}}$ and since $\tau_{1}\gg 1,$ we take $\log$ on both sides to find

[TABLE]

This, together with the estimate in (3.9), implies that

[TABLE]

This, together with (3.4), implies the desired results. ∎

4. Proof of Lemma 3.1

We use the same notations as in previous sections, where $\{M_{t}\}$ is the original flow that blows up at space-time origin $(0,0)$, and $t_{1}$ is a fixed time very close to the blowup time 0, and $\tau_{1}$ is the corresponding time for the rescaled flow, and hence $\tau_{1}=-\log(-t_{1})$.

Since $u(x,\theta,t)\rightarrow 0$ as $|x|,\ |t|\rightarrow 0^{+}$, to prove the desired Lemma 3.1 we have to compare two small quantities. To make our proof easier and more transparent, we rescale the flow such that the rescaled version of $u(x_{1},\theta,t_{1})$ is approximately $1,$ see the estimate in (4.13) below where $h$ is obtained by rescaling the function $u$.

To make this idea rigorous, we define a new flow $\{M_{1,s}\}$ by

[TABLE]

by rescaling, and translation, in time $t$. Here the new time variable $s\in[0,\tau_{1}^{-\frac{1}{10}})$ is defined such that

[TABLE]

We observe that the new flow $\{M_{1,s}\}$ will blow up at time $s=\tau_{1}^{-\frac{1}{10}}$, resulted by that, for a fixed $t_{1}<0,$ the rescaled flow $\{\frac{1}{\sqrt{-t_{1}}}M_{t_{1}+\frac{t}{-t_{1}}}\}$ will blowup time at time $t=1$. The initial condition for the new flow, i.e. $s=0,$ is

[TABLE]

The flow $M_{1,s}$ can be parametrized by

[TABLE]

in a neighborhood of the singularity, where the variable $z$ and the function $h$ are defined as

[TABLE]

and

[TABLE]

Recall the definition of $x_{1}\in\mathbb{R}^{3}$ in (3.1). Its corresponding part for $\{M_{1,s}\}$, denoted by $z_{1}$, is

[TABLE]

whose length satisfies

[TABLE]

The identities in (4.2) and (4.5) -(4.7) imply that

[TABLE]

Consequently Lemma 3.1 is equivalent to the following result about the flow $M_{1,s}$: recall that it will blowup at the (small) time $s=\tau_{1}^{-\frac{1}{10}},$

Proposition 4.1**.**

For any $s\in[0,\tau_{1}^{-\frac{1}{10}})$ we have

[TABLE]

where $\|o(1)\|_{L^{\infty}_{y,\theta}}\rightarrow 0$ as $\tau_{1}\rightarrow\infty$.

Proof.

Here to estimate the function $h$ we need the detailed estimates on the function $v$, which is a rescaled version of $u$. Thus it is convenient to derive some identity between $h$ and $v$, through their definitions in (2.4) and (4.5).

Compute directly to obtain

[TABLE]

where in the last step we used the definitions of $z$ and $s$ in (4.5) and (4.2) respectively.

Based on the identity above, we will prove the desired (4.10) in two steps: (1) derive informations for $h$ and its derivatives when $s\in[-1,0]$ and \big{|}|z|-|z_{1}|\big{|}\leq 2, then (2), by quasi-locality or local smooth extension of MCF, see [10, 13], we study the new flow in the small time interval $[0,\tau_{1}^{-\frac{1}{10}}]$ and find the desired estimate (4.10).

For the first step, it is not hard to verify that Theorem 2.1, which provides estimates for $v(y,\theta,\tau)$ when $|y|\leq 3\tau^{\frac{1}{2}+\frac{1}{20}}$, are applicable to here by the identity in (4.11) with $s\in[-1,0]$ and z\in\{z|\ \Big{|}|z|-|z_{1}|\Big{|}\}\leq 2.

Thus by the asymptotic (2.6), in the set $s\in[-1,0]$ and $z$ satisfying \Big{|}|z|-|z_{1}|\Big{|}\leq 2,

[TABLE]

It is easy to see that, in the set \Big{|}|z|-|z_{1}|\Big{|}\leq 2 and $s=0$,

[TABLE]

and when $s\in[-1,0]$,

[TABLE]

Next, we estimate the derivatives of $h$. We observe that the derivatives of $h$ satisfy the estimates,

[TABLE]

since the decomposition of $v$ in (2.6), the estimates on the remainder in (2.12) and the estimates on the scalar functions in (2.10) and (2.11) imply that

[TABLE]

Higher order derivatives follows by [10], Proposition 2.17.

Recall that, by the parametrization of the hyersurface $\{M_{1,s}\}$ in (4.4), $h(z,\theta,s)$ has the interpretation of being the radius of the hypersurface at the cross section $z$ and at the angel $\theta$. Thus the estimates (4.12) and (4.15) above imply that, for any $s\in[-1,0]$ and \Big{|}|z_{1}|-|z|\Big{|}\leq 2, this part of the hypersurface is close to a cylinder, up to second order derivatives.

Now we are ready for the second step.

By the Pseudo-locality property of mean curvature flow (see Theorem 1.5 of [13]), we have for $s\in[0,\tau_{1}^{-\frac{1}{10}}]$,

[TABLE]

Here we need that the time $\tau_{1}$ is large, so that $\tau_{1}^{-\frac{1}{10}}$ is small, to make Pseudo-locality property applicable.

Thus we have the desired estimate

[TABLE]

∎