Singularity formation in axially symmetric mean curvature flow with Neumann boundary

TL;DR

This paper investigates the behavior of axially symmetric surfaces evolving under mean curvature flow with Neumann boundary conditions, demonstrating that any initial singularity occurring is necessarily of type I.

Contribution

It establishes that all first singularities in such flows are of type I, providing a classification of singularity types under these boundary conditions.

Findings

All first singularities are of type I.

The analysis applies to smooth, axially symmetric surfaces.

Results contribute to understanding singularity formation in geometric flows.

Abstract

We study mean curvature flow of smooth, axially symmetric surfaces in $\mathbb{R}^3$ with Neumann boundary data. We show that all singularities at the first singular time must be of type I.

Equations

$$\frac{\partial }{\partial t}\mathbf{x}(l,t)=-H(l,t)\nu (l,t),l\in M^n,t\ge 0,$$

$$G=\{\mathbf{x}\in {\mathbb{R}}^3:a<x_1<b\},0<a<b,$$

$$⟨{\tau }_2(t),{\mathbf{i}}_1⟩=0,\text{and}⟨{\tau }_1(t),{\mathbf{i}}_1⟩>0.$$

$$y=⟨\mathbf{x},\omega ⟩\text{and}v={⟨\omega ,\nu ⟩}^{-1}.$$

$$p=⟨{\tau }_1,{\mathbf{i}}_1⟩y^{-1}\text{mbox}andq=⟨\nu ,{\mathbf{i}}_1⟩y^{-1},$$

$$p^2+q^2=y^{-2}.$$

$$p=\frac{1}{\rho \sqrt{1+{\rho }^{\prime 2}}}$$

$$k=⟨{\overline{\nabla }}_1\nu ,{\tau }_1⟩=\frac{-{\rho }^{\prime \prime }}{(1+{\rho }^{\prime 2}{)}^{3/2}}.$$

$$\Omega :=\bigcup _{0\le t<T}{M}_t\times \{t\}\subset {\mathbb{R}}^3\times {\mathbb{R}}^{+}.$$

$${\Omega }_t^{-}:=\{\mathbf{x}(l,t)\in M_t:H(l,t)<-c\}\subset M_t.$$

$$\underset{{\Omega }^{-}}{\inf }y=\underset{{\Gamma }_{{\Omega }^{-}}}{\inf }y\ge c.$$

$$\frac{dy}{dt}=-Hpy>0.$$

$$\underset{{\Omega }^{-}}{\inf }y=\underset{{\Gamma }_{{\Omega }^{-}}}{\inf }y.$$

$$\frac{|k|}{p}=\frac{k}{p}\le c.$$

$$\frac{|k|}{p}\le 1.$$

$$\frac{|k|}{p}\le 1+\frac{c}{p}.$$

$$\left(\frac{d}{dt}-\Delta \right)g\le -2\mu g^2-2\lambda \phi v^{-3}⟨\nabla v,\nabla g⟩-\frac{2\lambda \mu }{(\lambda -\mu v^2{)}^2}|\nabla v{|}^{2}g+\frac{2(n-1)}{y^2}{v}^2{\phi }^{\prime }|A{|}^2.$$

$$|A{|}^2\phi (v^2)\le \max \left(\underset{{\tilde{\Omega }}_0^{-}}{\max }|A{|}^2\phi (v^2),\underset{\begin{array}{l}\partial {\tilde{\Omega }}_t^{-}\\ t<T\end{array}}{\max }|A{|}^2\phi (v^2),c\right).$$

$$|A{|}^2$$

$$|A|(l_i,t_i)=\underset{l\in M^2,t\le T-\frac{1}{i}}{\max }|A|(l,t)$$

$${\tilde{\mathbf{x}}}_i(l,\tau )={\alpha }_i\left(\mathbf{x}(l,{\alpha }_i^{-2}\tau +t_i)-⟨{\mathbf{x}}_i,{\mathbf{i}}_1⟩{\mathbf{i}}_1\right),$$

$$\widetilde{H_i}(\cdot ,\tau )={\alpha }_i^{-1}H(\cdot ,{\alpha }_i^{-2}\tau +t_i)\text{and}|{\tilde{A}}_i|(\cdot ,\tau )={\alpha }_i^{-1}|A|(\cdot ,{\alpha }_i^{-2}\tau +t_i).$$

$${\alpha }_i^{-1}|A|(\cdot ,{\alpha }_i^{-2}\tau +t_i)\le 1.$$

$$\frac{\partial }{\partial \tau }{\tilde{\mathbf{x}}}_i=-{\alpha }_i^{-1}H\nu =-{\tilde{H}}_i\nu .$$

$$|A|=\sqrt{k^2+p^2}\le cp\le c{y}^{-1}.$$

$$|{\tilde{A}}_i|\le c({\alpha }_{i}y{)}^{-1}=c{\tilde{y}}_i^{-1}.$$

$$\underset{i\to \infty }{\lim }{\tilde{H}}_i=0.$$

$$\hat{v}(l)\hat{y}(l)-ϵ\le {\tilde{v}}_i(l,{\tau }_0){\tilde{y}}_i(l,{\tau }_0)\text{for all}i>I_0.$$

$$\frac{c}{2{\alpha }_i}\left(\cosh (2c^{-1}{\hat{x}}_1)+1\right)-\frac{ϵ}{{\alpha }_i}\le c\text{for all}i>I_0.$$

$$\underset{M_t\cap N_{ϵ}}{\max }|A{|}^2\le C\frac{1}{T-t}$$

Full text

1 Introduction

Consider a smooth, $n$-dimensional hypersurface immersion $\mathbf{x}_{0}:M^{n}\rightarrow\mathbb{R}^{n+1}$. The solution of mean curvature flow generated by $\mathbf{x}_{0}(M^{n})$ is the one-parameter family $\mathbf{x}:M^{n}\times[0,T)\rightarrow\mathbb{R}^{n+1}$ of smooth immersions satisfying

[TABLE]

with $\mathbf{x}(\cdot,0)=\mathbf{x}_{0}$. Here $\nu(l,t)$ represents a choice of unit normal – the outward-pointing unit normal in the closed setting – and $H(l,t)$ is the mean curvature. According to our choice of signs, the right-hand side is the mean curvature vector and the mean curvature of the round sphere is positive. We henceforth write $M_{t}=\mathbf{x}(\cdot,t)(M^{n})$.

In [9] Huisken initiated a formal investigation of the classical evolution (1.1), establishing that any compact and uniformly convex hypersurface of dimension at least two must contract smoothly to a point in finite time and in an asymptotically round fashion.

Altschuler, Angenent and Giga [1] studied generalized viscosity solutions of mean curvature flow in the axially symmetric setting. They showed in particular that there is a finite set of singular times outside of which the evolving hypersurfaces are smooth. In addition, they showed that at each of the singular times, a finite number of necks pinch off at isolated points along the axis of rotation (or else the entire connected component of the hypersurface shrinks to a point on the axis of rotation). The flow therefore produces a smooth family of smooth hypersurfaces away from the axis of rotation.

In the setting of closed two-convex hypersurfaces of dimension at least three, Huisken and Sinestrari [11] introduced a surgery-based algorithm for modifying high curvature regions in a topologically controlled way, thereby constructing a non-canonical continuation of the classical evolution which is compatible with the well-established theory of weak solutions in a precise quantitative sense.

In this paper we consider a smooth, compact, 2-dimensional hypersurface $M_{0}$ in $\mathbb{R}^{3}$ with boundary $\partial M_{0}\neq\emptyset$. We assume that $M_{0}$ is smoothly embedded in the domain

[TABLE]

and that the free boundary satisfies the constraint $\partial M_{0}\subset\partial G$. Moreover, we assume that $M_{0}$ is axially symmetric and that the surface meets the planes $x_{1}=a$ and $x_{1}=b$ orthogonally.

In this setting, the authors [8] proved that if the mean curvature is uniformly bounded on any finite time interval, then no singularity can develop during that time.

Of course, the well-known comparison principle guarantees that singularities must develop in finite time, motivating an analysis of the types of singularities that can occur. In [10] Huisken showed that if $M_{0}$ has positive mean curvature, then all singularities must be of type I. Moreover, they behave asymptotically like shrinking cylinders after appropriate parabolic rescaling.

In this paper we obtain a complete classification of singularities without any restriction on the mean curvature of the initial data. We emphasize that we henceforth restrict our attention to 2-dimensional surfaces in $\mathbb{R}^{3}$.

Theorem 1.1**.**

(Singularity Classification)* Consider a smooth, axially symmetric solution $M_{t}$ of mean curvature flow (1.1) in $\mathbb{R}^{3}$ with Neumann boundary on the maximal time interval $[0,T)$, where $T>0$ denotes the first singular time. Then all singularities that develop as $t\to T$ must be of type I.*

It follows as an immediate consequence of Theorem 1.1 that any axially symmetric surface with Neumann boundary cannot have $H<0$ everywhere. This property is independent of mean curvature flow.

If there exists some $0<t_{0}<T$ such that $H(l,t)>0$ for all $l\in M^{2}$ and $t>t_{0}$, then Theorem 1.1 follows from the work of Huisken, see section 5 of [10]. Our proof covers the cases in which points of negative mean curvature persist up to the singular time $T$.

Outline. The results in this paper are organised as follows. In section 2 we establish notation and introduce the requisite definitions. Section 3 contains preliminary height, gradient and curvature estimates. In section 4 we use these a priori estimates to prove directly that no singularities can develop in regions of negative mean curvature.

In section 5 we recall the parabolic rescaling techniques adopted in [8]. Section 6 uses this rescaling procedure to rule out singularities in certain regions of the hypersurface. Finally, in section 7, we combine these results with the work of Huisken in [10] to establish that all singularities must be of type I, completing the proof of the main theorem.

We point out that the estimates in section 7 rely on parabolic maximum principles for non-cylindrical domains. The results employed in this section go somewhat beyond standard theory and have therefore been included in an Appendix. We refer the reader to [5] for further details.

2 Notation and preliminaries

In this paper we follow the notation used in [8]. This agrees in particular with the notation used by Huisken in [10] and by Athanassenas in [2].

Let $\rho_{0}:[a,b]\rightarrow\mathbb{R}$ be a smooth, positive function on the bounded interval $[a,b]$ with $\rho^{\prime}_{0}(a)=\rho^{\prime}_{0}(b)=0$. Consider the surface $M_{0}$ in $\mathbb{R}^{3}$ obtained by rotating the graph of $\rho_{0}$ around the $x_{1}$-axis. We evolve $M_{0}$ according to (1.1) with Neumann boundary conditions at $x_{1}=a$ and $x_{1}=b$. Equivalently, we can consider the evolution of a periodic surface defined on the entire $x_{1}$ axis. This deformation process preserves axial symmetry. We denote by $T>0$ the extinction time of the smooth evolution.

Let $\mathbf{i}_{1},\mathbf{i}_{2},\mathbf{i}_{3}$ be the standard basis vectors in $\mathbb{R}^{3}$ associated with the $x_{1},x_{2},x_{3}$ axes respectively. We introduce a local orthonormal frame $\tau_{1}(t),\tau_{2}(t)$ on the evolving surfaces $M_{t}$ such that

[TABLE]

Let $\omega=\frac{\hat{\mathbf{x}}}{|\hat{\mathbf{x}}|}\in\mathbb{R}^{3}$ be the outward-pointing unit normal to the cylinder intersecting $M_{t}$ at the point $\mathbf{x}(l,t)$. Here $\hat{\mathbf{x}}=\mathbf{x}-\left\langle\mathbf{x},\mathbf{i}_{1}\right\rangle\mathbf{i}_{1}$. We additionally define

[TABLE]

Following convention we call $y$ the height function and $v$ the gradient function. We emphasize that $\rho:[a,b]\times[0,T)\rightarrow{\mathbb{R}}$, whereas $y:M^{2}\times[0,T)\rightarrow{\mathbb{R}}$. Note also that $v$ is a geometric quantity related to the inclination angle. More precisely, $v$ corresponds to $\sqrt{1+\rho^{\prime 2}}$ in our setting.

We denote by $g=\{g_{ij}\}$ the induced metric and by $A=\{h_{ij}\}$ the second fundamental form at the space-time point $(l,t)\in M^{2}\times[0,T)$. Following [10], we define the quantities

[TABLE]

which satisfy

[TABLE]

The second fundamental form has eigenvalues

[TABLE]

and

[TABLE]

We recall the following evolution equations, see [6, 9].

Lemma 2.1**.**

(Evolution Equations)* We have the evolution equations:*

• (i)

$\frac{\partial}{\partial t}y=\Delta y-\frac{1}{y}\,;$**

• (ii)

$\frac{\partial}{\partial t}v=\Delta v-|A|^{2}v+\frac{v}{y^{2}}-\frac{2}{v}|\nabla v|^{2}\,;$**

• (iii)

$\frac{\partial}{\partial t}k=\Delta k+|A|^{2}k-2q^{2}(k-p)\,;$**

• (iv)

$\frac{\partial}{\partial t}p=\Delta p+|A|^{2}p+2q^{2}(k-p)\,;$**

• (v)

$\frac{\partial}{\partial t}q=\Delta q+|A|^{2}q+q\left(p^{2}-q^{2}-2kp\right)\,;$**

• (vi)

$\frac{\partial}{\partial t}H=\Delta H+H|A|^{2}\,.$**

Finally, we establish notation for the smooth space-time hypersurface

[TABLE]

Let $c>0$. For each $0\leq t<T$ we define

[TABLE]

We let $\Omega^{-}:=\cup_{t<T}\Omega_{t}^{-}\times\{t\}\subset\Omega$ and we denote by $\Gamma_{\Omega^{-}}$ the non-cylindrical parabolic boundary of $\Omega^{-}$.

3 A priori estimates

We establish a priori height, gradient and curvature estimates. In the first step, we show that the height function $y$ has a lower bound in $\Omega^{-}$.

Lemma 3.1**.**

(Height Bound)* There exists a constant $c>0$ depending on $M_{0}$ and $\Gamma_{\Omega^{-}}$ such that*

[TABLE]

Proof.

In $\Omega^{-}$ we have

[TABLE]

That is to say, the height function is increasing in $\Omega^{-}$, from which we deduce that

[TABLE]

Now suppose that $y$ reaches zero on $\Gamma_{\Omega^{-}}$ at some time $t_{*}$. In particular, it must therefore have decreased immediately before $t_{*}$. Consider a constant $0<\bar{c}<c$ and the corresponding domain $\bar{\Omega}^{-}\supset\Omega^{-}$. On this new domain we once again have $\frac{dy}{dt}>0$, yielding a contradiction. ∎

Lemma 3.2**.**

(Gradient Estimate)* There exists a constant $c>0$ depending only on the initial hypersurface $M_{0}$ such that $yv\leq c$.*

Proof.

See Lemma 5.2 in [8]. ∎

In particular, Lemma 3.2 controls the gradient function away from the axis of rotation. This result provides the first indication that type II singularities can’t develop in our setting. Combining this with Lemma 3.1 we can therefore find a constant $c>0$ depending on $M_{0}$ and $\Gamma_{\Omega^{-}}$ such that $v|_{\Omega^{-}}\leq c\,.$

Proposition 3.3**.**

(Curvature Estimate)* There is a constant $c>0$ depending only on the initial hypersurface $M_{0}$ such that $\frac{k}{p}\leq c$.*

Proof.

See section 5 of [10]. ∎

The next result is a generalisation of Proposition 5.4 in [8]. For the convenience of the reader, we include the full proof below.

Proposition 3.4**.**

(Absolute Curvature Estimate)* Let $c>0$. Suppose that $l_{0}\in M^{2}$ and $t_{0}\in[0,T)$ are such that $H(l_{0},t_{0})\geq 0$ or $|H(l_{0},t_{0})|\leq c$. There exists a constant $C>0$ depending only on $c$ and the initial hypersurface $M_{0}$ such that $\frac{|k(l_{0},t_{0})|}{p(l_{0},t_{0})}\leq C$.*

Proof.

If both $H$ and $k$ are positive, Proposition 3.3 yields the pointwise estimate

[TABLE]

If $H\geq 0$ and $k<0$ , then from $k+p\geq 0$ we obtain $-|k|+p\geq 0$ and

[TABLE]

If $|H|\leq c$ and $k>0$ then $H>0$ and the result holds, so it remains to consider $|H|\leq c$ and $k<0$. We have $-|k|+p\geq-c$ so that

[TABLE]

From Lemma 3.2 we have $\frac{1}{p}=vy\leq c$. This completes the proof. ∎

4 Negative mean curvature

We use direct a priori estimates to establish that no singularities can develop in regions of negative mean curvature. This section has some overlap with parts of [5], which studies the first singular time for volume preserving mean curvature flow.

Let $\tilde{c}_{0}>0$ and consider the corresponding sets $\tilde{\Omega}_{t}^{-}\subset M_{t}$ such that $H<-\tilde{c}_{0}$. In addition, we define $\tilde{\Omega}^{-}:=\cup_{t<T}\tilde{\Omega}_{t}^{-}\times\{t\}$.

The following result is a generalisation of Proposition 4.6 in [5].

Proposition 4.1**.**

(Curvature away from Axis)* There exists a constant $c>0$ such that $|A|^{2}\leq c$ in $\tilde{\Omega}^{-}$.*

Proof.

Consider the product $g=|A|^{2}\varphi(v^{2})$, where $\varphi(r)=\frac{r}{\lambda-\mu r}$ and $\lambda,\mu>0$ are free constants. The evolution equation for $g$ yields the estimate

[TABLE]

Following Proposition 6.2 of [4] with $\mu>\frac{3}{4}$ and $\lambda>\mu\max v^{2}$ we can find a constant $c>0$ depending on $\mu$, $\lambda$, $\tilde{c}_{0}$ and $M_{0}$ such that

[TABLE]

Note that by construction we have $|H|=\tilde{c}_{0}$ on $\partial\tilde{\Omega}^{-}_{t}$ for all $0<t<T$. In [5], there is a positive mean curvature restriction on the boundary. Proposition 3.4 now yields a constant $c>0$ depending only on $\tilde{c}_{0}$ and $M_{0}$ such that on $\partial\tilde{\Omega}^{-}_{t}$ we have $\frac{|k|}{p}\leq c$ for all $t<T$. On $\partial\tilde{\Omega}^{-}_{t}$ we have

[TABLE]

for all $t<T$. The final estimate follows from Lemma 3.1. Since $v$ is bounded in $\tilde{\Omega}^{-}$, $\varphi(v^{2})$ is bounded from above. The product $|A|^{2}\varphi(v^{2})$ is therefore bounded on $\Gamma_{\tilde{\Omega}^{-}}$ and on $\tilde{\Omega}^{-}$ courtesy of (4.1). By our choice of $\lambda$, and since $v\geq 1$, we have a bound on $(\varphi(v^{2}))^{-1}$. This completes the proof.

∎

As in Proposition 4.8 of [5] we obtain as a consequence that $H$ is bounded from below in $\tilde{\Omega}^{-}$. That is, $H$ cannot go to $-\infty$.

Corollary 4.2**.**

(Mean Curvature Bound)* There exists a constant $c>0$ independent of time such that $H(l,t)\geq-c$ for all $l\in\ M^{2}$ and $t\in[0,T)$.*

Proof.

By construction we have $H<0$ in $\tilde{\Omega}^{-}$ and $H\geq-\tilde{c}_{0}$ on $\Omega\setminus\tilde{\Omega}^{-}$. Using Proposition 4.1, we can find a constant $c>0$ such that $|A|^{2}\leq c$ in $\tilde{\Omega}^{-}$. The trivial inequality $H^{2}\leq 2|A|^{2}$ completes the proof. ∎

As a result of Proposition 4.1, singularities can only develop in $\Omega\setminus\tilde{\Omega}^{-}$, which we investigate in the remaining sections.

5 Rescaling

We established in Section 4 that singularities are restricted to $\Omega\setminus\tilde{\Omega}^{-}$. It is well-known from [1, 3] that singularities of axially symmetric mean curvature flow are finite and discrete. Let $T$ be the first singular time for the smooth evolution and let $(x_{*},T)\in\Omega\subset\mathbb{R}^{3}\times\mathbb{R}^{+}$ be a singular point in space-time. It is therefore possible to analyse a space-time neighbourhood $N_{\epsilon}\subset\mathbb{R}^{3}\times\mathbb{R}^{+}$ centred at $(x_{*},T)$ such that the flow is smooth inside $N_{\epsilon}\backslash(x_{*},T)$. There are three possible cases:

N_{\epsilon}\subset\mathbb{R}^{3}\times\mathbb{R}^{+}$$|H|<c$$H\to\infty$$\frac{|A|^{2}}{H^{2}}\to\infty$$\frac{|A|^{2}}{H^{2}}<c

Next we introduce the parabolic rescaling techniques which will be used in section 6 to analyse the cases $|H|<c$ and $H\to\infty$ with $|A|^{2}/H^{2}\to\infty$. In particular, we use a standard contradiction argument in section 6 to show that a singularity cannot develop in either of these cases. Singularities can therefore only occur if $H\to\infty$ and $|A|^{2}/H^{2}$ is bounded. This remaining case is covered in section 7.

We employ the rescaling procedure introduced in [8]. Consider the smooth solution $M_{t}$ of (1.1) for $t\in[0,T)$, and suppose that a singularity forms at the centre of the space-time neighbourhood $N_{\epsilon}$ at the singular time $T$. In particular, $|A|^{2}\rightarrow\infty$ as $t\rightarrow T$. For integers $i\geq 1$, we select times $t_{i}\in\left[0,T-\frac{1}{i}\right]$ and points $l_{i}\in M^{2}$ such that:

$t_{i}\to T$ 2. 2.

$\mathbf{x}(l_{i},t_{i})$ lies on the $x_{1}x_{3}$ plane 3. 3.

[TABLE]

We write $\alpha_{i}=|A|(l_{i},t_{i})$ and $\mathbf{x}_{i}=\mathbf{x}(l_{i},t_{i})\,.$ Note that for $i$ sufficiently large, $\mathbf{x}_{i}$ is contained in $N_{\epsilon}$. We now rescale $M_{t}$ to obtain the family $\tilde{M}_{i,\tau}$ defined by

[TABLE]

where $\tau\in[-\alpha_{i}^{2}t_{i},\alpha_{i}^{2}(T-t_{i}-\frac{1}{i})]\,.$

Note that we rescale from a point on the axis of rotation corresponding to the point of maximum curvature, preserving axial symmetry. We define $\tilde{\rho}_{i,\tau}$ to be the generating curves of $\tilde{M}_{i,\tau}$. We denote by $|\tilde{A}_{i}|$ and $\tilde{H_{i}}$ the second fundamental form and mean curvature of $\tilde{M}_{i,\tau}\,$, respectively. By definition

[TABLE]

For $t\leq T-\frac{1}{i}$ we have

[TABLE]

Note that

[TABLE]

The rescaled flows cannot drift away to infinity: applying Proposition 3.4 we can find a constant $c>0$ depending only on $\tilde{c}_{0}$ and $M_{0}$ such that

[TABLE]

After rescaling, this becomes

[TABLE]

Since $|\tilde{A}_{i}|(l_{i},0)=1$ for all $i$, we have a bound on $\tilde{y}$ and we can therefore extract a convergent subsequence of points on the $x_{3}$ axis.

Along the sequence of rescalings we have the uniform curvature bound $|\tilde{A}_{i}|^{2}\leq 1$. Since each rescaled flow again satisfies (5.4), this gives rise to uniform bounds on all covariant derivatives of the second fundamental form, see for example [9].

Using the Arzela-Ascoli theorem we can therefore find a further subsequence which converges uniformly in $C^{\infty}$ on compact subsets of $\mathbb{R}^{3}\times\mathbb{R}$ to a non-empty smooth limit flow which exists on an interval $(-\infty,\beta)$ where $\beta\in[0,\infty]$. The crucial step is to analyse the properties of this limit flow, which we label $\tilde{M}_{\infty,\tau}$.

6 No singularities

In this section we use a standard contradiction argument to show that no singularities can develop as long as the mean curvature remains bounded. In addition, we show that no singularity can develop if both $H\to\infty$ and $|A|^{2}/H^{2}\to\infty$. The remaining scenario is analysed in the next section.

Theorem 6.1**.**

(Bounded Mean Curvature)* Consider a smooth, axially symmetric solution $M_{t}$ of mean curvature flow (1.1) with Neumann boundary on the time interval $[0,T)$ for some $T>0$. Then no singularity can develop if $H$ remains bounded.*

Proof.

Suppose in order to obtain a contradiction that a singularity forms at the point $x_{*}$ on the axis of rotation and at time $t=T$; in particular, $|A|^{2}\rightarrow\infty$ as $t\rightarrow T$. We assume in addition that the mean curvature remains bounded in a space-time neighbourhood around the point $(x_{*},T)$. We rescale using the procedure outlined in section 5 and analyse the properties of the resultant limit flow $\tilde{M}_{\infty,\tau}$. Since by assumption $|H|<c$ for some $c>0$, we have

[TABLE]

The limit flow $\tilde{M}_{\infty,\tau}$ is a stationary solution and must therefore be the catenoid. We relabel this solution $\hat{M}$ and henceforth use a ‘hat’ to indicate that a geometric quantity is associated with the catenoid.

The catenoid is obtained by rotating $\hat{y}=c\cosh(c^{-1}\hat{x}_{1})$ around the $x_{1}$ axis. For any $\epsilon>0$ and for any $l\in M^{2}$ we can find $I_{0}\in\mathbb{N}$ such that for any fixed $\tau_{0}\in(-\alpha_{I_{0}}^{2}t_{I_{0}},0)$ we have

[TABLE]

On the catenoid, $\hat{v}=\sqrt{1+\hat{y}^{\prime 2}}=\cosh(c^{-1}\hat{x}_{1})$. It therefore follows from Lemma 3.2 that

[TABLE]

For fixed $i$, the left-hand side can be made as large as we like, yielding the desired contradiction. We can therefore find a constant $c>0$ such that $|A|^{2}\leq c$ for all $t\in[0,T)$. Using standard theory, see for example [9], we obtain estimates on all covariant derivatives of $|A|$, allowing us to extend the flow beyond $T$. This completes the proof. ∎

We next consider the case in which $H\to\infty$ and $|A|^{2}/H^{2}\to\infty$.

Theorem 6.2**.**

Consider a smooth, axially symmetric solution $M_{t}$ of mean curvature flow (1.1) with Neumann boundary on the time interval $[0,T)$ for some $T>0$. Then no singularity can develop if both $H\to\infty$ and $|A|^{2}/H^{2}\to\infty$.

Proof.

We proceed as in the proof of Theorem 6.1: suppose in order to obtain a contradiction that a singularity forms at the point $x_{*}$ on the axis of rotation and at time $t=T$. We again rescale the flow. If $|A|^{2}/H^{2}\to\infty$ then $\alpha_{i}^{-1}H\to 0$, once again giving us a stationary limit flow, which must be the catenoid. The rest of the proof goes through unchanged. ∎

7 Type I singularities

We prove that all singularities must be of type I:

Proposition 7.1**.**

(Type I Singularities)* Consider a smooth, axially symmetric solution $M_{t}$ of mean curvature flow (1.1) in $\mathbb{R}^{3}$ with Neumann boundary data on the maximal time interval $[0,T)$ for some $T>0$. Suppose that a singularity forms on the axis of rotation at $x_{*}\in\mathbb{R}^{3}$ at time $T$. Assume in addition that there exists a neighbourhood $N_{\epsilon}\subset\mathbb{R}^{3}\times\mathbb{R}^{+}$ centred at $(x_{*},T)$ and a constant $c_{0}>0$ such that $|A|^{2}/H^{2}\leq c_{0}$ in $N_{\epsilon}$. Then there exists a constant $C>0$ such that*

[TABLE]

for all $t<T$.

We proceed as in section 5 of [10]. We emphasize that our setting will generate additional boundary terms.

Proof.

From Lemma 2.1 we have

[TABLE]

Following [10] we have bounds on the final term and deal with our different boundary terms by applying the non-cylindrical maximum principle, Proposition 7.3, to obtain

[TABLE]

Now note that $|q|=\left|\langle\nu,\mathbf{i}_{1}\rangle y^{-1}\right|\leq y^{-1}$. By assumption, $|A|^{2}/H^{2}$ is bounded in $N_{\epsilon}$ so we can find a constant $c>0$ such that $H|_{\Gamma_{N_{\epsilon}}}\geq c$. In addition, it is well-known (see for example Lemma 5.2 in [1]) that $y$ is bounded from below away from the singular point. In particular therefore we have a constant $c>0$ such that $y|_{\Gamma_{N_{\epsilon}}}\geq c$, giving us a bound on $|q|/H$. Applying Proposition 3.3 we find

[TABLE]

in $N_{\epsilon}$. The rest of the proof of Proposition 5.3 in [10] goes through unchanged in $N_{\epsilon}$. ∎

Together with Theorem 6.1 and Theorem 6.2, this gives Theorem 1.1.

Remark 7.2**.**

As a direct application of our main result we note that any axially symmetric surface with Neumann boundary cannot have $H<0$ everywhere. Indeed, it follows from section 4 that no singularity can develop under mean curvature flow in the negative mean curvature setting. However, an enclosing cylinder of radius $y_{max}+1$ must collapse onto a line under mean curvature flow at time $T=(y_{max}+1)^{2}/2$. This yields a contradiction with the well-known barrier principle for mean curvature flow.

Appendix: non-cylindrical maximum principle

In this section we state the maximum principle for non-cylindrical domains which was required for the proof of Theorem 1.1. In particular this extends work of Ecker [7] and Lumer [12] to our setting. Note that in [12] these are discussed in an operator theoretic setting.

Let $\Lambda=M^{n}$. Let $V\subset\Lambda\times(0,T)$ be an open non-cylindrical domain. Let $\Lambda_{t}=\Lambda\times\{t\}\,,$ and for $t\neq 0$ let $V_{t}=\Lambda_{t}\cap V$, the cross sections of $V$ for constant $t$. Let $\overline{V}$ denote the closure of $V$ and $V_{0}=\Lambda_{0}\cap\overline{V}\,.$ The boundary of $V$   is $\partial V=\overline{V}\backslash V\,.$ The parabolic boundary is $\Gamma_{V}=\partial V\backslash\Lambda_{T}\,.$ To describe the horizontal parts of the boundary of $V$ in the space-time diagram, we define the following: let $Z_{t}$ be the largest subset of $\Lambda_{t}\cap\partial V$ that is open in $\partial V$ and can be reached from “below” (with $t$ the vertical axis) in $V$ . Let $Z_{V}=\bigcup_{0<t<T}Z_{t}$ and $\delta_{V}=\Gamma_{V}\backslash Z_{V}\,.$

Proposition 7.3**.**

(Non-Cylindrical Maximum Principle)* Let $\left(M_{t}\right)_{t\in(0,T)}$ be a solution of the mean curvature flow (1.1) consisting of hypersurfaces $M_{t}=\mathbf{x}_{t}(\Lambda)$, where $\mathbf{x}_{t}=\mathbf{x}(\cdot,t):\Lambda\times[0,T)\rightarrow{\mathbb{R}}^{n+1}$ and $\Lambda$ is compact. Suppose $f\in C^{2,1}(V)\cap C(\overline{V})$ satisfies an inequality of the form*

[TABLE]

where the Laplacian $\Delta$ and the gradient $\nabla$ are computed on the manifold $M_{t}$. For the vector field $a:V\rightarrow{\mathbb{R}}^{n+1}$ we only require that it is continuous in a neighbourhood of all maximum points of $f$ . Then

[TABLE]

*for all $t\in[0,T)$.

Assuming $f$ to have a positive supremum in $V$ then*

[TABLE]

for all $t\in[0,T)\,.$