Singularities of the area preserving curve shortening flow with a free boundary condition

Abstract

We consider the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain or at a straight line. We give a criterion on initial curves that guarantees the appearance of a singularity in finite time. We prove that the singularity is of type II. Furthermore, if these initial curves are convex, then an appropriate rescaling at the finite maximal time of existence yields a grim reaper or half a grim reaper as limit flow. We construct examples of initial curves satisfying the mentioned criterion.

1 Introduction

The area preserving curve shortening flow (APCSF) for closed plane curves was introduced by Gage [7]. It is the “steepest descent flow” for the length functional under the constraint that the enclosed area is constant. For a family of simple closed curves $\gamma :{\mathbb {S}}^1\times [0,T)\rightarrow {\mathbb {R}}^2$, the evolution equation turns out to be

$$\begin{aligned} \frac{d}{dt}\gamma =\left( \kappa -\frac{\int \kappa ds}{L}\right) \nu =\left( \kappa -\frac{2\pi }{L}\right) \nu , \end{aligned}$$

where we use the following notation: $\nu =J\tau$ is the normal of the curves, where J is the rotation by $+\frac{\pi }{2}$; $\kappa$ is the curvature with respect to $\nu$, L is the length of the curves and ds denotes integration by arclength. Gage proved in [7] that a strictly convex simple closed curve remains strictly convex under the APCSF. The curves converge for $t\rightarrow \infty$ smoothly to a circle enclosing the same enclosed area as $\gamma _0$. Thus, the flow converges to the solution of the isoperimetric problem in ${\mathbb {R}}^2$. This problem consists in finding the shortest closed curve enclosing a fixed area. The analog result for n-surfaces in $\mathbb {R}^{n+1}$, $n\ge 2$ was proved by G. Huisken [12]: a uniformly convex, embedded surface moving according to the volume preserving mean curvature flow stays uniformly convex and exists for all times $t\in [0,\infty )$. The moving surfaces converge smoothly to a sphere enclosing the same volume as the initial surface.

We consider the APCSF in a free boundary setting and want to know when and how singularities develop. But at first we recall what is known about the existence of singularities in the closed situation.

Escher and Ito considered in [6] immersed closed curves possibly with self-intersections. Then the evolution equation is $\frac{d}{dt}\gamma =(\kappa -\frac{2\pi m}{L})\nu$ where $m\in \mathbb Z$ is the index (or turning number) of the immersed closed curves. The index m is independent of time, and by possibly changing the orientation it is non-negative. Escher and Ito proved that an immersed curve with $m\ge 1$ and enclosed area $A_0<0$ or $m\ge 2$ and $L_0^2<4\pi m A_0$ develops a singularity in finite time. The proof is inspired by the work of Chou on the surface diffusion flow for curves [3].

Wang and Kong also studied immersed closed curves moving according to the APCSF [19]. They proved that the flow exists for all times and converges smoothly to an m-fold circle when the initial curve is convex and has so-called “n-fold rotational symmetry” and index m ($n>2m$). On the other hand, “Abresch–Langer type” curves either converge to a multiple cover of a circle (when $A_0>0$) or the curvature blows up at finite time (when $A_0<0$) or the curvature blows up at the maximal time of existence (when $A_0=0$), see [19, Theorem 1.2]. Note that there are examples where only a slight change is necessary to deform an initial curve with $A_0<0$ into one with $A_0=0$ and then into one with $A_0>0$.

We now explain the free boundary setting of the APCSF which was studied by the author in [14, 15]. Let $\Sigma \subset {\mathbb {R}}^2$ be a convex simple closed curve in the plane and orient it positively. We call $\Sigma$ a support curve. It is not moving in time. An initial curve $\gamma _0:[a,b]\rightarrow {\mathbb {R}}^2$ is a curve with endpoints $\gamma _0(a),\gamma _0(b)\in \Sigma$ where we prescribe the angle to be $90^\circ$. We consider the “outer situation” which means that the curve $\gamma _0$ goes into the “exterior domain” with respect to $\Sigma$ and also comes back to $\Sigma$ “from the outside” at the endpoints. In formulas, this means

$$\begin{aligned} \tau _0(a) = - \nu _\Sigma (\gamma _0(a)), \quad \tau _0(b) = \nu _\Sigma (\gamma _0(b)), \end{aligned}$$

(1)

where $\tau _0:[a,b]\rightarrow \mathbb {R}^2$ is the tangent of $\gamma _0$ and $\nu _\Sigma :\Sigma \subset \mathbb {R}^2\rightarrow \mathbb {R}^2$ is the inner unit normal to $\Sigma .$¹

We now let the curve $\gamma _0$ flow according to the APCSF such that these conditions are preserved, i.e. $\gamma :[a,b]\times {[}0,T)\rightarrow {\mathbb {R}}^2$ satisfies $\gamma (a,t),\gamma (b,t)\in \Sigma$ and (1) for each time $t\in {[}0,T)$ and

$$\begin{aligned} \frac{d}{dt}\gamma =\left( \kappa -\frac{\int \kappa ds}{L}\right) \nu . \end{aligned}$$

As the curves are not closed the quantity $\int \kappa ds$ is not an integer times $2\pi$ in general. It is in fact the first step to find conditions that guarantee a bound of ${\bar{\kappa }}:= \frac{\int \kappa ds}{L}$ independent of t. In [15], the author proved that the flow in this setting does not develop a singularity when the initial curve satisfies four conditions:

• (i) $\gamma _0$ is strictly convex,

• (ii) it is embedded,

• (iii) it is contained in the exterior domain with respect to $\Sigma$ and

• (vi) it satisfies $L_0< \frac{4}{5\max |\kappa _\Sigma |}\arcsin (\frac{A_0}{L_0^2})$,

where $A_0$ is the enclosed area of the domain enclosed by $\gamma _0$ and the part of $\Sigma$ connecting $\gamma _0(b)$ and $\gamma _0(a)$. Furthermore, the curves $\gamma (\cdot ,t)$ subconverge under these conditions smoothly for $t\rightarrow \infty$ to an arc of a circle sitting outside of $\Sigma$ and meeting $\Sigma$ perpendicularly.

In this paper we answer the following questions that naturally arise when studying this setting:

• Are there curves that develop a singularity under the APCSF in the free boundary setting?

• Are there convex initial curves developing a singularity?

• Does the singularity appear in finite time?

• Of what type are the singularities?

• What does a blowup at the singular time look like?

For our main theorem we explain some preliminaries. As $\Sigma$ is a smooth convex closed curve, every $x\in \Sigma$ has an “antipodal point” $x'\in \Sigma$ which is a point in $\Sigma$ with $\tau _\Sigma (x)=-\tau _\Sigma (x')$, where $\tau _\Sigma :\Sigma \subset \mathbb {R}^2\rightarrow \mathbb {R}^2$ is the tangent of $\Sigma$. Note that this point is not unique as the curve is not strictly convex. The minimum width of $\Sigma$ is

$$\begin{aligned} d_\Sigma := \min \{|x-x'|:x, x'\in \Sigma , x' \text { antipodal to } x\}. \end{aligned}$$

This is the least distance of two parallel lines touching $\Sigma$.

We consider $\gamma _0:{[}a,b{]}\rightarrow \mathbb {R}^2$, an initial curve with $L_0< d_\Sigma$, where $L_0$ is the length of $\gamma _0$. By definition of $d_\Sigma$ the points $\gamma _0(a)$ and $\gamma _0(b)$ can not be antipodal to each other. We let the curve $\gamma _0$ flow by the APCSF with Neumann free boundary conditions as described above. As this flow is the “steepest descent flow” of the length functional (under a constraint), the length does not increase under the flow. As a consequence we get that all endpoints of the evolving curves $\gamma (a,t),\gamma (b,t)$ are not antipodal to each other. Note that for each time $t\in [0,T)$ the curve $\Sigma {\setminus }\{\gamma (a,t),\gamma (b,t)\}$ is divided into two pieces. At one piece the angle of the normal $\nu _\Sigma$ turns more than $\pi$. The angle of the unit normal of the other part, we call it the short piece, turns less than $\pi$.

For each $t\in [0,T)$ we append the “short piece” of $\Sigma$ to $\gamma (\cdot ,t)$ in order to close the curve $\gamma (\cdot ,t)$: Define a family $\sigma (t):[\alpha (t),\beta (t)]\rightarrow \Sigma$ by connecting $\gamma (b,t)$ and $\gamma (a,t)$ by following $\Sigma$ along the “short piece”. Note that $\sigma (t)$ is just a point if $\gamma (a,t)=\gamma (b,t)$. We use the notation $\sigma (0){=}{:}\sigma _0$. Since the endpoints of our curves are never antipodal and as the endpoints of $\gamma (\cdot ,t)$ vary continuously in t, the family $\sigma$ is continuous in t. We will see that it is actually $C^1$ in t. We denote the assembled closed curve by $\gamma (\cdot ,t) + \sigma (t)$. The boundary conditions imply that $\int _{\gamma (\cdot ,t)}\kappa d s\not \in 2\pi \mathbb {Z}$ for all $t\in [0,T)$, in particular $\int _{\gamma _0}\kappa d s\ne 0$. The (oriented) enclosed area $A(\gamma (\cdot ,t) + \sigma (t))$ is preserved under the APCSF, and we can state our main theorem:

Theorem 1.1

Let $\gamma _0:{[}a,b{]}\rightarrow {\mathbb {R}}^2$ be an initial curve satisfying $L_0<d_\Sigma$. Choose the orientation of $\gamma _0$ such that $\int _{\gamma _0}\kappa d s>0$. Fix $l\in \mathbb {N}$ such that $(2l-2)\pi<\int _{\gamma _0}\kappa ds<2l\pi$. We further assume

• (i) either $A(\gamma _0 +\sigma _0)>0$ and $\frac{L_0^2}{A(\gamma _0 +\sigma _0)} \le \pi \frac{(2l-1)^2}{l}$,

• (ii) or $A(\gamma _0 +\sigma _0)<0$,

where $\gamma _0 + \sigma _0$ is the extension of $\gamma _0$ along the “short piece” described above.

In these cases the solution of the area preserving curve shortening flow with Neumann free boundary conditions outside of $\Sigma$ develops a singularity in finite time, i.e. $T_{\text {max}}<\infty$. Furthermore, the finite time singularity is of type II in the sense that

$$\begin{aligned}&\max _{p\in {[}a,b{]}}|\kappa |(p,t)\rightarrow \infty \ (t\rightarrow T_{\text {max}}) \text { and }\\&\max _{p\in {[}a,b{]}}\left( |\kappa |^2(p,t)(T_{\text {max}}-t)\right) \text { is unbounded}. \end{aligned}$$

If $\gamma _0$ is convex, we can say what the limit flow looks like after a suitable rescaling procedure.

Corollary 1.2

Let $\gamma _0:[a,b]\rightarrow \mathbb {R}^2$ be an initial curve satisfying the conditions from Theorem 1.1. Assume further that $\gamma _0$ is convex, $\kappa _0\ge 0$. Then the “Hamilton blow-up” at $T_{max}<\infty$ yields either a grim reaper without boundary or half a grim reaper at a straight line.

Remark

• (i) The “Hamilton blow-up” was defined in [9]. We will explain it in the proof of Corollary 1.2.

• (ii) There is numerical evidence given by Mayer [16] that there are embedded closed curves that first get a self-intersection and then develop a singularity under the APCSF. In the free boundary setting, it seems to be the case that there are initially embedded curves that stay embedded but develop a singularity in finite time, see Example Three in Sect. 3. We think that these curves develop a singularity at the boundary.

We also study the situation at a straight line. The result is as follows.

Theorem 1.3

Let $\gamma _0:{[}a,b{]}\rightarrow \mathbb {R}^2$ be an initial curve at a straight line $\Sigma$. Let $\delta _0$ be the closed curve obtained by reflecting $\gamma _0$ at $\Sigma$. Let $\mathrm{ind}(\delta _0)=:m$ be the index of $\delta _0$. Then m is odd. Choose the orientation of $\delta _0$ such that m is positive.

Then the area preserving curve shortening flow with Neumann free boundary conditions at the line $\Sigma$ develops a singularity in finite time if one of the following conditions is satisfied:

• (i) Either $A(\delta _0)<0$.

• (ii) Or $m\ge 3$ and $L(\delta _0)^2 < 4\pi m A(\delta _0)$.

The singularity is of type II.

The structure of this paper is as follows. In Sect. 2 we recall some results from [14, 15] that we use in the proof of Theorem 1.1. We explain again how strongly the condition $L_0<d_\Sigma$ influences the behavior of $\int _{\gamma (\cdot ,t)}\kappa d s$ along the flow. A bound on $|{\bar{\kappa }}|$ independent of $T_{max}$ is a consequence. If $T_{max}=\infty$, then the bound on $|{\bar{\kappa }}|$ together with [15] imply subconvergence to a part of a circle that is possibly (partly) multicovered. We study the geometry of the limiting arc and get a contradiction to the assumptions. We refine results from [15] to show that the singularity is of type II. If the initial curve is convex we showed in [15] that the “Hamilton blowup” yields a grim reaper or half a grim reaper at a straight line.

In Sect. 3, we give examples of curves that do satisfy the conditions of Theorem 1.1 and Corollary 1.2.

Section 4 contains the proof of Theorem 1.3. We reflect the curves at the line $\Sigma$ and apply the results from [6]. We combine this with results from [15] to show that the singularity is of type II.

2 Singularities of type II in finite time

Notations

Let $\gamma :{[}a,b{]}\rightarrow {\mathbb {R}}^2$ be a piecewise smooth, regular curve and let $h:[a,b]\rightarrow \mathbb {R}^n$, $n\in \{1,2\}$, be a $C^1$-map, $h=h(p)$. We denote by $\partial _sh\,{:=}\, \frac{1}{|\partial _p\gamma |}\partial _ph$ the derivative with respect to arclength of h. We define $d s \,{:=}\, |\partial _p \gamma |d p$. We recall the formula for the curvature of $\gamma$

$$\begin{aligned} \kappa (p) = \langle \partial ^2_s \gamma (p),\nu (p)\rangle , \end{aligned}$$

where $\nu = J\tau =J\partial _s\gamma$ is the normal of the curve $\gamma$, J is the rotation by $+\frac{\pi }{2}$ in the plane.

Definition 2.1

We call a smooth, regular, convex, simple and smoothly closed curve $f:\mathbb S^1\rightarrow {\mathbb {R}}^2$ a support curve. We assume $f$ to be parametrized by arclength. We orient f positively so that $\kappa _\Sigma \ge 0$. We use the notation

$$\begin{aligned} \Sigma \,{:=}\, f(\mathbb {S}^1). \end{aligned}$$

The curve $\Sigma$ separates ${\mathbb {R}}^2$ into a bounded and an unbounded domain. The bounded domain is enclosed by $\Sigma$ and is denoted by $G_\Sigma$.

We define $d_\Sigma \, {:=}\, \min \{|x-y|:x,y\in \Sigma , \tau _\Sigma (x)=-\tau _\Sigma (y)\}$, the smallest distance between two parallel lines in ${\mathbb {R}}^2$ that touch $G_\Sigma$ (the minimum width).

Definition 2.2

A planar, smooth, regular curve $\gamma _0:[a,b]\rightarrow {\mathbb {R}}^2$ is called initial curve if it satisfies the conditions

$$\begin{aligned}&\gamma _0(a),\gamma _0(b)\in \Sigma \\&\tau _0(a) = -\nu _\Sigma (\gamma _0(a)), \quad \tau _0(b) = \nu _\Sigma (\gamma _0(b)), \end{aligned}$$

where $\tau _0=\partial _s\gamma _0$ is the tangent of $\gamma _0$ and $\nu _\Sigma = J\, \partial _sf \circ f^{-1}:\Sigma \rightarrow \mathbb {R}^2$ is the inner unit normal of $\Sigma$ (defined on the image $\Sigma = f(\mathbb S^1)$).

Definition 2.3

Let $\gamma _0:[a,b]\rightarrow {\mathbb {R}}^2$ be an initial curve. A smooth family of smooth, regular curves $\gamma :[a,b]\times [0,T)\rightarrow {\mathbb {R}}^2$ that satisfies

$$\begin{aligned} \begin{aligned} \frac{\partial \gamma }{\partial t} (p,t) = ( \kappa (p,t)&- \bar{\kappa }(t))\nu (p,t)&\forall (p,t)&\in [a,b]\times [0,T),\\ \gamma (p,0)&=\gamma _0(p)&\forall p&\in [a,b],\\ \gamma (a,t),\gamma (b,t)&\in \Sigma&\forall t&\in [0,T),\\ \tau (a,t) = - \nu _\Sigma (\gamma (a,t)),&\ \ \tau (b,t) = \nu _\Sigma (\gamma (b,t)),&\forall t&\in [0,T), \end{aligned} \end{aligned}$$

(2)

is called a solution of the area preserving curve shortening problem with Neumann free boundary conditions. Here, ${\bar{\kappa }}$ denotes the average of the curvature,

$$\begin{aligned} {\bar{\kappa }}(t)\,{:=}\, \frac{\int \kappa (p,t)d s}{\int d s}= \frac{\int \kappa (p,t)d s}{L(\gamma (\cdot ,t))}, \end{aligned}$$

and $\nu _\Sigma$ is the inner unit normal of $\Sigma$. In the rest of the article, we use the notation $\gamma _t\,{:=}\, \gamma (\cdot ,t)$.

Remark

For a smooth initial curve, existence and uniqueness of the solution of (2) is standard. One gets short time existence on a short time interval $[0,T_0]$. The solution can be extended up to a maximal time of existence $T_{max}\le \infty$. By regularity theory for parabolic Neumann problems the curves satisfy

$$\begin{aligned} \gamma \in C^{2+\alpha , 1 +\frac{\alpha }{2}}\left( [a,b]\times [0,T_{max}),\mathbb {R}^2\right) \cap C^\infty \left( [a,b]\times (0,T_{max}),\mathbb {R}^2\right) ,\ \alpha \in (0,1), \end{aligned}$$

where $C^{2+\alpha , 1 +\frac{\alpha }{2}}$ denotes the usual parabolic Hölder space. If $T_{max}<\infty$ then $\max _{{[}a,b{]}}|\kappa |(\cdot ,t)\rightarrow \infty$ ($t\rightarrow T_{max}$). A source for the existence for closed curves moving by a geometric flow with a constraint is for example [4]. The technique how to transform the free boundary problem into a standard Neumann boundary problem can be found in [17, 18]. For our specific situation a sketch of the existence and regularity result is in [15, Proposition 2.4].

Definition 2.4

Let $\Sigma$ be a support curve and let $\gamma :[a,b]\rightarrow {\mathbb {R}}^2$ be a curve with $\gamma (a),\gamma (b)\in \Sigma$. Then we call a curve $\sigma :[{\tilde{a}},{\tilde{b}}]\rightarrow \Sigma \subset {\mathbb {R}}^2$ with $\sigma ({\tilde{a}})=\gamma (b)$ and $\sigma ({\tilde{b}})=\gamma (a)$ a boundary curve on $\Sigma$ with respect to $\gamma$.

Definition 2.5

Let $\gamma :{[}a,b{]}\times [0,T)\rightarrow {\mathbb {R}}^2$ be a solution of (2). Consider a $C^1$-family of smooth curves $\sigma :[{\tilde{a}},{\tilde{b}}]\times [0,T)\rightarrow \Sigma$ with $\sigma ({\tilde{a}},t)=\gamma (b,t)$ and $\sigma ({\tilde{b}},t)=\gamma (a,t)$ for all $t\in [0,T)$, i.e. $\sigma _t\,{:=}\,\sigma (\cdot ,t)$ is a boundary curve on $\Sigma$ with respect to $\gamma _t = \gamma (\cdot ,t)$. Then for each $t\in [0,T)$, we call the following expression the oriented area enclosed by $\gamma _t$ and $\Sigma$:

$$\begin{aligned} A(\gamma _t + \sigma _t)\,{:=}\, \frac{1}{2} \int _{\gamma _t} p^1 dp^2 - p^2 dp^1 + \frac{1}{2} \int _{\sigma _t} p^1 dp^2 - p^2 dp^1. \end{aligned}$$

(3)

Remark

Our curves $\gamma _t$ are regular. But it can happen that a curve $\sigma _t$ is not regular. For our situation, this will only happen if $\gamma _t(a)=\gamma _t(b)$. Then $\sigma _t$ will be just the point $\sigma _t \equiv \gamma _t(a)=\gamma _t(b)$. This is not important for the definition of the enclosed area because in such a situation $\gamma _t$ is already closed and the second integral in (3) vanishes.

We recall some basic properties proved in [15].

Lemma 2.6

(Lemma 2.6, Lemma 2.11 and Corollary 2.14 [15]) Let $\gamma _0:{[}a,b{]}\rightarrow \mathbb {R}^2$ be a smooth initial curve. Then we have the following properties: the area preserving curve shortening flow is curve shortening and area preserving, i.e. $\frac{d}{dt}L(\gamma _t)\le 0$ and $\frac{d}{dt}A(\gamma _t,\sigma _t)=0$ on ${[}0,T)$, where $\gamma :{[}a,b{]}\times {[}0,T)\rightarrow \mathbb {R}^2$ is a solution of (2) and $\sigma :[{\tilde{a}},{\tilde{b}}]\times {[}0,T)\rightarrow \Sigma$ is a $C^1$-family of boundary curve on $\Sigma$ with respect to $\gamma$. As the domain $G_\Sigma$ is convex and as $\gamma _0$ goes into $\mathbb {R}^2{\setminus } G_\Sigma$ at $\gamma _0(a)$ and comes back to $\Sigma$ from $\mathbb {R}^2{\setminus } G_\Sigma$ at $\gamma _0(b)$ the flow improves convexity to strict convexity.² This is, $\kappa _0\ge 0$ for the initial curve implies $\kappa >0$ on ${[}a,b{]}\times (0,T)$.

Remark

• (i) If $\gamma _0$ is a smooth initial curve then a $C^1$-family of boundary curves $\sigma$ on $\Sigma$ with respect to $\gamma$ exists. This was proved in [15, Lemma 2.9]. Under the condition $L_0<d_\Sigma$ we will explain the construction of such a family below.

• (ii) We emphasize that it is allowed that one of the boundary curves $\sigma _t$ consists only of one point (namely of the endpoints $\gamma _t(a)=\gamma _t(b)$). Important in the proof of Lemma 2.6 is only that one has to find a family of boundary curves where the enclosed area is continuous in t.

Lemma 2.7

(Construction of the boundary curves) Let $\gamma _0:[a,b]\rightarrow \mathbb {R}^2$ be a smooth initial curve with $L_0< d_\Sigma$. Then the solution of (2) has the following property: the endpoints $\gamma _t(a), \gamma _t(b)$ divide $\Sigma$ into two pieces for each $t\in [0,T)$. The angle of the unit normal of one component of $\Sigma {\setminus }\{\gamma _t(a),\gamma _t(b)\}$ turns more than $\pi$ (and less or equal than $2\pi ).$ The unit normal of other component—we will call it the short piece—turns an angle of less than $\pi$. Note that the (degenerate) case where the short piece is just a point is possible. This only happens if $\gamma _t(a)=\gamma _t(b)$. We denote by $\sigma (t)$ the curve from $\gamma _t(b)$ to $\gamma _t(a)$ along the short piece of $\Sigma$. After reparametrizations we get a $C^1$-family of boundary curves $\sigma :[{\tilde{a}},{\tilde{b}}]\times [0,T)\rightarrow \Sigma$ with respect to $\gamma$, where $\sigma _t$ are regular smooth curves except in the degenerate case where $\sigma _t\equiv \gamma _t(a)=\gamma _t(b)$. As a consequence, the enclosed area $A(\gamma _t + \sigma _t)$ is constant along the flow.

Remark

The “short piece” is not the piece with the shorter length. It is the piece where the image of the unit normal on $\mathbb S^1$ is shorter.

Proof

The construction of the boundary curve is quite explicit. The only thing that we have to show is that $\sigma _t$ is $C^1$ (and in particular continuous) with respect to t. The continuity follows from that fact that $L(\gamma _t)\le L_0< d_\Sigma$. By this property the short piece cannot jump from time to time, i.e. the short piece of $\Sigma$ varies continuously in t. Since $\gamma$ is in fact $C^1$ in t and as $\Sigma$ is smooth, $\sigma$ is a $C^1$ family of boundary curves. $\square$

The following result comes from analyzing the geometric properties of a convex curve that satisfy the Neumann free boundary conditions outside a convex domain at the endpoints.

Proposition 2.8

Let $\Sigma \subset \mathbb {R}^2$ be a positively oriented convex smooth Jordan curve and let $\gamma :{[}a,b{]}\rightarrow \mathbb {R}^2$ be a $C^2$-curve with $\kappa >0$ and

$$\begin{aligned} \gamma (a),\gamma (b)&\in \Sigma ,\\ \tau (a) = - \nu _\Sigma (\gamma (a)),&\ \ \tau (b) = \nu _\Sigma (\gamma (b)), \end{aligned}$$

where $\nu _\Sigma$ is the inner unit normal of $\Sigma$. Then we have that $\int \kappa d s\ge \pi$.

Proof

In [15, Proposition 3.1], it was shown that the geometric situation of the curves imply $\int \kappa d s\ge \pi$. The statement there was formulated for a solution of (2). But the only properties of the curves that are used in the proof are strict convexity and the boundary conditions. $\square$

In order to be able to use results from [15] we need to show that ${\bar{\kappa }}(t)$ is bounded in $L^\infty$. As we want to show results about flows with infinite lifespan, we want the bound to be independent of the maximal time of existence $T_{max}$.

Proposition 2.9

Let $\gamma _0:{[}a,b{]}\rightarrow {\mathbb {R}}^2$ be an initial curve (not necessarily convex) with $L_0<d_\Sigma$. Consider the solution of the APCSF (2) on the maximal time interval of existence $[0,T_{max})$. Choose $l\in \mathbb {Z}$ such that $(2l - 2)\pi<\int _{\gamma _0}\kappa d s<2l\pi$. Then we have that

$$\begin{aligned} (2l-2) \pi< \int _{\gamma _t} \kappa d s < 2l\pi \quad \text { for all } t\in [0,T_{max}). \end{aligned}$$

Proof

By definition of $d_\Sigma$ and by the curve shortening property the points $\gamma _t(a)$ and $\gamma _t(b)$ are never “antipodal points”. This means that $\tau _\Sigma (\gamma _t(a))\not = -\tau _\Sigma (\gamma _t(b))$ for each $t\in [0,T)$. Taking into account the boundary conditions $\nu _\Sigma (\gamma _t(a)) = -\tau (a,t)$ and $\nu _\Sigma (\gamma _t(b)) =\tau (b,t)$ for the inner unit normal $\nu _\Sigma = J\tau _\Sigma$ we get that

$$\begin{aligned} \tau (a,t)\not =\tau (b,t) \end{aligned}$$

for each $t\in [0,T)$. This particularly implies that $\int _{\gamma _t}\kappa d s\not \in 2\pi \mathbb {Z}$ for each $t\in [0,T)$. The continuity of $\int _{\gamma _t}\kappa d s$ with respect to t implies the result. $\square$

Proposition 2.10

Let $\gamma :{[}a,b{]}\times [0,T_{\text {max}})\rightarrow {\mathbb {R}}^2$ be the solution of (2) where the initial curve $\gamma _0:{[}a,b{]}\rightarrow {\mathbb {R}}^2$ satisfies $L_0< d_\Sigma$. Furthermore, we assume that $\gamma _0$ satisfies

$$\begin{aligned} A(\gamma _0 +\sigma _0)\ne 0, \end{aligned}$$

where $\gamma _0 + \sigma _0$ is the extension of $\gamma _0$ via the “short piece” along $\Sigma$ defined in Lemma 2.7. Then there is a constant $\delta >0$ such that $L(\gamma _t)\ge \delta$ for all $t\in [0,T_{\text {max}})$.

Proof

We assume that there is a sequence $t_j\rightarrow T_{\text {max}}$ with $L(\gamma _{t_j})\rightarrow 0$ $(j\rightarrow \infty )$. Since $\Sigma$ is compact we get $x_0\in \Sigma$ and (after passing to a subsequence) $\gamma (a,t_j)\rightarrow x_0$, $\gamma (b,t_j)\rightarrow x_0$. This means that the curves $\gamma _{t_j}$ close up as $j\rightarrow \infty$. The boundary curves $\sigma (t_j)$ are the curves connecting the endpoints $\gamma _{t_j}(b)$ and $\gamma _{t_j}(a)$ along the part of $\Sigma$ where $\int _{\sigma _{t_j}}\kappa _\Sigma d s_\Sigma$ is smaller. This implies that $L(\sigma _{t_j})\rightarrow 0$ as $j\rightarrow \infty$. As a consequence, we also have that $A(\gamma _{t_j} + \sigma _{t_j})\rightarrow 0$ as $j\rightarrow \infty$. Due to the fact that $A(\gamma _0 + \sigma _0)=A(\gamma _{t_j} + \sigma _{t_j})$ for all $j\in \mathbb {N}$ we get a contradiction to our assumption. $\square$

Theorem 2.11

Let $\gamma :{[}a,b{]}\times [0,\infty )\rightarrow {\mathbb {R}}^2$ be a solution of (2) (without singularities in finite time) where the initial curve $\gamma _0:{[}a,b{]}\rightarrow {\mathbb {R}}^2$ satisfies $L_0< d_\Sigma$ and

$$\begin{aligned} A(\gamma _0 +\sigma _0)\ne 0. \end{aligned}$$

Here, $\gamma _0 + \sigma _0$ is the extension of $\gamma _0$ along the “short piece” of $\Sigma$ coming from Lemma 2.7. Choose $l\in \mathbb {Z}$ such that $(2l - 2)\pi<\int _{\gamma _0}\kappa d s<2l\pi$.

Then $\gamma _t$ $(t\rightarrow \infty )$ subconverges (after reparametrization) smoothly to a (possibly multicovered) arc of circle $\gamma _\infty$ sitting outside of $\Sigma$ at the endpoints. Note that the arc can be positively or negatively oriented. Each of the two contact angles at the endpoints of $\gamma _\infty$ is a $90^{\circ }$ angle. Furthermore, the limit curve satisfies

$$\begin{aligned} \int \kappa d s_\infty&\in \left[ (2l-1)\pi ,2l\pi \right) \text { if } l \ge 1, \end{aligned}$$

(4)

$$\begin{aligned} \int \kappa d s_\infty&\in \left( (2l-2)\pi ,(2l-1)\pi \right] \text { if } l\le 0. \end{aligned}$$

(5)

Proof

In [15, Theorem 7.15], subconvergence is proved under the conditions $L(\gamma _t)\ge c_1>0$ and ${\bar{\kappa }}(t)\in [\bar{c}, c_2]$ for all $t\in [0,\infty )$ for constants $c_1, \bar{c}, c_2>0$. But the proof in fact also works if we do not assume the lower bound ${\bar{\kappa }}\ge \bar{c}>0$. We only need $|{\bar{\kappa }}|\le c_2$ and $L(\gamma _t)\ge c_1>0$. We sketch this proof for the convenience of the reader: for any sequence $\tau _l\rightarrow \infty$ we reparametrize the original curves $\tilde{\gamma }(\cdot ,\tau _l)$ by constant speed and get a solution $\gamma _l:[0,1]\times [0,\infty )\rightarrow \mathbb {R}^2$ of (2) with $|\gamma _l'|=L(\tilde{\gamma }_{\tau _l})$ at the time $\tau _l$. Using Gagliardo–Nirenberg interpolation inequalities and integral estimates we proved in Corollary 7.14 from [15] a bound

$$\begin{aligned} \sup _{(p,t)\in [0,1]\times [1,\infty )}|\kappa _l(p,t)|\le C, \end{aligned}$$

where C does not depend on l. Using the graph representation of the curves, the lower bound on the length and the flow equation we get estimates $|\partial _t^i\partial _s^m\kappa |\le c$ on $[0,1]\times [\tau _l,\tau _l + \delta ]$ for any $\delta >0$. We split the derivatives $\partial _p\gamma _l$ into its tangential and normal part and use an induction argument together with the bound on $|\partial _s^m\kappa |$. This yields $|\partial _p^m\gamma _l|\le c$ on $[0,1]\times [\tau _l,\tau _l+ \delta ]$, where c depends on $m,\Sigma ,C,L_0$ and $\delta$. Choose $\tau _l\rightarrow \infty$ and $\delta >0$ such that $\bigcup \nolimits _{l\in \mathbb {N}}[\tau _l,\tau _l + \delta )=[1,\infty )$ then we have proved

$$\begin{aligned} |\partial _p^m\gamma _l|\le c \text { on }[0,1]\times [1,\infty ). \end{aligned}$$

The proof of these estimates can be found in [15, Proof of Proposition 4.7] or in [14, Section 5.3].

For any $t_l\rightarrow \infty$ we consider $\alpha _l:=\gamma _l(\cdot ,t_l)$. Using the theorem of Arzela–Ascoli the curves subconverge to a smooth curve $\gamma _\infty :[0,1]\rightarrow \mathbb {R}^2$ in every $C^m$ on [0, 1], $m\in \mathbb {N}_0$. This implies

$$\begin{aligned} \lim _{l'\rightarrow \infty }{\bar{\kappa }}(t_{l'})= \lim _{l'\rightarrow \infty }\frac{\int _{\alpha _{l'}}\kappa ds}{\int _{\alpha _{l'}} ds} ={\bar{\kappa }}(\gamma _\infty )\in [-c_2,c_2]. \end{aligned}$$

As a consequence we get that

$$\begin{aligned} \lim _{l'\rightarrow \infty }\int _{\alpha _{l'}}(\kappa -{\bar{\kappa }}(\gamma _\infty ))^2 d s = \lim _{l'\rightarrow \infty }\int _{\gamma (\cdot ,t_{l'})}(\kappa -{\bar{\kappa }})^2 d s =0, \end{aligned}$$

where we used $\lim \nolimits _{t\rightarrow \infty }\int \nolimits _{\gamma _t}(\kappa -{\bar{\kappa }})^2 ds = 0$, which was shown in Corollary 7.5 in [15]. Thus, the limit curve $\gamma _\infty$ satisfies $\kappa _\infty \equiv {\bar{\kappa }}(\gamma _\infty )\in [-c_2,c_2]$. By compactness of $\Sigma$ and by continuity we get that the endpoints of $\gamma _\infty$ lie in $\Sigma$, the curve goes into the “exterior” domain and comes back from the “exterior” domain at the endpoints. Is not possible that $\gamma _\infty$ is a part of a straight line by these geometric properties, which implies that ${\bar{\kappa }}(\gamma _\infty )\ne 0$. So we get that the limit curve $\gamma _\infty$ is a (possibly partly multicovered) arc of a circle. By reversing the orientation we can assume that $\gamma _\infty$ is positively oriented, thus $\kappa _\infty \equiv {\bar{\kappa }}(\gamma _\infty )>0$. Proposition 2.9 yields

$$\begin{aligned} \int \kappa d s_\infty \in \left[ (2l-2)\pi ,2l\pi \right] . \end{aligned}$$

We showed in Proposition 2.8 that for a strictly convex curve “outside” of $\Sigma$ at the endpoints we always have $\int \kappa ds \ge \pi$. Using this for the “last” open part of the arc $\gamma _\infty$ we get that $\int \kappa d s_\infty \in \left[ (2l-1)\pi ,2l\pi \right]$. The situation $\int \kappa d s_\infty =2\pi l$ is excluded by the geometric situation as well. If the arc was negatively oriented, estimate (5) is obtained by using (4) for the limiting arc with reversed orientation.

It remains to mention that the bounds $L(\gamma _t)\ge c_1>0$ and $|{\bar{\kappa }}|\le c_2$ are satisfied under the assumptions of the theorem. This follows from Propositions 2.9 and 2.10.

$\square$

We restate our result about the existence of finite time singularities.

Theorem 2.12

Let $\gamma _0:{[}a,b{]}\rightarrow {\mathbb {R}}^2$ be an initial curve with $L_0< d_\Sigma$. Choose the orientation of $\gamma _0$ such that $\int _{\gamma _0}\kappa ds >0$. Consider $l\in \mathbb {N}$ such that $\int _{\gamma _0}\kappa ds \in \left( (2l-2)\pi ,2l\pi \right)$. We further assume

• (i) either $A(\gamma _0 + \sigma _0)>0$ and $\frac{L_0^2}{A(\gamma _0 + \sigma _0)} \le \pi \frac{(2l-1)^2}{l}$,

• (ii) or $A(\gamma _0 + \sigma _0)<0$.

Here, $\gamma _0 + \sigma _0$ is the extension of $\gamma _0$ along the “short piece” of $\Sigma$ defined in Lemma 2.7. In both cases the solution of (2) develops a singularity in finite time, i.e. $T_{\text {max}}<\infty$.

Proof

Theorem 2.11 implies that $\gamma _t$ subconverges to an arc of a circle $\gamma _\infty$ sitting outside of $\Sigma$ at the endpoints. Property $l>0$ implies (4), which is $\int \kappa d s_\infty \in \left[ (2l-1)\pi ,2l\pi \right)$. This also gives us the information that the arc $\gamma _\infty$ is positively oriented. In particular, the enclosed area in the limit is positive, $A(\gamma _\infty + \sigma _\infty )>0$, which yields a contradiction in case (ii) because the flow is area preserving. We consider case (i): the quantities in the isoperimetric quotient satisfy

$$\begin{aligned} L(\gamma _\infty )&= 2(l-1)\pi r_\infty + \alpha _\infty r_\infty \ge (2l-1)\pi r_\infty \ \text { for some }\alpha _\infty \in [\pi ,2\pi )\text { and }\end{aligned}$$

(6)

$$\begin{aligned} A(\gamma _0 +\sigma _0)&= A(\gamma _\infty + \sigma _\infty )= (l-1) \pi r_\infty ^2 + {\tilde{A}}_\infty < l\pi r_\infty ^2, \end{aligned}$$

(7)

where $r_\infty$ is the radius of the arc $\gamma _\infty$ and $0<{\tilde{A}}_\infty < \pi r_\infty ^2$ is the area of the domain inside one full circulation of $\gamma _\infty$ without the positive area of $G_\Sigma$. We compute

$$\begin{aligned} \frac{L_0^2}{A(\gamma _0 + \sigma _0)}&\ge \frac{L(\gamma _{t_j})^2}{A(\gamma _0 +\sigma _0) } \rightarrow \frac{L(\gamma _\infty )^2}{A(\gamma _0 +\sigma _0) } \text { as } t_j\rightarrow \infty . \end{aligned}$$

(8)

We use (6) and (7) and the fact that the enclosed area is preserved and get

$$\begin{aligned} \frac{L_0^2}{A(\gamma _0 + \sigma _0)} > \frac{(2l-1)^2 \pi ^2 r_\infty ^2}{l \pi r_\infty ^2}= \pi \frac{(2l-1)^2}{l}, \end{aligned}$$

which contradicts our assumptions. $\square$

Remark

The result of the previous theorem can be improved by analyzing the geometric situation in the limit more carefully. Instead of using the estimate ${\tilde{A}}_\infty <\pi r_\infty ^2$ we can prove ${\tilde{A}}_\infty < \pi r_\infty ^2 (1-\frac{7}{20\pi })$. If $A(\gamma _0 + \sigma _0)>0$ we get that $\frac{L_0^2}{A(\gamma _0 +\sigma _0)}<\pi \frac{(2l-1)^2}{l-\frac{7}{20\pi }}$ implies a singularity in finite time. This is again not sharp because we estimated some geometric constants.

Corollary 2.13

Let $\gamma _0:{[}a,b{]}\rightarrow {\mathbb {R}}^2$ be an initial curve satisfying the conditions from Theorem 2.12. Then the finite time singularity is of type II in the sense that

$$\begin{aligned}&\max _{p\in {[}a,b{]}}|\kappa |(p,t)\rightarrow \infty \ (t\rightarrow T_{\text {max}}) \text { and }\\&\max _{p\in {[}a,b{]}}\left( |\kappa |^2(p,t)(T_{\text {max}}-t)\right) \text { is unbounded}. \end{aligned}$$

The proof of this corollary is based on the following lemma:

Lemma 2.14

Let $\gamma :[a,b]\times [0,T)\rightarrow \mathbb {R}^2$ be a solution of (2) with $T<\infty$ is a time such that $\{\max _{[a,b]}\kappa ^2(\cdot ,t): t\in [0,T)\}$ is unbounded. Then we have that

$$\begin{aligned} \kappa _{max}^2(t)\,{:=}\,\max _{[a,b]}\kappa ^2(\cdot ,t)\ge \frac{1}{4(T-t)} \quad \forall t\in (0,T). \end{aligned}$$

Proof

A bound $\max _{[a,b]}\kappa ^2(\cdot ,t) \ge \frac{1}{2(T-t)}$ was proved in [15, Proposition 4.1] for a convex initial curve. We refine this proof for a general initial curve: We compute the evolution equation of $\kappa ^2$ and estimate

$$\begin{aligned} \partial _t\kappa ^2= & {} \partial _s^2 \kappa ^2 - 2(\partial _s\kappa )^2 + 2\kappa ^4 - 2 \kappa ^3{\bar{\kappa }}\\\le & {} \partial _s^2 \kappa ^2 + 2\kappa ^4 + 2\left( \max _{[a,b]}|\kappa |(\cdot ,t)\right) ^3|{\bar{\kappa }}|\\\le & {} \partial _s^2 \kappa ^2 + 4\left( \max _{[a,b]}|\kappa |(\cdot ,t)\right) ^4, \end{aligned}$$

where we used $-\max _{[a,b]}|\kappa |\le {\bar{\kappa }}\le \max _{[a,b]} |\kappa |$ in the last step. As $\kappa ^2$ is $C^2$ the function $t\mapsto \kappa ^2_{max}(t)$ is Lipschitz and hence differentiable almost everywhere. At a point of differentiability we can compute the time derivative as $\frac{d}{dt} \kappa ^2_{max}(t) = \frac{\partial \kappa ^2(p,t)}{\partial t}$, where $p\in [a,b]$ is a point where the maximum is attained. This approach is sometimes called “Hamilton’s trick”. It goes back to [11]. We get that

$$\begin{aligned} \frac{d}{dt} \kappa ^2_{max}(t) \le \partial _s^2\kappa ^2(p,t) + 4 \left( \kappa ^2_{max}(t)\right) ^2, \end{aligned}$$

(9)

where $p\in [a,b]$ is a point where the maximum of $\kappa ^2(\cdot ,t)$ is attained. We now prove that

$$\begin{aligned} \partial _s^2\kappa ^2(p,t)\le 0 \end{aligned}$$

(10)

holds for such a point $p\in [a,b]$. If $p\in (a,b)$, we simply have a maximum in the inner part of [a, b]. Thus, inequality (10) is clear. So we assume that $p=a$. Case (i): $\kappa (a,t)>0$: Then $\kappa (a,t)=\max _{[a,b]}\kappa (\cdot ,t)$. So we have the inequality $\partial _s\kappa (a,t)\le 0$. In [15, Lemma 2.12] we proved by differentiating the boundary conditions that $\partial _s\kappa (a,t) = \left( \kappa (a,t)-{\bar{\kappa }}(t)\right) \kappa _\Sigma ( \gamma (a,t))$ for all $t\in (0,T)$. In our specific situation we get that

$$\begin{aligned} 0 \ge \partial _s\kappa (a,t)= \left( \kappa (a,t)-{\bar{\kappa }}(t)\right) \kappa _\Sigma ( \gamma (a,t)) \ge 0, \end{aligned}$$

where we used ${\bar{\kappa }}(t)\le \max _{[a,b]}\kappa (\cdot ,t) = \kappa (a,t)$ in the last inequality. We hence get that $\partial _s\kappa (a,t)=0$ and therefore $\partial _s\kappa ^2(a,t) =2\kappa (a,t)\partial _s\kappa (a,t)=0$. A positive sign of the second derivative $\partial _s^2\kappa ^2(a,t)>0$ would now imply a strict local minimum of $\kappa ^2(\cdot ,t)$ in a, which is a contradiction. As a consequence we get that (10) is satisfied. Case (ii): $\kappa (a,t)<0$: In this case we know that $\kappa (a,t)=\min _{[a,b]}\kappa (\cdot ,t)$. So we get that $\partial _s\kappa (a,t)\ge 0$ and

$$\begin{aligned} 0 \le \partial _s\kappa (a,t)= \left( \kappa (a,t)-{\bar{\kappa }}(t)\right) \kappa _\Sigma ( \gamma (a,t)) \le 0 \end{aligned}$$

because of ${\bar{\kappa }}(t)\ge \min _{[a,b]}\kappa (\cdot ,t)= \kappa (a,t)$. Thus, we also have $\partial _s\kappa ^2(a,t)=0$. As in the first case, we get that $\partial _s^2\kappa ^2(a,t)\le 0$. Case (iii): $\kappa (a,t)=0$: Here, we immediately get that $\partial _s\kappa ^2(a,t)= 2\kappa (a,t)\partial _s\kappa (a,t)=0$. As in the other two cases, this implies $\partial _s^2\kappa ^2(a,t)\le 0$ because a is a maximum point of $\kappa ^2(\cdot ,t)$. If $p=b$, (10) follows analogously as $\partial _s\kappa (b,t)= - \left( \kappa (b,t)-{\bar{\kappa }}(t)\right) \kappa _\Sigma ( \gamma (b,t))$ [15, Lemma 2.12].

We now use (9) and (10) and get

$$\begin{aligned} -\frac{d}{dt} \left( \frac{1}{\kappa ^2_{max}(t)}\right) \le 4 \end{aligned}$$

at all times $t\in (0,T)$ where $\kappa ^2_{max}$ is differentiable. Integrating and using the existence of a sequence $t_j\rightarrow T$ such that $\kappa _{max}^2(t_j)\rightarrow \infty$ yields the result. $\square$

Definition 2.15

We keep the notation of a type I singularity as in the (classical) curve shortening flow: A singular time $T<\infty$ is of type I if there is a constant $c>0$ such that

$$\begin{aligned} \max _{[a,b]}\kappa ^2(\cdot ,t)\le \frac{c}{T-t} \quad \forall t\in [0,T). \end{aligned}$$

Otherwise, the singularity is of type II.

Proof

(of Corollary 2.13). In [15, Theorem 4.16], the author proved that a convex initial curve cannot develop a type I singularity in finite time if $|{\bar{\kappa }}|\le c_2$ and $L(\gamma _t)\ge c_1>0$. We are able to generalize this result for general initial curves under the same bounds on the total curvature and on the length. Almost all steps of the proof of Theorem 4.16 in [15] are already formulated for the general case, see Section 4 in [15]. We sketch the most important steps: Assume that the flow develops a singularity of type I in finite time. We do a parabolic rescaling

$$\begin{aligned} \tilde{\gamma }_j(p,\tau )\,{:=}\, Q_j \left( \gamma \left( p,\tfrac{\tau }{Q_j^2} + T\right) - x_0\right) \quad \text { for } (p,\tau )\in [a,b]\times [-Q_j^2T,0), \end{aligned}$$

where $x_0\in \mathbb {R}^2$ is a “blowup point” of the flow, which means $t_j\rightarrow T$, $p_j\rightarrow p_0\in [a,b]$, $Q_j= |\kappa |(p_j,t_j) =\max _{[a,b]}|\kappa |(p,t_j)|\rightarrow \infty$, $\gamma (p_j,t_j)\rightarrow x_0$. Using the gradient estimates from Stahl [17, 18] we adapted the convergence procedure from [5, Remark 4.22 (2)] to the area preserving flow. This is similar to the procedure in Theorem 2.11 (but it is not necessary to use integral estimates because $T<\infty$). We get smooth subconvergence (after reparametrization) to a limit flow $\gamma _\infty :I\times (-\infty ,0)\rightarrow \mathbb {R}^2$, where I is an interval containing 0. Because of the $L^\infty$ bound on ${\bar{\kappa }}(t)$ the term ${\bar{\kappa }}_j(t)$ is scaled away in the limit. Thus, the limit flow satisfies $\partial _t\gamma _\infty =\kappa _\infty \nu _\infty$, it is an ancient solution of the curve shortening flow. The lower bound on the length implies that each curve $\gamma _\infty (\cdot ,t)$ has infinite length. If the singularity develops at the boundary then the curve $\gamma _\infty (\cdot ,t)$ meets a straight line perpendicularly at the endpoint. We reflect it and can consider a complete, unbounded solution of the curve shortening flow. A monotonicity formula for the free boundary situation yields the key properties of the limit flow: Each curve $\gamma _\infty (\cdot ,t)$ is proper and $\gamma _\infty$ is self-similarly shrinking, i.e. $\kappa _\infty (p,\tau )= \frac{\langle \gamma _\infty (p,\tau ),\nu _\infty (p,\tau )\rangle }{2 \tau }$. For plane curves, all the self-similarly shrinking solutions are classified. It turns out that the curvature of these solutions does not change sign, see [8]. We get that $\gamma _\infty$ is one of the following:

• (i) The line $\mathbb {R}\times \{0\}$,

• (ii) the shrinking sphere $\mathbb S^1_{\sqrt{-2\tau }}$, where the curves can also be negatively oriented,

• (iii) one of the closed “Abresch–Langer curves” [1], positively or negatively oriented,

• (iv) a curves whose image is dense in an annulus of $\mathbb {R}^2$.

The solutions (i), (ii) and (iii) are excluded because of the unbounded length and the properness of the curves. It remains to exclude i): We rescaled at points of maximal curvature which implies for $\tau _j{:=}-Q_j^2(T-t_j)$

$$\begin{aligned} \kappa _{\tilde{\gamma }_j}(p_j,\tau _j)= \frac{1}{Q_j}\kappa \left( p_j,\tfrac{-Q_j^2(T-t_j)}{Q_j^2} + T\right) = \frac{1}{Q_j}\kappa (p_j,t_j)=1 \quad \forall j\in \mathbb {N}. \end{aligned}$$

We reparametrize in the spatial component such that $\tilde{\kappa }_j(0,\tau _j)=1$ for all $j\in \mathbb {N}$. By the type I property we get that

$$\begin{aligned} \tau _j = -\kappa ^2(p_j,t_j)(T-t_j) \ge - \frac{c}{T-t_j} (T-t_j)= - c>-\infty . \end{aligned}$$

The blowup rate from Lemma 2.14 yields

$$\begin{aligned} \tau _j = -\kappa ^2(p_j,t_j)(T-t_j) \le -\, \frac{1}{4}<0. \end{aligned}$$

Thus, there is a time $\tau \in [-c,-\,\frac{1}{4}]$ such that $\kappa _\infty (0,\tau )=1$. This excludes the line as a limit flow. $\square$

Corollary 2.16

Let $\gamma _0:{[}a,b{]}\rightarrow \mathbb {R}^2$ be a convex initial curve satisfying the conditions from Theorem 2.12. Then the “Hamilton blow-up” at $T_{max}$ yields either a grim reaper (we call this situation an “inner singularity”) or half a grim reaper at a plane (a “boundary singularity”).

Proof

The situation of a finite type II singularity was treated in [15, Section 6]. We repeat the important steps for the sake of completeness. We recall the “Hamilton blow-up” [9]: define $T\,{:=}\, T_{max}$. For $j\in \mathbb N$ choose $t_j\in [0,T-\frac{1}{j}]$ and $p_j\in [a,b]$ such that

$$\begin{aligned} |\kappa |^2(p_j,t_j) \left( T-\tfrac{1}{j} - t_j\right) = \max \left\{ \left( |\kappa |^2(p,t) \left( T-\tfrac{1}{j} - t\right) \right) : t\in \left[ 0,T-\tfrac{1}{j}\right] , p\in [a,b]\right\} . \end{aligned}$$

Then define $Q_j{:=}|\kappa |(p_j,t_j)$ and

$$\begin{aligned} \tilde{\gamma }_j(\cdot ,\tau )\,{:=}\, Q_j\left( \gamma \left( \cdot ,\tfrac{\tau }{Q_j^2} + t_j\right) - \gamma (p_j,t_j)\right) \text{ for } \tau \in \left[ -Q_j^2t_j,Q_j^2\left( T-t_j-\tfrac{1}{j}\right) \right] \text{ on } [a,b]. \end{aligned}$$

As the singularity is of type II, one can show certain properties of the rescaled flow. The most important ones are $\tilde{\kappa }_j(p_j,0)=0 \ \forall j$, $|\tilde{\kappa }_j| (\cdot ,\tau )\le 1 \ \forall j$ and

$$\begin{aligned} \forall \epsilon>0 \ \forall \bar{\tau }>0 \ \exists j_0(\epsilon ,\bar{\tau })\in \mathbb N,\ \forall j\ge j_0:&|\tilde{\kappa }_j|^2(p,\tau ) \le 1 + \epsilon \\&\forall \tau \in [-Q_{j_0}^2t_{j_0},\bar{\tau }], \forall p\in [a,b]. \end{aligned}$$

Then there exist reparametrizations $\psi _j: I_j \rightarrow [a,b]$ with $|I_j|\rightarrow \infty$ ($j\rightarrow \infty$) such that a subsequence of the rescaled curves

$$\begin{aligned} \gamma _j\,{:=}\,\tilde{\gamma }_j(\psi _j,\cdot ): I_j\times [-Q_j^2t_j,Q_j^2(T-t_j-\tfrac{1}{j})]\rightarrow {\mathbb {R}}^2\end{aligned}$$

converges locally smoothly to a limit flow $\tilde{\gamma }_\infty : {\tilde{I}} \times (-\infty ,\infty )\rightarrow {\mathbb {R}}^2$ (where ${\tilde{I}}$ is an unbounded interval containing 0). The proof of this subconvergence can be found in [15, Proposition 6.2, Proposition 4.7]. It is again similar to the proofs of Theorem 2.11 and Corollary 2.13.

The limit flow $\tilde{\gamma }_\infty$ is a smooth solution of the curve shortening flow and satisfies $0<\tilde{\kappa }_\infty \le 1$ everywhere and $\tilde{\kappa }_\infty = 1$ at least at one point. If ${\tilde{M}}_\tau ^\infty {:=}\tilde{\gamma }_\infty ({\tilde{I}},\tau )$ has a boundary, then $\partial {\tilde{M}}^\infty _\tau \subset \Sigma _\infty$, where $\Sigma _\infty$ is a line through $0\in {\mathbb {R}}^2$, and $\langle \tilde{\nu }_\infty , \nu _{ \Sigma _\infty }\rangle = 0$ on $\partial {\tilde{M}}_\infty$. By reflecting at the line $\Sigma _\infty$ one gets an eternal solution of the curve shortening flow with bounded curvature where the maximal curvature is attained at least at one point. Due to [10, Theorem 1.3], the limit flow must be a translating solution, and the only translating solution in the case of curves is the “grim reaper” which is the flow of curves given by $x=-\log \cos y + \tau$ for $y\in (-\frac{\pi }{2},\frac{\pi }{2})$. In the situation where the limit flow does have a boundary it must be “half the grim reaper” at $\Sigma _\infty$ because the grim reaper has only one symmetry axis. $\square$

In [3, 6] the blowup-rate at the singularity was characterized for the $L^2$-norm of the curvature, and not for the $C^0$-norm as above. This $L^2$-rate can also be proved for the free boundary setting:

Proposition 2.17

Let $\gamma :[a,b]\times [0,T_{max})\rightarrow {\mathbb {R}}^2$ be a solution of (2) with $T_{max}<\infty$ and $|{\bar{\kappa }}|\le c<\infty$. Then there is a constant $C>0$ and a sequence of times $t_k\rightarrow T_{max}$ such that

$$\begin{aligned} \int |\kappa (\cdot ,t_k)|^2d s_{t_k} \ge C(T_{max}-t_k)^{-\frac{1}{2}} \end{aligned}$$

Proof

The proof is due to [3, Proposition A] and [6, Proposition 5]. Since $T_{max}<\infty$ we have that $\left\{ \int \kappa ^2 ds: t\in [0,T_{max})\right\}$ is unbounded. As it was pointed out in [6, Proof of Proposition 5], this comes from the fact that the proof of the short time existence only depends on the $C^{1,\alpha }$-norm of the initial data for all $\alpha \in (0,1)$. For the Neumann boundary condition setting the estimates behind this argument can be found in [14, Lemma 5.3.2]. In order to follow the proof of [6, Proposition 5] we only have show that

$$\begin{aligned} \frac{d}{dt} E(t)\le C\left( E(t) + E(t)^3\right) \end{aligned}$$

for $E(t):= \int (\kappa -{\bar{\kappa }})^2 ds$. In [15, Corollary 7.4], the inequality

$$\begin{aligned} \frac{d}{dt} E(t)\le C\left( E(t) + E(t)^{\frac{5}{3}} + E(t)^3\right) \end{aligned}$$

is proved under the condition $|{\bar{\kappa }}|\le c<\infty$. We have

$$\begin{aligned} E^{\frac{5}{3}} \le \begin{Bmatrix} E,&\text { if } 0\le E\le 1\\ E^3,&\text { if } 1\le E \end{Bmatrix} \le E^3 + E. \end{aligned}$$

This was also used in [6, Proof of Proposition 5]. $\square$

The following corollary is immediate.

Corollary 2.18

Under the conditions of Theorem 2.12 there is a sequence of times $t_k\rightarrow T_{max}<\infty$ such that $\int |\kappa (\cdot ,t_k)|^2d s_{t_k} \ge C(T_{max}-t_k)^{-\frac{1}{2}}$.

3 Examples

It remains to show that there are curves that satisfy the conditions from Theorem 2.12 or Corollary 2.13.

Example 1

Let us consider a convex curve $\Sigma$ that almost looks like a circle with $d_\Sigma > 2\pi$. Then one can construct an initial curve $\gamma _0:[0,1]\rightarrow \mathbb {R}^2$ with $L_0<\frac{4}{3}\pi$, $l=2$ and $A(\gamma _0 + \sigma _0)>\frac{\pi }{2}$. An example is drawn in Fig. 1. Note that $\sigma _0$ is the connection of $\gamma _0(1)$ and $\gamma _0(0)$ along $\Sigma$ that is visible in the picture. We check the isoperimetric quotient of that initial curve and compare it to the conditions of Theorem 2.12:

$$\begin{aligned} \frac{L_0^2}{A(\gamma _0 + \sigma _0)}< \frac{2}{\pi } \left( \frac{4}{3}\pi \right) ^2 = 2 \frac{16}{9} \pi < \frac{9}{2} \pi . \end{aligned}$$

Fig. 1

An initial curve, where the flow develops a singularity in finite time, see Example One

Thus, this curve develops a type II singularity in finite time. This is somehow not surprising as it was shown in [6, Proposition 9] that a curve looking like the described $\gamma _0$ but closed on the “lower part” (a so-called “limaçon”) develops a singularity in finite time under the area preserving curve shortening flow without boundary. And the “limaçon” is the classical example where the curve shortening flow (without boundary) develops a type II singularity [2]. These type II singularities are usually expected when there is a self-intersection.

But there are examples satisfying the conditions from case (ii) in Theorem 2.12 that seem to behave differently, see Example Two.

Example 2

We construct $\gamma _0:[0,1]\rightarrow \mathbb {R}^2$ as shown in Fig. 2. Again, $\sigma _0$ is the connection of $\gamma _0(1)$ to $\gamma _0(0)$ along $\Sigma$. As in the first example we have that $l=2$. We construct $\gamma _0$ such that $L_0<d_\Sigma$, $L_0< \frac{4}{3}\pi$ and $A(\gamma _0 + \sigma _0)>\frac{\pi }{2}$. We conclude again

$$\begin{aligned} \frac{L_0^2}{A(\gamma _0 + \sigma _0) }< \frac{2}{\pi } \left( \frac{4}{3}\pi \right) ^2 = 2 \frac{16}{9} \pi < \frac{9}{2} \pi . \end{aligned}$$

For this particular $\gamma _0$ we conjecture that the curves stay embedded under the flow (2) and that the type II singularity forms at the boundary.

Fig. 2

Another initial curve, where the flow develops a singularity in finite time, see Example Two

Example 3

The conditions of Theorem 2.12, case (ii) are satisfied by a curve $\gamma _0:[0,1]\rightarrow \mathbb {R}^2$ as shown in Fig. 3. We choose $G_\Sigma$ big enough such that $L_0< d_\Sigma$. We have that $\kappa _0>0$ and $l=2$. We have constructed $\gamma _0$ in such a way that $A(\gamma _0 +\sigma _0)<0$. By Theorem 2.12 we get a singularity in finite time that is of type II.

Fig. 3

Another initial curve, where the flow develops a singularity in finite time, see Example Three

Example 4

As Theorem 2.12 gives the existence of singularities also for non-convex curves, we provide such an example, see Fig. 4. The initial curve $\gamma _0:[0,1]\rightarrow \mathbb {R}^2$ satisfies $\int _{\gamma _0}\kappa ds \in (-2\pi ,0)$ but $A(\gamma _0 + \sigma _0)>0$. After changing the orientation case (ii), Theorem 2.12 applies, and the flow develops a singularity in finite time.

Fig. 4

A non-convex initial curve, where the flow develops a singularity in finite time, see Example Four

4 The area preserving curve shortening flow at a straight line

In this section, we consider the area preserving curve shortening flow (APCSF) at a straight line. We prove that there are initial curves that develop a singularity in finite time. The situation is somehow easier than in the previous section. The strategy is to reflect the curves at the line and to use the results from [6] for the closed case. First we have to specify some notation for the case that $\Sigma$ is a straight line.

Definition 4.1

Consider the map $f: s\mapsto (-s,0) \in \mathbb {R}^2$, $s\in (-\infty ,\infty )$. The map f parametrizes the line $\Sigma \,{:=}\, \{(x,y)\in \mathbb {R}^2:x\in \mathbb {R},y=0\}$. A smooth, regular curve $\gamma _0:[a,b]\rightarrow {\mathbb {R}}^2$ is called initial curve if it satisfies the conditions

$$\begin{aligned} \gamma _0(a),\gamma _0(b)&\in \Sigma \\ \tau _0(a)&= e_2, \\ \tau _0(b)&= -e_2, \end{aligned}$$

where $\tau _0=\partial _s\gamma _0$ is the tangent of $\gamma _0$ and $e_2= (0,1)\in \mathbb {R}^2$ is the second standard vector in $\mathbb {R}^2$.

Definition 4.2

Let $f:[a,b]\rightarrow {\mathbb {R}}^2$ be a piecewise smooth, regular and closed curve. The number

$$\begin{aligned} \mathrm{ind}(f)\,{:=}\, n(\partial _pf,0) \in \mathbb Z \end{aligned}$$

is called the index (or turning number) of f. Here, $n(\partial _p f,0)$ denotes the winding number of the curve $\partial _p f:[a,b]\rightarrow {\mathbb {R}}^2$ with respect to $0\in {\mathbb {R}}^2$.

Theorem 4.3

Let f be a piecewise smooth, regular and closed curve, defined on intervals $[a_j,b_j]$, $j=1,\ldots ,k$, and with exterior angles $\alpha _j$, $j=1,\ldots ,k$. Then

$$\begin{aligned} \mathrm{ind}(f)=\frac{1}{2\pi }\sum _{j=0}^k\int _{a_j}^{b_j}\kappa _fds_f + \frac{1}{2\pi } \sum _{j=0}^k\alpha _j \quad \in \mathbb Z. \end{aligned}$$

Proof

See [13, Theorem 2.1.6]. $\square$

Lemma 4.4

Let $\gamma _0:{[}a,b{]}\rightarrow \mathbb {R}^2$ be an initial curve. Reflect the curve $\gamma _0$ at the line $\Sigma$ into the lower half space of $\mathbb {R}^2$. Then the resulting closed curve $\delta _0$ is a $C^2$ curve with $\mathrm{ind}(\delta _0)=: m \in \mathbb {Z}$. The number m is odd.

Proof

As the curves meet $\Sigma$ perpendicularly and because of reflection at a line the reflected curves are $C^2$. We treat two cases:

Case 1: $f^{-1}(\gamma _0(a))\le f^{-1}(\gamma _0(b))$.

Then consider $\gamma _0 + \sigma _0$ where $\sigma _0$ is the line segment from $\gamma _0(b)$ to $\gamma _0(a)$. The exterior angles at the points where $\tau _0$ is not continuous are $+\frac{\pi }{2}$ (or $+\pi$ if $\gamma _0(a)=\gamma _0(b)$). Thus, we have $l\,{:=}\,\mathrm{ind}(\gamma _0 +\sigma _0)= \frac{1}{2\pi }\left( \int _{\gamma _0}\kappa d s + \pi \right) \in \mathbb {Z}$ or equivalently $\int _{\gamma _0}\kappa ds = 2\pi l - \pi$. After reflecting we get $\mathrm{ind}(\delta _0)= \frac{2\int _{\gamma _0}\kappa d s}{2\pi } = 2l - 1$.

Case 2: $f^{-1}(\gamma _0(a)) > f^{-1}(\gamma _0(b))$.

We denote by $\sigma _0$ the line segment from $\gamma _0(b)$ to $\gamma _0(a)$. Note that this is oriented in the opposite direction compared to f. Now the exterior angles of $\gamma _0 + \sigma _0$ are $-\frac{\pi }{2}$. This implies $\int \kappa ds = 2\pi l +\pi$ for $l\in \mathbb {Z}$. By reflection we conclude $\mathrm{ind}(\delta _0)= \frac{2\int _{\gamma _0}\kappa d s}{2\pi } = 2l +1.$ $\square$

Remark

The APCSF preserves the reflection symmetry with respect to the x-axis. It hence does not matter whether we start at the straight line the APCSF with Neumann free boundary conditions and then reflect at $\Sigma$ or if we reflect at first and then consider the APCSF for closed curves. Thus, we recover the APCSF with Neumann free boundary conditions from the flow of the closed curves.

Proposition 4.5

Let $\gamma _0:{[}a,b{]}\rightarrow \mathbb {R}^2$ be an initial curve. Reflect $\gamma _0$ at $\Sigma$ and denote the closed curve by $\delta _0$. Choose the orientation of $\delta _0$ such that $\mathrm{ind}(\delta _0){=}{:}m\ge 0$. Lemma 4.4 shows that m is odd. Then the area preserving curve shortening flow with Neumann free boundary conditions at the line $\Sigma$ develops a singularity in finite time if one of the following conditions is satisfied:

• (i) Either $A(\delta _0)<0$.

• (ii) Or $m\ge 3$ and $L(\delta _0)^2 < 4\pi m A(\delta _0)$.

Proof

We use Lemma 4.4 to get that m is odd, so $m\ge 1$ is always satisfied. Use [6, Proposition 9] for the flow of the reflected curve to get that that $T_{max}<\infty$. $\square$

Corollary 4.6

The finite time singularity appearing in Proposition 4.5 is of type II.

Proof

Denote by $\delta _t$, $t\in [0,T_{max})$, the closed curves and with $\gamma _t$, $t\in [0,T_{max})$, the curves with boundary. By the isoperimetric inequality for $\delta _t$ we get that $L(\delta _t)^2\ge 4\pi |A(\delta _t)| = 4\pi |A(\delta _0)|$. This implies $L(\gamma _t)^2\ge \pi |A(\delta _0)|>0$. Thus the length is bounded from below uniformly in t. We have that $2\int _{\gamma _0}\kappa d s = \int _{\delta _0}\kappa d s=2\pi m\in \mathbb {Z}$. Continuity yields $\int _{\gamma _t}\kappa d s= \pi m$ for all $t\in [0,T_{max})$. Thus $|{\bar{\kappa }}_{\gamma _t}(t)|\le c_2<\infty$ uniformly in t. A blowup argument as in [15, Theorem 4.16] or as in the proof of Corollary 2.13 implies that the singularity is of type II. $\square$