A compact non-convex ancient curve shortening flow

Jumageldi Charyyev

Abstract. An example of a compact, non-convex, embedded ancient solution for
the curve shortening flow, which is asymptotic to Yin-Yang curve, is
constructed.

##  1\. Introduction

As with other geometric flows, the analysis of the ancient solutions of the
curve shortening flow is one of the several central problems in geometric
analysis. The curve shortening flow was first introduced in [7] and in the
same paper the first examples of nontrivial ancient solutions were provided,
including Grim Reaper and Yin-Yang. The following table lists the convexity
and compactness properties of previously known embedded ancient solutions of
the curve shortening flow:

Compact | Convex |  Remarks
---|---|---
Yes | Yes |  All such solutions are classified in [4]: shrinking circles or ancient “paperclip” solutions (Angenent Ovals)
No | Yes |  All such solutions are classified in [3]: stationary line or Grim Reaper
No | No |  An example of such solution with an explicit parametrization was given in [8], where the solution looks like a sine curve. A family of examples, for which the solutions have finite total curvature, was given in [2]. The examples are constructed through particular gluing of several Grim Reaper curves.

The main result of this paper completes the missing row in the above table, by
giving an example of a compact, non-convex, ancient solution.

###### Main Theorem.

There exists a compact ancient solution of the curve shortening flow
asymptotic to Yin-Yang solution as $t\rightarrow-\infty$.

The same result was obtained independently in [1].

The strategy of the proof of the theorem is indicated below.

The proof of the existence of such solution is based on constructing a
sequence of the curve shortening flows
$\\{\alpha_{t}^{n}\\}_{n=1}^{n=\infty}$, $t\in[-T_{n},0]$,
$T_{n}\rightarrow\infty$, such that the maximum of the curvature of
$\alpha_{t}^{n}$ is uniformly bounded in $n$ at any given time
$t\in(-\infty,0)$. Then by smooth compactness there exists a subsequence
converging to an ancient solution $\alpha_{t}^{\infty}$, $t\in(-\infty,0]$.

The closed initial curves $\\{\alpha_{-T_{n}}^{n}\\}_{n=1}^{n=\infty}$ are
constructed by “attaching” a chunk of a Grim Reaper to a truncated Yin-Yang
curve further and further away from the origin. Since the Grim Reaper is a
soliton, it is reasonable to expect that the flow $\alpha^{n}_{t}$ will have a
“tip” region which remains close to a Grim Reaper for a long time. Likewise,
since $\alpha^{n}_{-T_{n}}$ is identical to the Yin-Yang away from the tip
region, it is plausible that the portion of $\alpha^{n}_{t}$ lying away from
its tip region will be very close to a Yin-Yang, for a long time. The actual
proof is based on a formalization of these two heuristic ideas.

Here is a brief description of the approach in [1]. The authors begin by
constructing an approximation solution – an evolving curve violating the curve
shortening flow equation by an error which satisfies an $L^{1}$ spacetime
bound – by gluing Grim Reapers to truncated Yin-Yangs. The curve shortening
flow with the same initial condition remains close to the approximate
solution, in the sense that, the two evolving curves co-bound a region of
small area. Using a decomposition of the curve shortening flow into graphical
pieces, $L^{1}$ estimates, and a contradiction argument, the authors obtain
uniform curvature bounds on time intervals $[-T-\frac{1}{4},-T]$ for all
sufficiently large $T$.

When compared, the construction of older and older solutions and an extraction
of an ancient solution as a subsequential limit is standard and the same for
the two approaches. However, apart from the standard ingredients such as
parabolic regularity and the fact that the number of inflection points is non-
increasing in time, there is a little overlap between the two arguments. Both
the form of the estimates and the methods for proving them are different.

Organization of the paper. First, in Section 2 the properties and the
asymptotics of the Yin-Yang curve are introduced. Section 3 contains the
construction of the initial curves. Section 4 defines a class of barrier
curves. Section 5 formulates and proves the “stability” properties mentioned
in the preceding paragraph (Theorem 5.1), as well as the proof of Main
Theorem.

Acknowledgments. The author wishes to thank his advisor Prof. Bruce Kleiner
for his support and guidance during the course of this work.

##  2\. The Yin-Yang solution

This section presents some properties of the Yin-Yang solutions and introduces
some notation which will be used later.

$\bf{\hat{\gamma_{t}}:}$ Yin-Yang curve shortening flow – a spiral-like
counterclockwise rotating soliton with the unit angular speed centered at the
origin and arclength parametrized on the complex plane as follows:

$\displaystyle\hat{\gamma_{t}}(s)$ $\displaystyle=$ $\displaystyle e^{it}F(s)$
$\displaystyle F(s)$ $\displaystyle=$ $\displaystyle x(s)+iy(s)$
$\displaystyle x^{\prime}(s)$ $\displaystyle=$
$\displaystyle\cos((x(s)^{2}+y(s)^{2})/2)$ (1) $\displaystyle y^{\prime}(s)$
$\displaystyle=$ $\displaystyle\sin((x(s)^{2}+y(s)^{2})/2)$ $\displaystyle
T(s)$ $\displaystyle=$ $\displaystyle
x^{\prime}(s)+iy^{\prime}(s)=F^{\prime}(s)$ $\displaystyle N(s)$
$\displaystyle=$ $\displaystyle-y^{\prime}(s)+ix^{\prime}(s)=iT(s)$
$\displaystyle H(s)$ $\displaystyle=$ $\displaystyle
x(s)x^{\prime}(s)+y(s)y^{\prime}(s)$

where $s$ is clearly an arclength parametrization such that $F(0)=0$, $H(s)>0$
for $s>0$, and $N(s)$ is the inward unit normal vector. This solution first
appeared in [7]. All the self-similar solutions of the curve shortening flow
were classified in [6].

One of the key properties of the curve is that $H=\langle F,T\rangle$, which
justifies the fact that Yin-Yang is a rotating soliton:

$\displaystyle\langle\partial_{t}\hat{\gamma_{t}}(s),N(\hat{\gamma_{t}}(s))\rangle=\langle
ie^{it}F(s),e^{it}N(s)\rangle=\langle iF(s),N(s)\rangle=$ (2)
$\displaystyle=\langle F(s),-iN(s)\rangle=\langle F(s),T(s)\rangle=H(s)$
$\displaystyle\langle\partial_{ss}\hat{\gamma_{t}}(s),N(\hat{\gamma_{t}}(s))\rangle=\langle
e^{it}F^{\prime\prime}(s),e^{it}N(s)\rangle=\langle
F^{\prime\prime}(s),N(s)\rangle=H(s)$

Here for $z,w\in\mathbb{C}$ the inner product is $\langle
z,w\rangle:=\Re(z\bar{w})$, which coincides with the usual dot product of the
real vectors. The fact that $H=\langle F,T\rangle$ follows quickly from the
following identities:

$\displaystyle
x^{\prime\prime}=-(xx^{\prime}+yy^{\prime})\sin((x^{2}+y^{2})/2)=-(xx^{\prime}+yy^{\prime})y^{\prime}$
(3) $\displaystyle
y^{\prime\prime}=(xx^{\prime}+yy^{\prime})\cos((x^{2}+y^{2})/2)=(xx^{\prime}+yy^{\prime})x^{\prime}$
$\displaystyle
H=x^{\prime}y^{\prime\prime}-x^{\prime\prime}y^{\prime}=xx^{\prime}+yy^{\prime}=\langle
F,T\rangle$

The portions of the curve with the positive and negative curvature
occasionally will be referred as the $\bf{``positive\ wall"}$ and the
$\bf{``negative\ wall"}$, respectively.

In [6] it was proved that $H(s)$ is an odd function and $H(s)>0$ for $s>0$.
The function has one critical point in $[0,\infty)$, at which it has the
global maximum; afterwards $H(s)$ is decreasing asymptotically to 0 and is
convex in the ray $[s_{i},\infty)$ for some $s_{i}>0$ (a positive inflection
point of $H(s)$ function). Thus, $H^{\prime}(s)<0$ for $s\in[s_{i},\infty)$
and $H^{\prime}(s)\xrightarrow{s\rightarrow\infty}0$ .

From the parametrization of Yin-Yang at time $t=0$, it can be observed that
$\hat{\gamma}_{0}$ is an integral curve of the unit vector field
$V(x,y)=\left(\cos((x^{2}+y^{2})/2),\sin((x^{2}+y^{2})/2)\right)$ which passes
through the origin. The integral curves of the vector field $V(x,y)$ foliate
the plane. From the above equations it can be observed that each integral
curve is another rotating soliton (counter-clockwise with unit angular speed)
of the curve shortening flow. Among all these integral curves only Yin-Yang is
symmetric with respect to the origin. For the different characterization of
these curves see [6].

$\bf{\hat{D}_{t}:}$ the $\bf{``corridor"}$ – the closure of one of the
connected components of $\mathbb{R}^{2}\setminus\hat{\gamma_{t}}$, for which
$N(\hat{\gamma}_{t}(s))$, the unit normal of $\hat{\gamma_{t}}$ at $s$, points
towards the inside of $\hat{D}_{t}$ when $s>0$. So, $\langle F,N\rangle<0$
when $s>0$ and $\langle F,N\rangle>0$ when $s<0$. Since $\hat{\gamma}_{t}$ is
a proper embedding of $\mathbb{R}$ into $\mathbb{R}^{2}$, then by the Jordan
separation theorem, the complement has two connected components; the boundary
of each is the image of $\hat{\gamma}_{t}$ .

$\bf{w(s):}$ defined for $s<0$, is the radial width of the corridor
$\hat{D}_{t}$ at the point $\hat{\gamma}_{t}(s)$ – the length of the line
segment $[\hat{\gamma}_{t}(s),\hat{\gamma}_{t}(\sigma)]$, where $\sigma$ is
such that the latter line segment lies completely inside the corridor and
passes through the origin when extended. Such $\sigma$ exist uniquely because
it is the first instance when the ray $[0,\hat{\gamma}_{t}(s))$ intersects the
positive wall after $\hat{\gamma}_{t}(s)$.

######  Lemma 2.1.

As $|s|\rightarrow\infty$ the mutual ratios of the following quantities

$(3|s|)^{1/3},\indent\dfrac{1}{|H(s)|},\indent|F(s)|,\indent\dfrac{\pi}{w(s)}$

converge to 1 i.e., the quantities are asymptotic.

###### Proof of Lemma 2.1.

It is straightforward to justify that the following identities hold true:

$\displaystyle x^{\prime\prime}=-Hy^{\prime},\indent
y^{\prime\prime}=Hx^{\prime}$
$\displaystyle(x^{\prime}y-xy^{\prime})^{\prime}=x^{\prime\prime}y-xy^{\prime\prime}=-H^{2}$
$\displaystyle x^{\prime}y-xy^{\prime}=-\int_{0}^{s}H^{2}$ $\displaystyle
x^{\prime}y-xy^{\prime}=\langle F,N\rangle=-\int_{0}^{s}H^{2}$ $\displaystyle
H^{\prime}=1+xx^{\prime\prime}+yy^{\prime\prime}=1+H(x^{\prime}y-xy^{\prime})$
(4) $\displaystyle H^{\prime}=1-H\int_{0}^{s}H^{2}=1+H\langle F,N\rangle$

Consider the case when $s_{i}<s$. Since
$H^{\prime}(s)\xrightarrow{s\rightarrow\infty}0$, from the relation (4) it
will follow that $1/H(s)$ and $\int_{0}^{s}H^{2}$ are asymptotic (i.e., there
ratio converges to 1). Since both functions are increasing in $s$ and are
asymptotic, then their respective derivatives, $-H^{\prime}/H^{2}$ and
$H^{2}$, must be asymptotic. So, $-H^{\prime}\cong H^{4}$ (i.e., the ratio
converges to 1), which implies that $H(s)\cong(3s)^{-1/3}$. The similar result
is true for $s<0$.

Since $\partial_{s}(\frac{1}{2}|F(s)|^{2})=\langle F(s),T(s)\rangle=H(s)$,
after integrating it follows that
$\frac{1}{2}|F(s)|^{2}\cong\frac{1}{2}(3|s|)^{2/3}$ and hence the desired
outcome $|F(s)|\cong(3|s|)^{1/3}$.

For the lifted polar angle $\theta=\arctan\dfrac{y}{x}$ the following identity
is true:

(5)
$\partial_{s}\theta=\dfrac{xy^{\prime}-x^{\prime}y}{x^{2}+y^{2}}=-\dfrac{\langle
F,N\rangle}{|F|^{2}}$

For the given $s>0$ let $\sigma(s)>0$ be such that
$\theta(\sigma(s))-\theta(s)=2\pi$ (i.e., one full circular turn along the
positive wall). It should be noted that $\sigma(s)=s+O(s^{1/3})$, because of
the almost one circular turn of radius $|F(s)|$. Then one has the following:

$\displaystyle
2\pi=\mathop{{\int}}_{s}^{\sigma(s)}\partial_{s}\theta=-\mathop{{\int}}_{s}^{\sigma(s)}\dfrac{\langle
F,N\rangle}{|F|^{2}}=\mathop{{\int}}_{s}^{\sigma(s)}\dfrac{\sqrt{|F|^{2}-\langle
F,T\rangle^{2}}}{|F|^{2}}=\mathop{{\int}}_{s}^{\sigma(s)}\dfrac{1}{|F|}\sqrt{1-\dfrac{H^{2}}{|F|^{2}}}$
(6)
$\displaystyle\cong\mathop{{\int}}_{s}^{\sigma(s)}\dfrac{1}{|F|}\cong\mathop{{\int}}_{s}^{\sigma(s)}(3\tau)^{-1/3}d\tau=\frac{1}{2}\left[(3\sigma(s))^{2/3}-(3s)^{2/3}\right]=$
$\displaystyle=\frac{1}{2}\left[(3\sigma(s))^{1/3}+(3s)^{1/3}\right]\left[(3\sigma(s))^{1/3}-(3s)^{1/3}\right]\cong$
$\displaystyle\cong(3s)^{1/3}\left[(3\sigma(s))^{1/3}-(3s)^{1/3}\right]\cong(3s)^{1/3}|F(\sigma(s))-F(s)|$

where $a\cong b$ means that $a$ and $b$ are asymptotic, so that
$|F(\sigma(s))-F(s)|\cong 2\pi(3s)^{-1/3}$. The approximations above are
easily justifiable, because the integrands are asymptotic as
$s\rightarrow\infty$. Now for the radial width it can be deduced that
$w(s)\cong\pi(3s)^{-1/3}$, because $|F(\sigma(s))-F(s)|\cong 2w(s)$. The lemma
is proved.

∎

##  3\. Admissible Curves

In this section particular examples of the curves will be constructed, which
look like a portion of Grim Reaper glued to Yin-Yang. These curves will be
referred to as $\bf{admissible}$ $\bf{initial}$ $\bf{conditions}$ and any
curve obtained from them by the curve shortening flow will be called an
$\bf{admissible}$ $\bf{curve}$.

Before proceeding, it should be noted that a Grim Reaper curve is a curve
congruent to the graph of the function $y=-\frac{1}{\lambda}\log\cos\lambda x$
for some $\lambda>0$. The parameter $\lambda$ corresponds to the speed of the
translation of the corresponding Grim Reaper solution of the curve shortening
flow. The case $\lambda=1$ is referred as the standard Grim Reaper (curve).

Using the notation and terminology from the previous section, the construction
of admissible curves goes as follows:

  1. (1)

Take $s<0<\sigma$ such that $w(s)=|F(s)-F(\sigma)|$ (two points on the
opposite walls of the corridor) and the line segment $[F(s),F(\sigma)]$ is
radial. For the sake of definiteness assume that the ray $OA$ coincides with
the positive $y-$axis. Denote $A=F(s)$ and $B=F(\sigma)$. $|s|\gg 1$ is chosen
to be big enough, which will be specified later.

  2. (2)

Let $a$ and $b$ be the lines perpendicular to the ray $OA$ at the points $A$
and $B$, respectively. It should be noted that, since the function
$|F(\bar{s})|$ is increasing as $|\bar{s}|$ increases, there exists
$\bar{\sigma}>0$ such that $F(\bar{\sigma})$ lies on the line $a$, and the
portion of the positive wall between the points $F(\sigma)$ and
$F(\bar{\sigma})$ lies entirely between the parallel lines $a$ and $b$.
Moreover, the portion of the negative wall between the points $F(s)$ and
$F(0)=O$ lies below the line $a$.

  3. (3)

Let the curve $\rho$ be the Grim Reaper such that $(i)$ it is a graph over the
line segment $AB$; $(ii)$ has the tip at the midpoint of the line segment
$AB$; $(iii)$ has the width equal to $|AB|$; $(iv)$ is expanding in the
direction of $-iF(s)$ i.e., in this case, in the positive $x-$axis direction.

  4. (4)

Translate the tip of the curve $\rho$ and itself towards the origin along the
line segment $AB$ until it touches the negative wall. Denote the new Grim
Reaper to be $\bar{\rho}$. Let $\bar{\rho}$ be tangent to the negative wall at
the point $P=F(s_{0})$, $s<s_{0}<0$.

  5. (5)

Let the curve $\bar{\rho}$ intersect the positive wall for the first time at
the point $Q=F(\sigma_{0})$, $0<\sigma_{0}<\sigma$.

  6. (6)

Make a convex $C^{2}$ “gluing” of Grim Reaper $\bar{\rho}$ with the positive
wall near the point $Q$ as follows: $(i)$ choose the point $U$ on $\bar{\rho}$
such that the distance from $U$ to $Q$ along $\bar{\rho}$ is equal to $w(s)$
and such that the point $B$ is closer to $U$ than to $Q$; $(ii)$ choose the
point $V$ on the positive wall such that the distance from $V$ to $Q$ along
the positive wall is also equal to $w(s)$ and now $B$ is closer to $Q$ than to
$V$ (after rescaling by the factor of $\pi/w(s)$ the distances will be equal
to $\pi$); $(iii)$ let the tangent line $u$ at $U$ tangent to $\bar{\rho}$ and
the tangent line $v$ at $V$ tangent to the positive wall intersect at the
point $W$; $(iv)$ let $\bar{\Lambda}(s)w(s)/\pi$ be equal to the distance
between $Q$ and the ray $OB$; $\bar{\Lambda}(s)\rightarrow\infty$ when
${|s|\rightarrow\infty}$, as it will be explained below; $(v)$ make a $C^{2}$
gluing of $\bar{\rho}$ and the positive wall at the points $U$ and $V$ such
that the new curve connecting $U$ and $V$ lies completely inside the
triangular-like region $WUV$ bounded by the lines $u$ and $v$, and the
portions of $\bar{\rho}$ and the positive wall; moreover, the gluing must be
such that the curvature of the new curve connecting the points $U$ and $V$ is
bounded by 1 after rescaling by $\pi/w(s)$; such gluing is possible, because
after rescaling the curvatures at the points between $U$ and $V$ on
$\bar{\rho}$ and the positive wall converge to zero uniformly as
$|s|\rightarrow\infty$, and the angle between the tangent lines $u$ and $v$ is
converging to $\pi$ as $|s|\rightarrow\infty$. This specifies how large $|s|$
should be.

  7. (7)

Similar to the above construction, using the partition of unity make a $C^{2}$
gluing of $\bar{\rho}$ with the negative wall near the point $P$ such that the
curvature of the newly obtained curve near the point $P$ is bounded by 1 after
rescaling by $\pi/w(s)$, and such that there is only one inflection point (it
can be even arranged so that the point $P$ itself is the only inflection
point). This also specifies how large $|s|$ should be.

By the virtue of the above construction, it should be noted that, the
curvature of every admissible initial condition is bounded by some universal
constant, say, $M$ after rescaling by the factor of $\pi/w(s)$.

Since $\hat{\gamma}_{0}(s)$ has a unique inflection point at $s=0$, an
admissible initial condition $\alpha$ also has an inflection point at the
origin. Let $\\{\alpha_{t}\\}_{t\in[0,T)}$ be the evolution of the curve
$\alpha=\alpha_{0}$ under the curve shortening flow. Clearly, $\alpha_{t}$
lies inside $\hat{D}_{t}$ due to the avoidance principle of the curve
shortening flow. Because the number of the inflection points decreases under
the curve shortening flow, the curves $\alpha_{t}$ have at most two inflection
points.

In the above construction let $\Lambda(s)w(s)/\pi$ be equal to the distance
between the point $P$ and the ray $OA$; after rescaling by the factor of
$\pi/w(s)$ the quantity $\Lambda(s)$ represents the “height” of the portion of
the attached Grim Reaper curve $\bar{\rho}$, which is closer to the negative
wall (after rescaling the Grim Reaper has the standard width $\pi$). Clearly,
$\Lambda(s)\rightarrow\infty$ as $|s|\rightarrow\infty$, because after
rescaling the distance between the parallel lines $a$ and $b$ becomes exactly
$\pi$ and the distance between the points $A$ and $P$ is now $\Lambda(s)$, the
rescaled curve $\bar{\rho}$ is now the standard Grim Reaper, and in the
vicinity of the point $A$ the negative wall is getting exponentially close to
the circle centered at the origin of the radius $\pi|F(s)|/w(s)$ as
$|s|\rightarrow\infty$. It can be further estimated that
$\Lambda(s)e^{\Lambda(s)}\cong\pi^{2}/w(s)^{2}$ and hence after rescaling the
curvatures near the points $P$ and $Q$ of the Grim Reaper and Yin-Yang are
asymptotic to $e^{-\Lambda}$ and $e^{-\Lambda}/\Lambda$, respectively. An
analogous result can be derived for $\bar{\Lambda}(s)$, the “height” of the
portion of the attached Grim Reaper curve $\bar{\rho}$, which is closer to the
positive wall i.e., the distance from the point $Q$ to the ray $OA$, where
$\bar{\Lambda}(s)^{2}e^{\bar{\Lambda}(s)}\cong 2\pi^{2}/w(s)^{2}$. It can be
easily deduced that $\Lambda(s)\cong\bar{\Lambda}(s)$. The estimation of the
asymptotics of $\bar{\Lambda}(s)$ can be found in the Appendix.

Let $\hat{\gamma}^{\theta}_{t}:=e^{i\theta}\hat{\gamma}_{t}$ for $t\geq 0$ and
for $\theta\in(0,\pi)$, the rotated families of Yin-Yang curves. The family
$\\{\hat{\gamma}^{\theta}_{t}\\}_{\theta\in(0,\pi)}$ foliates the interior of
$\hat{D}_{t}$. For each $\theta\in(0,\pi)$ the curve
$\hat{\gamma}^{\theta}_{t}$ intersects the curve $\alpha_{t}$ in at most two
points, since initially $\hat{\gamma}^{\theta}_{0}$ intersects $\alpha_{0}$ in
exactly two points.

Consider the angle function $\theta:\alpha_{t}\mapsto S^{1}$, which has a lift
$\hat{\theta}:\alpha_{t}\mapsto\mathbb{R}$, since $\alpha_{t}$ is the boundary
of the disc lying inside the corridor $\hat{D}_{t}$. The curvature function
$H$ of the admissible initial condition $\alpha_{0}$ has exactly two zeros and
both are transverse. Moreover, $\alpha_{0}$ has exactly two radial tangencies.

######  Lemma 3.1.

Let $[0,t_{0})$ be the maximal time interval such that for any $t$ in that
interval the curvature function $H$ of $\alpha_{t}$ has exactly two transverse
zeros. Then for every $t\in[0,t_{0})$ there are exactly two radial tangencies
of $\alpha_{t}$, which are the non-degenerate critical points of the angle
function $\hat{\theta}$ (max-min).

###### Proof of Lemma 3.1.

Assume the contrary and let $[0,t_{1})$ be the maximal time interval for which
the assertion is true, where $t_{1}<t_{0}$. A local max or min of the function
$\hat{\theta}$ must be non-degenerate, because the curvature is nonzero at
that point (otherwise, if the curvature vanishes, then it has a transverse
zero at that point and locally there $\alpha_{t}$ will lie on both sides of
the radial tangency, hence contradicting the local max-min assumption). From
the definition of $t_{1}$ it follows that there must be a radial tangency at
some point $p\in\alpha_{t_{1}}$, where $p$ is not a non-degenerate local max
or min. So, $p$ is in fact an inflection point. However, then there should be
a pair of radial tangencies close to $p$ on $\alpha_{t_{1}}$, contradicting
the definition of $t_{1}$. Thus, $t_{1}=t_{0}$. The lemma is proved. ∎

From Lemma 3.1 it will follow that at time $t=t_{0}$ the admissible curve
$\alpha_{t_{0}}$ has exactly one non-transverse zero of the curvature function
$H$ and for further times $t>t_{0}$ the curves $\alpha_{t}$ become strictly
convex.

$\bf{tip(\alpha_{t})}:$ the tip of the admissible curve $\alpha_{t}$, the non-
degenerate critical point corresponding to the max of the angle function
$\hat{\theta}$. The uniqueness follows from the lemma when $t<t_{0}$, and from
the convexity of $\alpha_{t}$ when $t\geq t_{0}$.

$\bf{tip(\alpha_{t})^{\pm}:}$ the radial projections of $tip(\alpha_{t})$ on
the positive and negative walls of $\hat{\gamma_{t}}$.

$\bf{L_{\pm}(t):}$ the distances from the origin along the walls of
$\hat{\gamma}_{t}$ of the points $tip(\alpha_{t})^{\pm}$ i.e.,
$tip(\alpha_{t})^{+}=\hat{\gamma}_{t}(L_{+}(t))$ and
$tip(\alpha_{t})^{-}=\hat{\gamma}_{t}(-L_{-}(t))$.

$\bf{w(t)}:$ the “width” at the tip, the radial distance between
$tip(\alpha_{t})^{+}$ and $tip(\alpha_{t})^{-}$ or $w(-L_{-}(t))$ in
correspondence with the notation from the previous section.

##  4\. Hyperbolic Supersolution

A family of the open curves $\Gamma(s,t)$, where $t\in[0,T)$ and
$s\in[-L(t),R(t)]$ for some functions $L(t),R(t)\in C^{1}([0,T))$, is defined
to be a $\bf{hyperbolic\ supersolution}$, if it has the following properties
(where $\hat{\gamma}_{t}$ and $\hat{D}_{t}$ are as in Section 2):

  1. (1)

$\Gamma(s,t)=e^{it}(F(s)+\phi(s,t)N(s))$ and the curve is completely inside
the region $\hat{D}_{t}$ (a graph over Yin-Yang $\hat{\gamma}_{t}$ at each
time $t$);

  2. (2)

$\phi(s,t)=\lambda^{-}w(t)\exp[-a(t)(L(t)+s)]+\lambda^{+}w(t)\exp[-a(t)(R(t)-s)]$,
where $w(t),a(t)\in C^{1}([0,T))$ and $\lambda^{\pm}>0$ (a hyperbolic function
over the Yin-Yang);

  3. (3)

$\langle\partial_{t}\Gamma,\tilde{N}\rangle\geq\tilde{H}$, where $\tilde{N}$
and $\tilde{H}$ are the unit inward normal and the curvature of the
$\Gamma-$curve, respectively (a supersolution to the curve shortening flow).

From the definitions of the unit normal and the curvature the following
statement is true:

(7)
$\langle\partial_{t}\Gamma,\tilde{N}\rangle\geq\tilde{H}\Longleftrightarrow\langle\partial_{t}\Gamma,\dfrac{i\partial_{s}\Gamma}{|\partial_{s}\Gamma|}\rangle\geq\dfrac{\langle\partial_{s}\partial_{s}\Gamma,i\partial_{s}\Gamma\rangle}{|\partial_{s}\Gamma|^{3}}$

The last inequality, the supersolution condition, can be rewritten as:

(8)
$\displaystyle|\partial_{s}\Gamma|^{2}\langle\partial_{t}\Gamma,i\partial_{s}\Gamma\rangle-\langle\partial_{s}\partial_{s}\Gamma,i\partial_{s}\Gamma\rangle\geq
0$ (9) $\displaystyle A(\phi)+Q(\phi)+P(\phi)\geq 0$

Where $A$, $Q$, and $P$ contain solely the linear, quadratic, and higher order
terms in $\phi$, respectively. In the Appendix it is computed that

(10) $\displaystyle A(\phi)$ $\displaystyle=$
$\displaystyle\phi_{t}-\phi_{ss}+\phi_{s}\langle F(s),N(s)\rangle-\phi
H(s)^{2}$ (11) $\displaystyle Q(\phi)$ $\displaystyle=$ $\displaystyle-H(3\phi
A(\phi)+2\phi\phi_{ss}+\phi_{s}^{2}+\phi^{2}H^{2})$

##  5\. The Main Argument

This section gives the main estimate on the evolution of admissible curves,
and uses it to prove the Main Theorem.

######  Theorem 5.1.

For any $\epsilon>0$ there exists $R_{\epsilon}\gg 1$ such that if
$\alpha_{0}$ is an admissible initial condition with
$\|tip(\alpha_{0})\|>R_{\epsilon}$ and $\alpha_{t}$ is the evolution of
$\alpha_{0}$ under the curve shortening flow with $\|tip(\alpha_{t})\|\geq
R_{\epsilon}$ for $t\in[0,T)$, then

  1. (1)

(Stability near the tip) In the “neighborhood of the tip$(\alpha_{t})"$ i.e.,
the connected component of $\alpha_{t}$ containing $tip(\alpha_{t})$ inside
the ball of the radius $1000w(t)$ centered at tip$(\alpha_{t})$, the curve
$\alpha_{t}$ is $\epsilon-$close to the standard Grim Reaper in the pointed
$C^{\infty}-$topology after rescaling the curve by the factor of $\pi/w(t)$ ;

  2. (2)

(Exponentially close to the Yin-Yang) There exists a hyperbolic supersolution
$\Gamma(s,t)=e^{it}(F(s)+\phi(s,t)N(s))$ lying inside the closed curve
$\alpha_{t}$ for $t\in[0,T)$ and $s\in[-L(t),R(t)]$ (the “barrier curve”),
where

(12)
$\displaystyle\phi(s,t)=\lambda^{-}w(t)\exp[-a(t)(L(t)+s)]+\lambda^{+}w(t)\exp[-a(t)(R(t)-s)]$
(13) $\displaystyle a(t)=\dfrac{\pi(1-\epsilon)}{w(t)},\indent\indent
L(t)=L_{-}(t)-\lambda w(t),\indent R(t)=L_{+}(t)-\lambda w(t)$

for some $\lambda^{\pm},\lambda>0$ and $w(t)$ is the width at the tip.

Moreover, there exists a constant $M_{\epsilon}\gg 1$ such that the curvature
of $\alpha_{t}$ away from the neighborhood of the tip (see the condition (1)
above) is bounded by $M_{\epsilon}$ after rescaling by the factor of
$\pi/w(t)$ as long as the conditions (1) and (2) hold i.e., for $t\in[0,T)$
(the control of the curvature away from the tip).

###### Proof of Main Theorem Theorem.

Fix any $\epsilon>0$ and consider a sequence of time-shifted admissible curves
$\\{\alpha_{t}^{n}\\}_{n=1}^{n=\infty}$, $t\in[-T_{n},0)$ (the extinction
occurs at $t=0$), with $\|tip(\alpha_{-T_{n}}^{n})\|\rightarrow\infty$ (and
hence $T_{n}\rightarrow\infty$). Let $\\{T_{n}^{\epsilon}\\}_{n=1}^{n=\infty}$
be the sequence of times such that
$\|tip(\alpha_{-T_{n}^{\epsilon}}^{n})\|=R_{\epsilon}$.

Clearly, $\\{T_{n}^{\epsilon}\\}_{n=1}^{n=\infty}$ must be bounded, because
the admissible curves will be sweeping an area of size $2\pi T_{n}^{\epsilon}$
starting from the moment when they cross the threshold $R_{\epsilon}$ until
their extinction, which is clearly a finite area. Passing to a subsequence it
can be taken that
$T_{n}^{\epsilon}\xrightarrow{n\rightarrow\infty}T^{\epsilon}$.

For any fixed $t\in(-\infty,-T^{\epsilon})$ the sequence
$\\{\|tip(\alpha_{t}^{n})\|\\}_{n=1}^{n=\infty}$ must be bounded. Otherwise,
there will be a sequence of admissible curves defined for the time interval
$[t,0]$, where each curve goes extinct in finite amount of time $-t$ and
sweeps an area of size $-2\pi t$ in the process, while the tips of the curves
at the time $t$ can get arbitrarily far away from the origin. Because of
exponential proximity of the admissible curve $\alpha_{t}^{n}$ to the walls of
$\hat{D_{t}}$, the area of the region enclosed by the hyperbolic supersolution
or the barrier curve $\Gamma(s,-t)$, after connecting its endpoints by the
line segment, gets arbitrarily large as $n\rightarrow\infty$, since
$\|tip(\alpha_{t}^{n})\|\rightarrow\infty$. For every $n$ this region lies
completely inside $\alpha_{t}^{n}$, which contradicts with the above fact of
$\alpha_{t}^{n}$ enclosing a region of the area $-2\pi t$ for any $n$. So,
say, $\|tip(\alpha_{t}^{n})\|\leq R^{t}<\infty$ for any $n$.

Fix $t\in(-\infty,-T^{\epsilon})$. In the vicinity of their tips the
curvatures of $\alpha_{t}^{n}$ are bounded by $(1+\epsilon)R^{t}$ for any any
$n$, because of being $\epsilon-$close to the standard Grim Reaper after
rescaling by the factor of $R^{t}$. Away from the tips the maximum of the
curvatures of $\alpha_{t}^{n}$ are bounded uniformly in $n$, in fact, they are
bounded by $M_{\epsilon}R^{t}$.

Thus, for every $t\in(-\infty,-T^{\epsilon})$, the maximum of the curvature
function of $\alpha_{t}^{n}$ is bounded uniformly in $n$. So, there exists a
subsequence of $\\{\alpha_{t}^{n}\\}_{n=1}^{n=\infty}$ which converges to some
curve shortening flow $\chi_{t}$, $t\in(-\infty,-T^{\epsilon})$. The curve
shortening flow $\chi_{t}$, $t\in(-\infty,0)$, is clearly ancient, non-convex,
and asymptotic to Yin-Yang as $t\rightarrow-\infty$. The theorem is proved.

∎

###### Proof of Theorem 5.1.

For the hyperbolic supersolution it can be further specified that one can take
$\lambda^{\pm}=\frac{1}{\pi(1-\epsilon)}$ and
$\lambda=\frac{100}{\pi(1-\epsilon)}$, so that the function $\phi$ will have a
simpler form:

(14)
$\displaystyle\phi(s,t)=\dfrac{1}{a}\exp[-a(L+s)]+\dfrac{1}{a}\exp[-a(R-s)]$
(15) $\displaystyle a=\dfrac{\pi(1-\epsilon)}{w},\indent\indent
L=L_{-}-\dfrac{100}{a},\indent R=L_{+}-\dfrac{100}{a}$

Assume the contrary. Then for some $\epsilon>0$ there is a sequence of the
admissible initial conditions $\\{\alpha^{j}_{0}\\}_{j=1}^{j=\infty}$ with the
corresponding evolutions $\\{\alpha^{j}_{t}\\}_{j=1}^{j=\infty}$,
$t\in[0,T_{j}]$, under the curve shortening flow, where the time $T_{j}$ is
the first instance when one of two properties in Theorem 5, either “stability”
or “proximity”, fail and one has that
$\|tip(\alpha_{T_{j}}^{j})\|=:R_{j}\rightarrow\infty$. This leads to the
analysis of two cases.

Case 1∘: Assume that the stability fails for infinitely many indices $j$
(hence after passing to the subsequence can be assumed for all $j$). Consider
$\beta^{j}_{t}$, the rescaled by the factor of $\pi/w(T_{j})$ and time-shifted
curve shortening flows of $\alpha_{t}^{j}$, in the time intervals
$[-\pi^{2}T_{j}/w(T_{j})^{2},0]$. Passing to a subsequence it can be argued
that $-\pi^{2}T_{j}/w(T_{j})^{2}=:-K_{j}\rightarrow-K$ for some $0\leq
K\leq\infty$.

Case 1∘a:  First consider the case when $K=\infty$. Below it will be proved
that, at any given time $t\in(-\infty,0)$ the curves
$\\{\beta_{t}^{j}\\}_{j=1}^{j=\infty}$ have uniform bounds on their curvature:
the uniform bound near the tip due to the stability combined with the uniform
bound away from the tip owing to the fact of exponential proximity.

For the sake of contradiction, assume that away from their tips the maxima of
the curvature of the curves $\beta_{t}^{j}$ is an unbounded sequence in $j$.
Moreover, let $-\hat{K_{j}}$ be the first instance when the maximum of the
curvature of $\beta_{t}^{j}$ is at least $j$. There are two possibilities:
either $\limsup_{j\rightarrow\infty}(K_{j}-\hat{K_{j}})>0$ (“long enough
flow”) or $\limsup_{j\rightarrow\infty}(K_{j}-\hat{K_{j}})=0$ (“rapid increase
of curvature”). For any $j$ at the time $-\hat{K_{j}}$ the part of the curve
$\beta_{-\hat{K_{j}}}^{j}$ away from its tip is graphical over Yin-Yang, and
it is confined between Yin-Yang and the corresponding hyperbolic
supersolution. After interpolating between Yin-Yang and the barrier curve one
gets uniform in $j$ bound on the $C^{1}-$norms of the curves
$\beta_{-\hat{K_{j}}}^{j}$.

If $\limsup_{j\rightarrow\infty}(K_{j}-\hat{K_{j}})>0$, then, passing to a
subsequence if necessary, the uniform in $j$ bound on $C^{1}-$norm combined
with the parabolic regularity will yield a uniform in $j$ bounds on
$C^{k}-$norms. This will contradict with the fact that the maxima of the
curvatures of the curves $\beta_{-\hat{K_{j}}}^{j}$ away from the tips is
unbounded in $j$.

In the case of $\limsup_{j\rightarrow\infty}(K_{j}-\hat{K_{j}})=0$, the
uniform in $j$ bounds of $C^{k}-$norms of the curves $\beta_{-K_{j}}^{j}$ away
from the tips and the uniform in $j$ bound of $C^{0}-$norm of the curves
$\beta_{-\hat{K_{j}}}^{j}$ away from the tips will yield uniform in $j$ bounds
of $C^{k}-$norms of the curves $\beta_{\tau}^{j}$ for
$\tau\in[-K_{j},-\hat{K_{j}}]$. This causes another contradiction.

Thus, away from their tips the maxima of the curvature of the curves
$\beta_{t}^{j}$ must be a bounded sequence in $j$.

Due to the uniform in $j$ bound on the curvature, the pointed sequence of the
curve shortening flows $\beta_{t}^{j}$, pointed at the tips, converge in
Cheeger-Gromov-Hausdorff sense to some non-compact ancient solution, say
$\chi_{t}$. Clearly, all the curves $\chi_{t}$ must be restricted inside some
strip of width $\pi$. The curve $\chi_{t}$ can have at most one inflection
point, because of the properties of the admissible curves evolving under the
curve shortening flow. However, it can not have exactly one inflection point,
since otherwise it will fail to be contained inside the strip. Thus,
$\chi_{t}$ is a non-compact convex ancient solution. The only such curve is
the Grim Reaper (see [3] for the complete classification of the ancient convex
curve shortening flows). $\chi_{t}$ must be the standard Grim Reaper of width
$\pi$, because of the exponential proximity to the walls of Yin-Yang. However,
this causes a contradiction, because at the time $t=0$ the limit curve is the
standard Grim Reaper while the curves $\beta_{0}^{j}$ are no more close enough
to Grim Reaper near the tips.

Case 1∘b:  When $0\leq K<\infty$, again, the curves $\beta_{t}^{j}$ with
$t\in[-K_{j},0]$ converge to some limit flow $\chi_{t}$ with $t\in[-K,0]$. The
curves $\beta_{-K_{j}}^{j}$, pointed at the tips, converge to the standard
Grim Reaper, because they are rescaled admissible initial conditions and, as
$j\rightarrow\infty$, near the tips they are congruent to the ever growing
chunks of the standard Grim Reaper. So, $\chi_{-K}$ is the standard Grim
Reaper and thus $\chi_{t}$ are also the standard Grim Reaper. Similar to the
previous case, this causes another contradiction at the time $t=0$.

The above argumentation by contradiction shows that the stability property can
not fail for infinitely many indices $j$. Case 1∘ is studied completely.

Case 2∘: Now, it will be proved that, when the stability argument holds true,
then the proximity argument holds true as well. This will conclude that the
proximity property as well can not fail for infinitely many indices $j$.

First, it will be proved that $\Gamma(s,t)$ is a hyperbolic supersolution.

Define the functions $\psi$ and $\eta$:

(16)
$\psi(s,t):=\dfrac{1}{a(t)}\exp[-a(t)(L(t)+s)],\indent\eta(s,t):=\dfrac{1}{a(t)}\exp[-a(t)(R(t)-s)]$

Clearly, $\phi=\psi+\eta$. After computing for $A(\psi)/\psi$ and
$A(\eta)/\eta$ it follows that

(17)
$\displaystyle\dfrac{A(\psi)}{\psi}=-\dfrac{\dot{a}}{a}-\dot{a}(L+s)-a\dot{L}-a^{2}-a\langle
F(s),N(s)\rangle-H(s)^{2}$ (18)
$\displaystyle\dfrac{A(\eta)}{\eta}=-\dfrac{\dot{a}}{a}-\dot{a}(R-s)-a\dot{R}-a^{2}+a\langle
F(s),N(s)\rangle-H(s)^{2}$

Also due to Lemma 2.1 it can be assumed that for $t\in[0,T_{j})$ (for the tips
far enough away from the origin) the mutual ratios of the following quantities
are between $1-\epsilon$ and $1+\epsilon$:

(19) $(3L_{\pm}(t))^{1/3},\indent\dfrac{1}{|H(\pm L_{\pm}(t))|},\indent|F(\pm
L_{\pm}(t))|,\indent\dfrac{\pi}{w(-L_{-}(t)))}=\dfrac{\pi}{w(t)}=\dfrac{a(t)}{1-\epsilon}$

For the change in time of $L_{-}(t)$ one has that, $-\dot{L}_{-}(t)$ is at
least the amount of the drift of Yin-Yang due to its rotation (the drift is
$\langle-F_{t},T\rangle=\langle F,N\rangle$, the magnitude of the projection
of the velocity to the tangent space) plus the amount of the drift of the tip
along the walls, when projected, due to its flow inside the corridor.

The tip is moving in the direction of $-iF(-L_{-}(t))$ (the direction
perpendicular to the radial tangent at the tip) and its speed is at least
$\pi(1-\epsilon)/w(t)=a(t)$, because of the approximate Grim Reaper near the
tip. The amount of the drift of the tip along the walls is equal to the
magnitude of the projection of its velocity to the tangent vector of Yin-Yang.

Accounting the drift term of Yin-Yang and the drift of the tip along the walls
and the asymptotic behaviors of the related quantities in the equation (19)
one obtains the following sequence of inequalities:

$\displaystyle-\dot{L}_{-}(t)$ $\displaystyle\geq$ $\displaystyle\langle
F(-L_{-}(t)),N(-L_{-}(t))\rangle+a(t)\dfrac{\langle-
iF(-L_{-}(t)),T(-L_{-}(t))\rangle}{|F(-L_{-}(t))|}=$ $\displaystyle=$
$\displaystyle\langle F,N\rangle+a\dfrac{\langle F,N\rangle}{|F|}=\langle
F,N\rangle+a\dfrac{\sqrt{|F|^{2}-\langle F,T\rangle^{2}}}{|F|}=\langle
F,N\rangle+a\sqrt{1-\dfrac{H^{2}}{|F|^{2}}}\geq$ $\displaystyle\geq$
$\displaystyle\langle F,N\rangle+a(1-(1+\epsilon)H^{4})\geq a+\langle
F,N\rangle-(1+\epsilon)H^{3}$

Due to the asymptotics of $w(t)\cong\pi(3L_{-}(t))^{-1/3}$ and from the
equation (5) it follows that

(21)
$\displaystyle\dot{w}(t)\geq\dfrac{-\pi(1-\epsilon)\dot{L}_{-}}{(3L_{-})^{4/3}}\geq\dfrac{\pi(1-\epsilon)^{2}}{3L_{-}}\geq\dfrac{w^{3}}{\pi^{2}}=\dfrac{(1-\epsilon)^{3}}{a^{3}}$

After combining the estimates in the equations (5) and (21) and using the
asymptotics the following lower bound is obtained:

(22) $\displaystyle-a\dot{L}\geq a^{2}+a\langle
F,N\rangle-(1+\epsilon)aH^{3}+100a\dfrac{w^{3}}{\pi^{2}}\geq a^{2}+a\langle
F,N\rangle+\dfrac{10}{a^{2}}$

Similar estimates are true for $-\dot{L}_{+}(t)$ and $-a\dot{R}(t)$ as well.

By differentiating twice, it is straightforward to check that the function
$a|\langle F(s),N(s)\rangle|+H(s)^{2}$ is convex and has the absolute minimum
at $s=0$. Hence, during the estimates for the lower bounds one may take $s=\pm
L_{\pm}(t)$ for the quantity $a|\langle F(s),N(s)\rangle|+H(s)^{2}$.

Clearly, because of all the above given relations (17) and (22) one has that

(23) $\displaystyle
A(\psi)\geq\dfrac{10}{a^{2}}\psi-\dot{a}(L+s)\psi\indent\indent\indent
A(\eta)\geq\dfrac{10}{a^{2}}\eta-\dot{a}(R-s)\eta$ (24) $\displaystyle
A(\phi)\geq\dfrac{10}{a^{2}}\phi-\dot{a}(L+s)\psi-\dot{a}(R-s)\eta$

Obviously, $-\dot{a}\geq 0$ and $R-s\geq 0$. Moreover, because of the estimate
in (21) on has that

(25)
$\displaystyle-\dot{a}=\dfrac{\pi(1-\epsilon)\dot{w}}{w^{2}}\geq\dfrac{(1-\epsilon)w}{\pi}\geq\dfrac{(1-\epsilon)^{2}}{a}$

Whenever $L+s\gtrsim\frac{\log a}{a}$ or $R-s\gtrsim\frac{\log a}{a}$, the
higher order terms in $\phi$ in the equation (9) can be dominated by the
linear terms $-\dot{a}(L+s)\psi$ or $-\dot{a}(R-s)\eta$, since $\phi$ is
exponentially decaying in the variables $L+s$ or $R-s$ with the decay rate
$a$.

In the case, when $0\leq R-s,L+s\lesssim\frac{\log a}{a}\ll 1$ (very close to
the endpoints of the barrier curve), effectively, $s=R$ or $s=-L$, $\eta=\phi$
or $\psi=\phi$, $H=\pm\frac{1}{a}$, $\phi_{s}=\pm a\phi$,
$\phi_{ss}=a^{2}\phi$, and $A(\phi)=\frac{10}{a^{2}}\phi$. After plugging the
values directly in the inequality (8), the supersolution condition, and
keeping the track of the “effectively large terms” one gets that

(26) $\displaystyle A(\phi)+Q(\phi)+P(\phi)\cong\dfrac{10}{a^{2}}\phi\mp
3a\phi^{2}+a^{4}\phi^{3}\mp a^{3}\phi^{4}=$ (27)
$\displaystyle=\phi\left(\dfrac{1}{a^{2}}+\left(\dfrac{3}{a}\mp\dfrac{1}{2}a^{2}\phi\right)^{2}+a^{3}\phi^{2}\left(\dfrac{3}{4}a\mp\phi\right)\right)\geq
0$

It should be noted that, in the equation (26) near the endpoints of the
$\Gamma$-curve the cubic term $a^{4}\phi^{3}$ corresponds to
$\phi_{s}^{2}\phi_{ss}$ and is of the order $O(a)$, while the linear term is
$O(1/a^{3})$ and the other two are $O(1/a)$. This remark justifies the
necessity for the careful analysis near the endpoints of the barrier curve and
is another way to prove the inequality (27).

This separate “endpoint analysis” concludes that $\Gamma(s,t)$ is indeed a
hyperbolic supersolution for $t\in[0,T_{j})$.

It must further be proved that the curve $\Gamma(s,t)$ lies inside the
admissible curve $\alpha_{t}$ for all times $t\in[0,T_{j})$ to completely
justify that the $\Gamma$-curve is a barrier curve. This is true, because

  1. (1)

for the admissible initial condition $\alpha_{0}$ the exponential decay rate
of the glued chunk of Grim Reaper, when viewed as a graph over Yin-Yang in the
neighborhood of $tip(\alpha_{0})$ until it coincides with the Yin-Yang, is
equal to $\pi/w(t=0)$, while for $\Gamma(s,0)$ it is smaller; it is
$\pi(1-\epsilon)/w(0)$ to be precise. This will guarantee that $\Gamma(s,0)$
is inside $\alpha_{0}$ given that $\|tip(\alpha_{0})\|\gg 1$ (the gluing took
place far away) and $\lambda_{\pm}>e^{-\lambda}$ (to ensure that the endpoints
of $\Gamma(s,0)$ are inside $\alpha_{0}$);

  2. (2)

by the virtue of the definition of the curve $\Gamma(s,t)$, its endpoints are
located in the vicinity of $tip(\alpha_{t})$ (the distance $\lambda w(t)$ away
from the tip), thus avoiding the intersection with any point away from the
vicinity; the endpoints avoid intersecting with any point in the vicinity of
the tip, since near the tip the admissible curve $\alpha_{t}$ looks
$\epsilon-$close to Grim Reaper after proper rescaling and therefore the
exponential decay rate of $\alpha_{t}$, when viewed as a graph over Yin-Yang
in the neighborhood of the tip, is at least $\pi(1-\epsilon)/w(t)=a(t)$, which
is exactly the decay rate of $\Gamma(s,t)$;

  3. (3)

the $\Gamma$-curves are the supersolutions for the curve shortening flow and
therefore can not be tangent to the $\alpha$-curves at some point in
spacetime.

Case 2∘ is completely analyzed.

Now the existence of the uniform bound on the curvature function will be
proved. Assume that for some given $\epsilon>0$ there is no such constant
$M_{\epsilon}$. Then there exists a sequence of rescaled admissible curves
$\\{\alpha_{t}^{j}\\}_{j=1}^{j=\infty}$ with $t\in[0,t_{j}]$, where $t_{j}$ is
the first instance when the maximum of the curvature of the curve
$\alpha_{t_{j}}^{j}$ is at least $j$ (clearly, the maximum is attained away
from the neighborhood of its tip, since near the tip the curve looks
$\epsilon-$close to the standard Grim Reaper). Similar to the earlier
discussion, there are two possibilities: either
$\liminf_{j\rightarrow\infty}t_{j}>0$ (“long enough flow”) or
$\liminf_{j\rightarrow\infty}t_{j}=0$ (“rapid increase of curvature”). For any
$j$ at time $t_{j}$ the part of the curve $\alpha_{t_{j}}^{j}$ away from its
tip is graphical over Yin-Yang, and it is confined between Yin-Yang and the
corresponding hyperbolic supersolution. After interpolating between Yin-Yang
and the barrier curve one gets uniform in $j$ bound on the $C^{1}-$norms of
the curves $\alpha_{t_{j}}^{j}$.

If $\liminf_{j\rightarrow\infty}t_{j}>0$, then, passing to a subsequence if
necessary, the uniform in $j$ bound on $C^{1}-$norm combined with the
parabolic regularity will yield a uniform in $j$ bounds on $C^{k}-$norms. This
will contradict with the fact that the maxima of the curvatures of the curves
$\alpha_{t_{j}}^{j}$ away from the tips is unbounded in $j$.

In the case of $\liminf_{j\rightarrow\infty}t_{j}=0$, the uniform in $j$
bounds of $C^{k}-$norms of the curves $\alpha_{0}^{j}$ away from the tips and
the uniform in $j$ bound of $C^{0}-$norm of the curves $\alpha_{t_{j}}^{j}$
away from the tips will yield uniform in $j$ bounds of $C^{k}-$norms of the
curves $\alpha_{\tau}^{j}$ for $\tau\in[0,t_{j}]$. This causes another
contradiction. Thus, there must exist such constant $M_{\epsilon}$. The
theorem is proved. ∎

##  6\. Appendix

###  6.1. The Asymptotics of $\bar{\Lambda}$

Consider the circle of radius $R$ centered at the origin and the following
Grim Reaper:

(28) $\displaystyle y=-\log(\cos(x-(R-\pi/2))),\indent R-\pi<x<R$

i.e., Grim Reaper of width $\pi$ with vertical $x=R-\pi$ and $x=R$ asymptotes.
Let $0<z<\pi/2$ be such that the point $(R-z,-\log(\cos(\pi/2-z))$ is the
intersection point of the circle and the Grim Reaper, which is the closest to
the $x$-axis. It can be easily verified that

$\displaystyle\sqrt{2Rz-z^{2}}$ $\displaystyle=$ $\displaystyle-\log\sin z$
$\displaystyle\sqrt{2Rz}$ $\displaystyle\cong$ $\displaystyle-\log z$ (29)
$\displaystyle\bar{\Lambda}$ $\displaystyle\cong$ $\displaystyle\sqrt{2Rz}$
$\displaystyle e^{\bar{\Lambda}}$ $\displaystyle\cong$ $\displaystyle e^{-\log
z}=z$ $\displaystyle\bar{\Lambda}^{2}e^{\bar{\Lambda}}$ $\displaystyle\cong$
$\displaystyle 2R$

when $R\rightarrow\infty$ and hence when $z\rightarrow 0$.

###  6.2. Linear and Quadratic Differential Operators $A$ and $Q$

The computations of the linear differential operator $A$ and the quadratic
terms $Q$ are provided here. For simplicity $\Gamma$ is written as
$\Gamma=F+\phi N$, where $F=e^{it}F(s)$, $N=e^{it}N(s)$, and $T=F_{s}$. In the
following computations only the linear and quadratic terms in $\phi$ are kept
while higher order terms are dropped:

$\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle F+\phi N$
$\displaystyle\Gamma_{t}$ $\displaystyle=$ $\displaystyle F_{t}+\phi_{t}N+\phi
N_{t}$ $\displaystyle\Gamma_{s}$ $\displaystyle=$ $\displaystyle(1-\phi
H)T+\phi_{s}N$ $\displaystyle i\Gamma_{s}$ $\displaystyle=$
$\displaystyle(1-\phi H)N-\phi_{s}T$ $\displaystyle\Gamma_{ss}$
$\displaystyle=$ $\displaystyle-(2\phi_{s}H+\phi H_{s})T+(H(1-\phi
H)+\phi_{ss})N$ $\displaystyle|\Gamma_{s}|^{2}$ $\displaystyle=$
$\displaystyle(1-\phi H)^{2}+\phi_{s}^{2}$ (30)
$\displaystyle\langle\Gamma_{t},i\Gamma_{s}\rangle$ $\displaystyle=$
$\displaystyle\phi_{s}\langle F,N\rangle+H(1-\phi H)+\phi_{t}(1-\phi
H)+\phi\phi_{s}$ $\displaystyle\langle F_{t},T\rangle$ $\displaystyle=$
$\displaystyle\langle iF,T\rangle=-\langle F,N\rangle$ $\displaystyle\langle
F_{t},N\rangle$ $\displaystyle=$ $\displaystyle\langle iF,N\rangle=\langle
F,T\rangle=H$ $\displaystyle\langle N_{t},T\rangle$ $\displaystyle=$
$\displaystyle\langle iN,T\rangle=\langle-T,T\rangle=-1$ $\displaystyle\langle
N_{t},N\rangle$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\langle\Gamma_{ss},i\Gamma_{s}\rangle$ $\displaystyle=$
$\displaystyle\phi_{ss}(1-\phi H)+\phi_{s}(2\phi_{s}H+\phi H_{s})+H(1-\phi
H)^{2}$
$\displaystyle|\Gamma_{s}|^{2}\langle\Gamma_{t},i\Gamma_{s}\rangle-\langle\Gamma_{ss},i\Gamma_{s}\rangle$
$\displaystyle\cong$ $\displaystyle\phi_{t}-\phi_{ss}+\phi_{s}\langle
F(s),N(s)\rangle-\phi H(s)^{2}-$ $\displaystyle-$ $\displaystyle\phi
H[3\phi_{t}+2\phi_{s}\langle F,N\rangle-2\phi
H^{2}-\phi_{ss}]+\phi_{s}[\phi-\phi_{s}H-\phi H_{s}]$

Hence for the linear part it follows that

(31) $\displaystyle A(\phi)=\phi_{t}-\phi_{ss}+\phi_{s}\langle
F(s),N(s)\rangle-\phi H(s)^{2}$

Using the relation $H_{s}=1+H\langle F,N\rangle$ and after the careful
rearrangements one gets that

(32) $\displaystyle A(\phi)+Q(\phi)=(1-3\phi
H)A(\phi)-H(2\phi\phi_{ss}+\phi_{s}^{2}+\phi^{2}H^{2})$

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Courant Institute of Mathematical Sciences, New York University, 251 Mercer
Street, New York, NY 10012

e-mail address<EMAIL_ADDRESS>


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