8
2. CURVE SHORTENING AND GRAYSON’S THEOREM
the equivalence class of x. Then the condition for embeddedness is simply
that X is injective on R/(AZ).
CHAPTER 2
Curve Shortening and Grayson’s Theorem
In this chapter and the next we discuss the curve shortening flow (CSF).
A number of important techniques in the field of geometric flows exhibit
themselves in the curve shortening flow in an elegant and less technical way.
The CSF was proposed in 1956 by Mullins to model the motion of
idealized grain boundaries. In 1978 Brakke studied the mean curvature
flow, of which the CSF is the 1-dimensional case, in the context of geometric
measure theory. Renewed interest in the CSF resulted from the works of
Gage and Hamilton in 1986 on convex plane curves and Grayson in 1987 on
embedded plane curves.
In this chapter and throughout the rest of the book we shall assume that
the objects we consider are smooth, i.e., C 1 . This assumption is made more
out of convenience than necessity and we refer the reader to the notes and
commentary at the end of this chapter for references to results with weaker
regularity hypotheses.
1. Basic geometric theory of planar curves
We consider parametrised immersed curves, described by smooth maps
X : I ⇢ R ! R2 satisfying the ‘immersion’ condition: |X 0 (u)| =
6 0 for every
u 2 I. Here I is a (possibly infinite) interval in the real line. In particular
we are interested in properties of such curves which are invariant under both
reparametrisation of the domain R and rigid motions of the range R2 .
1.1. Immersed, embedded and closed curves. A parametrised curve
X is called embedded if X is a homeomorphism to its image set (that is,
X is injective and maps closed sets to closed sets).
A particular case which will be important to us is that of closed curves,
which we understand for the purposes of this chapter to mean curves for
which the immersion X is defined on all of R and is periodic with some
period, so that there exists A > 0 such that X(x + A) = X(x) for all x 2 M .
In this case we say X is embedded if X is injective modulo periodicity, so
that X(x) = X(y) implies y x = kA for some k 2 Z. In this case we also
say that the curve is a simple closed curve.
For a closed curve given by a periodic immersion X : R ! R2 , we can
induce a map (which we also call X) from the quotient M = R/(AZ) to R2
by defining X([x]) = X(x) for each x, where [x] = {y 2 R : y x 2 AZ} is
7
Exercise 2.1. If X : R ! R2 is a simple closed curve, show that the
induced map X : R/(AZ) ! R2 is a homeomorphism to its image set, if we
place the quotient topology on R/(AZ) [Hint: A much more general result
holds: The inverse map of a continuous bijection from a compact topological
space is always continuous].
1.2. Arc length parametrisation, unit tangent and normal vectors. Invariance under reparametrisation can be guaranteed by finding a
canonical parametrisation. This is provided by the arc length parametrisation, which is uniquely defined (up to translations and reversal of direction)
by the condition that the derivative of the map X have length 1. It will
be convenient to restrict our attention largely to changes of parametrisation
which do not reverse direction, but we will remark on the e↵ects of reversing
direction as we proceed.
The arc length parametrisation gives us immediately an important invariant of the embedding: The unit tangent vector T, which is simply the
derivative of X with respect to an arc length parameter. Note that translation of the arc length parameter does not a↵ect T, while reversal of direction
changes T to T. By convention we will refer to arc length parameters as
s, and will use primes to denote derivatives with respect to an arbitrary
(increasing) parameter. We see that
(2.1)
T=
dX
X0
=
.
ds
|X 0 |
Let J : R2 ! R2 be the anticlockwise rotation through angle ⇡/2. Then we
can define a unit normal vector N to the curve in a canonical way by choosing
N = JT (our choice of convention here means that a simple closed curve
with anticlockwise parametrisation has an outward-pointing unit normal).
Again, reversal of parametrisation reverses the direction of N.
1.3. Curvature. Di↵erentiating the unit tangent and normal vectors1
gives the final invariant we need, the curvature : Since T has unit length,
the derivative of T with respect to an arc length parameter is orthogonal to
T, and hence is a multiple of N. We define the curvature by this relation,
and are led to the Serret-Frenet equations for a plane curve:
(2.2)
d
T = N;
ds
d
N = T.
ds
1Note that T and N are themselves vector-valued functions from I ⇢ R to R2 , and
can be di↵erentiated by di↵erentiating each component function.
1. BASIC GEOMETRIC THEORY OF PLANAR CURVES
9
Since reversal of direction of parametrisation reverses s and T and N, it also
reverses the sign of . However we note that the product N is una↵ected.
This is called the curvature vector of the curve.
We recall that the fundamental theorem of plane curve theory says that
for any smooth function  on an interval I ⇢ R, there exists a curve X : I !
2
R parametrised by arc length with curvature (s) = k(s) for every s 2 I.
Furthermore this curve is unique up to composition with a rigid motion
(rotation and/or translation). In particular a circle of radius r parametrised
anticlockwise has  constant and equal to 1/r, and these and straight lines
are the only curves with constant curvature.
2
1.4. Length and
R area. If X : I ! R is an immersion, the length
L is defined by L = I @X
@u du. This is invariant under reparametrisations.
In the case where X : R ! R2 is a closed
curve then we define the length
R x+B
@X
by integrating over a single period: L = x
@u du for any x, where B
is the period of X.
If X is a simple closed curve then we can also consider the enclosed area
A, which is defined as follows (for a curve parametrised anticlockwise):
Z
⇥
⇤
1 x+B
(2.3)
A=
det X(u) X 0 (u) du.
2 x
This expression does not change if we reparametrise the curve, but changes
sign if we reverse the direction. It is sometimes useful to express this in a
slightly di↵erent way: If we choose an arc length parameter, then X 0 = T
and the expression becomes det[X T] = X · N (again assuming anticlockwise
parametrisation), so we obtain the following:
Z
Z
1
1 x+B
(2.4)
A=
det [X T] ds =
X · N ds.
2 M
2 x
For a simple closed curve the length always dominates the area, since the
isoperimetric inequality states that L2 4⇡ A, with equality precisely when
the curve is a circle.
Exercise 2.2. Let ⌦ be the region enclosed by a simple closed curve parametrised
anticlockwise by an embedding X : R/(AZ) ! R2 . Using the divergence theorem and the fact that the Rdivergence of the vector field V (x, y) = 12 (x, y) is
equal to 1, prove that A = ⌦ dxdy is given by expression (2.3).
1.5. Angle parameter.
Let X : I ⇢ R ! R2 be an immersed planar curve. Let the function
✓ : I ! R be a smooth choice of angle that the unit normal N makes with
the positive x-axis, so that
(2.5)
N(u) = (cos ✓(u), sin ✓(u))
for each u 2 I. Since N = JT we then also have T = ( sin ✓, cos ✓). The
possible choices for ✓ di↵er by integer multiples of 2⇡, and in particular the
10
2. CURVE SHORTENING AND GRAYSON’S THEOREM
derivative of ✓ is well defined. Di↵erentiating (2.5) and using the SerretFrenet equation (2.2) we obtain
@N
@
@✓
@✓
=
(cos ✓, sin ✓) = ( sin ✓, cos ✓)
=
T
@s
@s
@s
@s
and we conclude that
@✓
(2.6)
= .
@s
It follows from this that integrating the curvature with respect to arc length
on any interval gives the change in the angle ✓:
Z y
(2.7)
 ds = ✓(y) ✓(x).
T =
x
In the case of a closed curve X : M = R/(AZ) ! R2 , the angle parameter
can be defined on R but not in general on the quotient R/(AZ): Integrating
over one period gives
Z
Z x+A
 ds =
 ds = ✓(x + A) ✓(x).
M
x
Since (cos(✓(x+A)), sin(✓(x+A))) = N(x+A) =R N(x) = (cos(✓(x)), sin(✓(x)))
1
we have ✓(x+A) ✓(x) 2 2⇡Z. The integer 2⇡
M  ds is called the turning
number of the curve. The theorem of turning tangents or Umlaufsatz of H. Hopf [] states that the turning number of a simple closed curve
is either 1 or 1, depending on the direction of parametrisation.
2. Definition of curve shortening flow
The curve shortening flow is a geometric heat equation which can be
used to deform curves in the plane. The idea is that we start from a curve
given by a smooth (periodic) map X0 : M = (RR/AZ) ! R2 with X00 6= 0
everywhere (i.e. an immersion), and produce from it a family of such maps
X : M ⇥ [0, T ) ! R2 satisfying a certain law of motion.
One way to motivate the curve shortening flow is to try to write down
as naively as possible a heat equation for X. We want the flow to be independent of the parametrisation of the curve, and we can achieve this by
first transforming to a canonical choice of parametrisation: The arc length
parameter s. Then we can simply write down the heat equation in this
parametrisation for X:
@
@2X
X=
.
@t
@s2
Although this looks like the usual heat equation, it is not: The arc length
parameter is defined using the map X, and so changes as the curve changes.
So even though the equation looks linear, in fact it is not. We can see this
more clearly if we write the flow in terms of derivatives with respect to an
(2.8)
3. CURVE SHORTENING FLOW AS A GRADIENT FLOW
11
arbitrary fixed parametrisation, instead of the arc length parametrisation:
X0
@
1
0
Since T = |X
0 | and @s f = |X 0 | f for any f , we have
✓ 0 ◆0
@
1
X
X 00
X 00 · X 0 0
(2.9)
X=
=
X.
@t
|X 0 | |X 0 |
|X 0 |2
|X 0 |4
This has the structure of a quasilinear heat equation: The right-hand side
is a linear function of X 00 , but the coefficients depend on X 0 .
We can write the equation in a more geometric way using the SerretFrenet formulae: By definition we have T = @X
@s , and the Serret-Frenet
@2
@
equations (2.2) give Ts = N, so we have @s
N, and the
2 X = @s T =
curve-shortening flow equation becomes
@
X = N.
@t
That is, the curve moves in the normal direction with speed equal to the
curvature at each point.
(2.10)
3. Curve shortening flow as a gradient flow
Curve shortening flow and the related higher dimensional mean curvature flow arise quite often in applications. One reason is an underlying
variational principle: The curve shortening flow acts to decrease the length
of the curve at the fastest rate possible (relative to the total speed of motion
as measured in the square-integral sense). Let us make this precise: Given
an immersion X0 : M ! R2 , we can consider smooth variations of the form
d
X(u, t) = V (u),
dt
where V is an arbitrary (smooth) map from M to R2 . We can decompose
V into its normal and tangential parts: V = f N + AT. Now compute the
rate of change of the length of the curve under this variation:
Z
d
d
L=
|Xu | du
dt
dt M
Z ⌧
Xu @
=
, Xu du
M |Xu | @t
Z ⌧
@
=
T,
(f N + AT) ds
@s
ZM
=
hT, fs N + f T + As T ANi ds
ZM
=
(f  + As ) ds
ZM
=
f  ds,
M
12
2. CURVE SHORTENING AND GRAYSON’S THEOREM
since A(L) = A(0) by periodicity. If we fix the ‘total speed’ by prescribing
RL
2
0 |V | ds, then we have by the Cauchy-Schwarz inequality
Z
d
L=
f  ds
dt
M
✓Z
◆1/2 ✓Z
◆1/2
|V |2 ds
2 ds
,
M
M
with equality if and only if A is zero and f is a negative multiple of , or
equivalently V is a negative multiple of N. We say that curve shortening
flow is the gradient flow of the length functional. In particular under CSF
the length evolves by
Z
d
(2.11)
L=
2 ds.
dt
M
4. Invariance properties
The curve shortening flow is our first example of a geometric heat equation. We explore here precisely the sense in which it is ‘geometric’. In fact
there are several important ways in which the flow is invariant:
4.1. Invariance under isometries of the plane. An important invariance principle states that the curve shortening flow is invariant under
isometries of the ambient space: Here we allow M to be either an interval
in the real line or a quotient R/(AZ).
Proposition 2.3. Let X : M ⇥ [0, T ) ! R2 be a smooth solution of curve
shortening flow, and let A : R2 ! R2 be an isometry. Then A X is also
a solution.
Proof. Denoting by TX the unit tangent vector of X, we have
TA
X
(A X)0
|A X|
A⇤ (X 0 )
=
|A⇤ (X 0 )|
✓ 0 ◆
X
= A⇤
|X 0 |
= A⇤ (TX ) ,
=
where we denoted by A⇤ the derivative (or linear part) of the isometry A.
It follows that the arc length parametrisation of A X is the same as for X,
and so since A is an affine transformation (and so has zero second derivative)
we have
✓
◆
✓
◆
@
@
@
(N)A X =
T
=
(A⇤ (TX )) = A⇤
TX = A⇤ ((N)X ) .
@s
@s
@s
A X
4. INVARIANCE PROPERTIES
Also we have
@
@t (A
X) = A⇤
@
(A X) =
@t
@
@t X
13
. Therefore
A⇤ (X NX ) =
A
X NA
X
as required.
⇤
This includes invariance under translations, rotations and reflections.
4.2. Invariance under reparametrisation. Extremely important is
the invariance of the curve shortening flow under reparametrisations: Let
X : M ⇥ [0, T ) ! R2 be a smooth solution of the curve shortening flow.
Let ' : N ! M be a di↵eomorphism (in the case where N and M are
quotients, so that N = R/(BZ) and M = R/(AZ), then we require ' :
R ! R to be a di↵eomorphism such that '(x + B) = '(x) + A). Then
(x, t) 7! X̃(x, t) = X('(x), t) is again a solution of curve shortening flow.
To see this, we can work with the expression (2.9) for curve shortening flow: Since X̃ 0 (x, t) = '0 (x, t)X 0 ('(x), t)) by the chain rule, we have
TX̃ (x, t) = sgn('0 )TX ('(x), t)), and so
@ X̃
@X
(x, t) =
('(x), t);
@t
@t
1
@
1
0
T
=
sgn('0 )TX ('(x), t)
|X̃ 0 | X̃ (x,t) |'0 (x)X 0 ('(x), t)| @x
'0 sgn('0 )
= 0
(TX )0 ('(x), t)
|' (x)| |X 0 ('(x), t)|
✓ 0 ◆0
1
X
=
.
|X 0 | |X 0 |
('(x),t)
The two right-hand sides are equal since X is a solution of curve-shortening
flow, so the left-hand sides are also. Therefore X̃ satisfies curve-shortening
flow. In this sense we can think of curve shortening flow as a flow of regular
curves (equivalence classes of parametrised curves under reparametrisation).
Note that the reparametrisation has to be time-independent for this
to work. If we have a more general reparametrisation given by (x, t) 7!
('(x, t), t), then the curve shortening flow is transformed by the addition of
a tangential term: We get
!0 ✓
◆
@ X̃
1
X̃ 0
1
@'
(2.12)
=
+
(x, t) X̃ 0 (x, t).
0
@t
' (x, t) @t
|X̃ 0 | |X̃ 0 |
This has a useful converse: If we have a solution of an equation which is
curve shortening flow with an additional tangential term, then we can always
solve for a time-dependent reparametrisation ' to transform this to curve
shortening flow.
4.3. Time translations. A simpler invariance is time translation: If
we relabel time by adding any constant, then we still have a solution of curve
shortening flow.
14
2. CURVE SHORTENING AND GRAYSON’S THEOREM
4.4. Scaling spacetime. Another invariance which will prove very
useful later is the invariance of the flow under dilations of spacetime. The
flow is not invariant under simple dilations of space, since this changes the
curvature of the curve. It is also not invariant under dilations of the time
parameter. However, a balanced combination of the two does provide an
important invariance: If we dilate space, and also dilate the time parameter
in the right way, then the result still satisfies the curve shortening flow.
Let us make this precise: Suppose X : M ⇥ [0, T ) ! R2 is a solution
of mean curvature flow. Then for any > 0 we can define a rescaled flow
X : M ⇥ [0, 2 T ) ! R2 by
2
X (x, t) = X(x,
t).
To see that this is again a solution, we check both sides of equation (2.9):
On the left we have
@
X (x, t) =
@t
2 @X
@t
(x,
2 t)
.
On the right we have X 0 = X 0 , so
✓ 0 ◆0
✓ 0 ◆0
X
1
1
X
=
|X 0 | |X 0 |
|X 0 | |X 0 |
Therefore
@X
=
@t
1 @X
@t
=
1
1
|X 0 |
✓
X0
|X 0 |
◆0
=
1
|X 0 |
✓
X0
|X 0 |
◆0
as required, and X satisfies curve shortening flow. This rescaling is very
important: If the flow approaches a singularity where the curvature becomes
large, then we can ‘zoom in’ to bring the curvature down to a manageable
level, as long as we slow down the time parameter also. In this way we
can understand the structure of singularities by investigating the limits of
rescaled flows as we zoom in to the singularity. We will investigate this
further in the next chapter.
5. The shrinking circle solution
Some solutions of CSF can be written down without difficulty: Most
obviously, if the curve is a straight line then  = 0 everywhere and the curve
does not move, so we obtain a stationary solution. Slightly less immediate
is the solution starting from a circle: By symmetry (given that solutions for
given initial data are unique) the evolution can only give a circle with the
same centre at any later time. If we substitute the form
X(✓, t) = r(t)(cos ✓, sin ✓)
6. EVOLVING GRAPHS
15
then we have X✓ = r( sin ✓, cos ✓), so |X✓ | = r and
@r
@X
(cos ✓, sin ✓) =
@t
@t
✓
◆
1 @
X✓
=
|X✓ | @✓ |X✓ |
1 @
=
( sin ✓, cos ✓)
r(t) @✓
1
=
(cos ✓, sin ✓).
r(t)
p
Therefore @r
r(0)2 2t. This solution is therefore a
@t = 1/r, so r(t) =
circle which shrinks to its centre after finite time T = 12 r(0)2 .
16
2. CURVE SHORTENING AND GRAYSON’S THEOREM
into the curve shortening flow. So now we ask: If we insist that X is a
graphical embedding (of the form X(x, t) = (x, u(x, t))), how can we choose
u to evolve in such a way that the embedding satisfies (2.14) for some A?
Since the T component A does not matter, this equation is satisfied provided
@X
@u
@t · N = (0, @t ) · N = .
0
To proceed we must find N and  in terms of u: Since T = p(1,u )0 2 , our
1+(u )
0
convention T = JN gives N = p(u ,
1+(u0 )2
(2.15)
=
Proposition 2.4. Let A be a simply connected open subset of R ⇥ [0, T ),
and let u : A ! R be a smooth solution of the quasilinear heat equation
@u
u00
(2.13)
=
.
@t
1 + (u0 )2
Let X(x, t) = (x, u(x, t)) be the corresponding graph embedding. Then there
exists an open set B ⇢ R ⇥ [0, T ) and a di↵eomorphism (p, t) 2 B 7!
('(x, t), t) 2 A (unique up to precomposition with a time-independent di↵eomorphism) such that X̃(p, t) = X('(x, t), t) is a solution of curve shortening
flow.
Conversely, if B is a simply connected open set in R ⇥ [0, T ) and X :
B ! R2 is a solution of curve-shortening flow such that X 0 (p, t) · (1, 0) 6= 0
for all (p, t) 2 B, then there exists a unique di↵eomorphism ' : A ⇢
R ⇥ [0, T ) ! B and a smooth function u : A ! R satisfying (2.13) such
that X('(x, t), t) = (x, u(x, t)) for all (x, t) 2 A.
Proof. We will prove this in detail in a more general context when
we discuss the mean curvature flow in Chapter 6. Now we only give an
explanation of why the equation (2.13) is the correct one.
We computed in equation (2.12) the e↵ects of allowing a time-dependent
reparametrisation of the curve in curve shortening flow. By choosing the way
the parametrisation changes in time, we can transform any flow of the form
@X
(2.14)
= N + AT
@t
. The curvature of X is given by
N0
·T
|X 0 |
0
1
1
@ @ (u0 , 1) A
(1, u0 )
q
=p
·p
2
1 + (u0 )2 @x
1
+ (u0 )2
1 + (u0 )
6. Evolving graphs
The curve shortening flow is a system of partial di↵erential equations.
However we can reduce it locally to a scalar equation by writing the evolving curves as graphs. The fact that the equation is equivalent to one for an
evolving graph is closely related to the geometric invariance of the flow: If
the moving curve is a graph, then there is a time-dependent reparametrisation which makes the curve move in the vertical direction, preserving the
graphical parametrisation.
1)
=⇣
u00
1 + (u0 )2
⌘3/2 .
Our equation therefore becomes
@u
p @t
=
1 + (u0 )2
so that
=
u00
,
(1 + (u)2 )3/2
@u
u00
=
@t
1 + (u0 )2
⇤
p
p
Example 2.5. If u(x, t) =
r2 2t x2 for |x| < r2 2t and 0  t <
r2
2 (the lower half of a shrinking circle of initial radius r), then
x
u0 = p
,
r2 2t x2
2
r
2t
u00 =
,
(r2 2t x2 )3/2
2
r
2t
2
1 + u0 = 2
.
r
2t x2
as claimed.
00
u
so that 1+(u
0 )2 =
(2.13) holds.
p
1
.
r 2 2t x2
We also have
@u
@t
=
p
1
,
r2 2t x2
so equation
7. The grim reaper
A particular solution of the graphical curve-shortening flow is the following: We look for a solution that simply translates vertically at some speed
v. That is, we seek a solution of (2.13) such that
u(x, t) = w(x) + vt.
8. THE PAPERCLIP AND THE HAIRCLIP
17
18
2. CURVE SHORTENING AND GRAYSON’S THEOREM
Substituting this form into the equation gives
w00
= (arctan w0 )0 .
1 + (w0 )2
v=
The solution therefore has w0 = tan(v(x x0 )), which integrates to give
1
w=
log cos(v(x x0 )) + w0 .
v
Rearranging gives u(x, t) = v1 log cos(v(x x0 )) + v(t t0 ), or equivalently
(writing y for u(x, t))
(2.16)
e
v(y y0 )
= cos(v(x
v2 t
x0 ))e
Figure 1. The paperclip solution of CSF (with y plotted
horizontally and x vertically)
.
Of course we can reverse the sign of v to obtain a solution which translates
downwards:
(2.17)
ev(y
y0 )
= cos(v(x
x0 ))e
v2 t
.
This solution is called the ‘grim reaper’ curve.
8. The paperclip and the hairclip
It is a surprising fact that despite the nonlinear nature of the curve
shortening flow, the solutions in (2.16) and (2.17) can be combined to give
another pair of solutions:
(2.18)
cosh(v(y
y0 )) = cos(v(x
x0 ))e
v2 t
sinh(v(y
y0 )) = cos(v(x
x0 ))e
v2 t
and
(2.19)
.
Figure 2. The hairclip solution of CSF
⇡
2)
The first of these (restricted to the range |x| <
describes a simple closed
curve which shrinks to a point (becoming circular in shape) at t = 0. These
exist for all t < 0, and as t ! 1 they look like two copies of the grim
reaper, one translating upwards from y = 1, the other translating downwards from y = +1, and joined smoothly in between. In the physics literature this is called the ‘paperclip’ solution, and is has been referred to also as
the Angenent oval. This is a beautiful illustration of the main result we will
prove in this chapter: The theorem of Grayson, which states that any simple
closed curve — even one which is highly non-circular such as the hairclip
for t very negative — will evolve under CSF to a single point, with circular
shape as the final time is approached. The hairclip is also an important
example of an ancient solution of CSF and we will discuss it further in the
next chapter.
The second is called the ‘hairclip’, and exists for all t 2 R. As t ! 1
it converges to a horizontal line. As t ! 1 it looks like an infinite row
of grim reapers, alternating between translating up and translating down,
joined smoothly in between. It illustrates an important result of Ecker and
Huisken for graphs evolving by curve shortening flow (or more generally
mean curvature flow): If the initial data for the flow is an entire graph, then
the solution exists for all time, and under some conditions on the behaviour
at infinity it eventually settles down to the graph of a constant function.
We will discuss this and related results in Chapter 8.
The reader may check that these are in fact solutions of the curve shortening flow. One can do this by solving the equations for y in terms of x and
checking that it satisfies the graphical curve shortening flow equation (except of course at vertical points in the case of the paperclip), or by checking
more geometrically that the normal speed of motion of the curve agrees with
the curvature at each point.
9. Evolution of geometric quantities under CSF
We wish to compute the evolution of various geometric quantities associated to a curve evolving by the CSF. For future reference we compute
first in a more general context, of curves moving in their normal direction:
Suppose
(2.20)
@X
=
@t
N.
9. EVOLUTION OF GEOMETRIC QUANTITIES UNDER CSF
19
For a function f , let ft +
By Clairaut’s theorem on the equality of
mixed partial derivatives, for any C 2 function f : M1 ⇥ [0, T ) ! R we have
@2f
@2f
@t@u = @u@t , i.e., fut = ftu for short. On the other hand, this formula does
not hold in general when we replace u by an arc length parameter s since
in general s will depend on time. Instead, we have the following.
@f
@t .
Lemma 2.6. Let X be a curve evolving by (2.20). For any C 2 function f :
M1 ⇥ [0, T ) ! R we have
fst = fts +  fs .
In other words, the commutator of the di↵erential operators

@ @
@2
@2
@
,
+
=  .
@t @s
@t@s @s@t
@s
1 @
@u ,
= @X
@u
⇣
fst = |Xu |
Proof. Using
@
@s
=
|Xu |
= |Xu |
=
=
1
3
fu
3
⌘
and
@
@s
and that the normal angle evolves by
(2.26)
✓ t = s .
Lemma 2.9 (Evolution of  under CSF). Under the curve shortening flow,
the curvature evolves by the heat-type equation:
t = ss + 3 .
t = ✓st = ✓ts + 2 ✓s
= ss + 3 .
⇤
10. Evolution of area
1
ftu
h Nu , Xu i fu + fts
⇤
and
@
@s
under CSF). Under the curve
fst = fts + 2 fs .
Conclude from this that the time derivative of the 1-form ds is
@
(2.22)
(ds) =
 ds.
@t
The same formula as (2.21) holds for Euclidean vector-valued functions,
so that we have
Xst = Xts + 2 Xs .
That is, under the CSF, the unit tangent vector field evolves by
(2.24)
hN, Ti = 0 we easily derive that
Proof. Using ✓s = , ✓t = s , and (2.21), we compute
hNs , Ti fs + fts
 fs + fts
@
@t
d
dt
N t = s T
(2.25)
(2.27)
is
Exercise 2.8. Let X be a curve evolving by (2.20). Show that the time
@
derivative of @s
, as a vector field, is given by
✓ ◆
@
@
@
=  .
@t @s
@s
(2.23)
since Ns = T by (2.2). From this and
the unit normal evolves by
t
hXtu , Xu i fu + |Xu |
Corollary 2.7 (Commutator of
shortening flow,
2. CURVE SHORTENING AND GRAYSON’S THEOREM
we compute that
since hN, Xu i = 0 and hNs , Ti = .
(2.21)
@
@t
20
Tt = (  N)s + 2 T
= s N
For a simple closed curve X : M ⇥ [0, T ) ! R2 parametrised anticlockwise and moving under CSF we can compute the evolution of the enclosed
area A by di↵erentiating the expression (2.3):
Z
dA
1
=
det [Xt Xu ] + det [X Xut ] du
dt
2 M
Z
1
=
det [Xt Xu ] + @u (det [X Xt ]) det [Xu Xt ] du
2
Z M
=
det [Xt Xu ] du
M
Z
=
 det [N T] |Xu | du
ZM
=
 ds
(2.28)
=
M
2⇡.
Here the second term in the second line integrated to zero, and we used the
antisymmetry of the determinant to reach the third line. In the fourth line,
our convention T = JN gives det[N T] = 1, and we have ds = |Xu | du. The
last equality is the Umlaufsatz (see section 1.5).
11. Smoothing in curve shortening flow
One of the most important properties of geometric heat equations (and
parabolic equations more generally) is smoothing: Even for highly irregular
initial data, solutions become smooth (infinitely many times di↵erentiable)
at any positive time. In this section we will show how to prove this (assuming
only a bound on the curvature) for curve shortening flow using maximum
11. SMOOTHING IN CURVE SHORTENING FLOW
21
principle estimates. A very similar procedure can be applied to other flows,
such as the higher-dimensional mean curvature flow and the Ricci flow.
We start with the first derivative of curvature estimate. The method
of proof is to apply the maximum principle to a suitable first derivative
quantity.
22
2. CURVE SHORTENING AND GRAYSON’S THEOREM
Note that Z0 = 3 . Then
(m+1)
(2.35)
=
=
Lemma 2.10. Suppose that we have a solution to the CSF on a closed curve
with
(2.29)
where t0 2 (0, K
2]
||  K
t
Z2 = 52 (2) + 8((1) )2 ,
Z3 = 62 (3) + 26(1) (2) + 8((1) )3 .
Exercise 2.11. Prove by induction on m that for m 1,
X
(2.36)
Zm = (m + 3) 2 (m) +
ca1 ,a2 ,a3 (a1 ) (a2 ) (a3 ) ,
= 2 ss + 3
= 2
2 (s )2 + 24 .
ss
Next, di↵erentiating (2.27) with respect to s while using the commutator
formula (2.21) yields
(s )t = ts + 2 s
(2.32)
= (s )ss + 42 s .
Thus
((s )2 )t = 2s (s )ss + 42 s
(2.33)
= ((s )2 )ss
where the ca1 ,a2 ,a3 are nonnegative integers depending only on a1 , a2 , a3 and
where the sum is over 0  a1  a2  a3  m 1 with a1 + a2 + a3 = m.
In the rest of this section constants depend only on their indices.
Theorem 2.12 (Derivative estimates for CSF). For each m 1 there exists
Cm < 1 such that if ||  K on M1 ⇥ [0, t0 ), where t0 2 (0, K 2 ] and
K < 1, then
Cm K
|(m) |  m/2
t
on M1 ⇥ (0, t0 ).
Proof. We will prove by induction on m that there exist positive constants {bm,i }1im and {Am }m 1 such that the expression
2 (ss )2 + 82 (s )2 .
From (3.72) and (3.74) we compute for any constant b > 0 that
(t (s )2 + b2 )t
(t (s )2 + b2 )ss =
2t (ss )2 + 8t2 + 1
Choosing b = 5 and by (3.70) and t0  K
(2.34)
(t (s )2 + 52 )t
2,
on M1 ⇥ [0, t0 ), which implies (3.71).
(m)
ss .
m
X
bm,i ti ((i) )2
i=0
(2.37)
2t (ss )2 + 10K 4
(
m )t
(
m )ss
⇤

2am tm ((m+1) )2 + Am K 4 ,
where am = bm,m . Firstly, let 1 = t (s )2 + 52 . By (2.34),
(2.37) with a1 = 1 and A1 = 10. Then define recursively
m
Next we consider the higher derivative estimates for the CSF. Given a
m
function f on M1 ⇥ [0, t0 ), let f (m) + @@smf for m 0. Define
(m)
+
satisfies on M1 ⇥ [0, t0 ) the inequality
on M1 ⇥ [0, t0 ). At t = 0 we have t (s )2 + 52  5K 2 . Thus, by the
maximum principle we conclude that
t (s )2 + 52  15K 2
m
2b (s )2 + 2b4 .
we obtain
(t (s )2 + 52 )ss 
Z m + t
(m+1)
+ 2 (m)
ss
s
Z1 = 42 (1) ,
4K
|s |  p .
t
2
(m+1)
ss
Using (3.75), we calculate for example
on M1 ⇥ [0, t0 ),
Proof. First note that from t = ss + 3 we may derive
(2.31)
(m)
st
(m)
ts
= (Zm )s + 2 (m+1) .
and K < 1. Then on M1 ⇥ (0, t0 )
(2.30)
(m+1)
ss
Zm+1 = t
= am tm ((m) )2 +
1
satisfies
m 1.
Assume by induction that ti/2 (i)  Ci K on M1 ⇥ [0, t0 ) for i  m and
that (2.37) holds. Using this and (2.36) we compute that
(
m+1 )t
(
m+1 )ss

m + 1 + 2 (m + 4) t2 am+1
2am tm ((m+1) )2
+ 2Bm am+1 K 3 t(m+1)/2 (m+1) + Am K 4
2am+1 tm+1 ((m+2) )2
12. SHORT TIME EXISTENCE
for some Bm < 1. Choose am+1 =
(
m+1 )t
(
m+1 )ss


2am+1 t
2am
3(m+4) ,
m+1
(
23
2. CURVE SHORTENING AND GRAYSON’S THEOREM
so that
(m+2) 2
) +
2am+1 tm+1 ((m+2) )2 +
2
Bm
am+1 K 6 t
Am+1 K 4 ,
13. Global existence
+ Am K
4
2 a
where Am+1 = Bm
m+1 + Am . That is, (2.37) holds with m replaced by
m + 1. Since m+1  bm+1,0 K 2 at t = 0, by the maximum principle we have
am+1 tm+1 ((m+1) )2 
24
 bm+1,0 K 2 +Am+1 K 4 t  (bm+1,0 + Am+1 ) K 2 ,
q
bm+1,0 +Am+1
so that t(m+1)/2 (m+1)  Cm+1 K on M1 ⇥[0, t0 ), where Cm+1 =
.
am+1
The theorem follows using induction.
⇤
m+1
12. Short time existence
Our analysis of the curve shortening flow requires the following short
time existence theorem:
Theorem 2.13 (Short time existence for the CSF). Let M1 be a closed
1-manifold and let X0 : M1 ! R2 be a smooth immersion. Then there
exists " > 0 and a smooth solution X (t) : M1 ! R2 to the CSF defined for
t 2 [0, ") and satisfying X (0) = X0 .
We will not prove this result here (see Chapter ?? for the corresponding
result for higher dimensional mean curvature flow, of which this is a special
case). We mention only some of the issues which arise: The equation (??)
is not strictly parabolic, since in a general parametrisation it becomes
@X
1
=
⇡N (X 00 )
@t
|X 0 |2
where ⇡N (v) = (v ·N)N is the orthogonal projection onto the subspace generated by the unit normal N. Since there is no dependence on the component
of X 00 tangent to the curve, the symbol is degenerate and the flow not parabolic. This means that standard existence theorems for partial di↵erential
equations cannot be applied. As we will see later, the degeneracy is a direct
consequence of the invariance of the flow under reparametrisation. There
are two approaches to overcoming this degeneracy: The first is to write the
evolving curves as graphs over some fixed curve (such as the initial curve, or
a nearby smooth curve if the initial curve is not sufficiently regular). This
produces an evolution equation for a scalar function, which is strictly parabolic. One can then produce from the graphical solution a solution of the
original curve shortening flow. The second approach is to apply the NashMoser inverse function theorem. For this approach (which applies also for
the higher dimensional mean curvature flow) see §2 of Gage and Hamilton
[90].
The following result, essentially due to Gage and Hamilton [90], is the
basic long-term existence theorem for curve shortening flow.
It is a special case of the corresponding result for mean curvature ??,
and a very similar result also holds for Ricci flow ??
Theorem 2.14. Let X : M ⇥ [0, T ) ! R2 be a smooth solution of CSF.
If supM⇥[0,T ) || = K < 1 then there exists > 0 such that the solution X
extends smoothly to M ⇥ [0, T + ). In particular, if T is the maximal time
of existence then lim supt!T supM⇥{t} || = 1.
Proof. The smoothing estimates of section 11, together with the short
time existence result, provide bounds in C k on , for any k. To prove the
global existence theorem we will show that the maps X(., t) converge in C 1
to a limit X(., T ) as t approaches T , and then apply the short time existence
result starting from X(., T ) to extend to a longer time interval.
Our first step is therefore to control all of the derivatives of the map X
under the curve shortening flow. We begin with a very simple supremum
bound:
Z t
@X
(2.38)
|X(x, t) X(x, 0)| 
(x, ⌧ ) d⌧  K|t|  KT.
@t
0
Note that the derivatives of the map X are not invariant under reparametrisation, so we fix a parameter x (for convenience, the arc length parameter
for the curve when t = 0) and control all of the derivatives of X with respect
to this fixed parametrisation.
A crucial step is to bound on the first derivatives of X both above and
below (the lower bound ensures that the limiting map is an immersion).
Lemma 2.15. For any x 2 M and t 2 [0, T ),
exp{ K 2 T }  X 0 (x, t)  1.
Proof. We compute directly
◆
@X 0
@t
@
= T · |X 0 | ( N)
@s
= |X 0 |T · s N 2 T
@
X0
|X 0 | =
·
@t
|X 0 |
(2.39)
=
✓
2 |X 0 |,
@
1
0
@s (·) = |X 0 | (·) . This gives
0
log |X (x, t)| log |X 0 (x, 0)|
where we used equation (2.2), and the relation
0
@
@t
K 2t
log |X 0 |
|X 0 (x, 0)|
K 2,
and by integration 0
K 2 T . The result follows since by our choice of coordinate we have
= 1 for all x.
⇤
13. GLOBAL EXISTENCE
25
Finally we bound the higher derivatives of X. In the following we denote
by X (k) the kth derivative of X with respect to the fixed spatial parameter,
0
so that f (0) = f and f (k) = f (k+1) for any function f . To establish bounds
on X (k) we first derive an evolution equation. This would be very messy to
write down precisely, but all we need is a general form:
Lemma 2.16. For each k
N·X
(k+1)
=
,...,pk
Apq01,...,q
k
1
k ⇣
Y
i=1
p1 +2p2 +···+kpk =k+1
1+q0 +2q1 +···+kqk 1 =p1 +···+pk
T·X
(i)
⌘pi kY1
q
@sj  j
;
j=0
and furthermore under curve shortening flow we have
X
@X (k)
=
@t
,...,pk
Cqp01,...,q
k
p1 +2p2 +···+kpk =k
q0 +2q1 +···+(k+1)qk =1+p1 +···+pk
+
X
,...,pk
Dqp01,...,q
k
p1 +2p2 +···+kpk =k
q0 +2q1 +···+(k+1)qk =1+p1 +···+pk
k ⇣
Y
i=1
k ⇣
Y
i=1
k
⌘p i Y
@sj 
qj
k
⌘p i Y
@sj 
qj
T · X (i)
T · X (i)
N
j=0
T,
j=0
,...,pk
p1 ,...,pk
p1 ,...,pk
where Apq00,...,q
k 1 , Cq0 ,...,qk and Dq0 ,...,qk are constants.
Remark 2.17. These expressions, while convenient for the proof, look rather
ugly. The reader will find it instructive to write out the meaning of the expressions in first few cases. For example, when k = 1 the only possibility in
the expression for N · X 00 is p1 = 2 and q0 = 1, yielding N · X 00 = A21 |X 0 |2 
(in fact by the definition of curvature we have N · X 00 = |X 0 |2 ). For small
values of k the expressions become
3,0
1,1
0 3 2
0 3
0
00
N · X 000 = A3,0
2,0 |X |  + A0,1 |X | s + A1,0 |X |T · X ;
4,0,0
4,0,0
0 4 3
0 4
0 4
N · X (4) = A4,0,0
3,0,0 |X |  + A1,1,0 |X | s + A0,0,1 |X | ss
2,1,0
0 2
00 2
0 2
00
+ A2,1,0
2,0,0 |X | T · X  + A0,1,0 |X | T · X s
0,2,0
0
000
00 2
+ A1,0,1
1,0,0 |X |T · X  + A1,0,0 (T · X ) .
3|X 0 |T · X 00 
|X 0 |3 s
and
N · X (4) =
|X 0 |4 (33 + ss )
and
⌘
⇣
⌘⌘
@X 00 ⇣ 0 2 ⇣ 2,0 3
2,0
2,0
0,1
0,1
s + C0,0,1
ss + T · X 00 C2,0,0
2 + C0,1,0
s N
= |X | C3,0,0  + C1,1,0
@t
⇣
⇣
⌘
⇣
⌘⌘
2,0
2,0
2,0
0,1
0,1
+ |X 0 |2 D3,0,0
3 + D1,1,0
s + D0,0,1
ss + T · X 00 D2,0,0
2 + D0,1,0
s T.
6|X 0 |2 T · X 00 s
3(T · X 00 )2 
@X 0
=
@t
|X 0 |s N
|X 0 |2 T
and
@X 00
= |X 0 |2 3 |X 0 |2 ss T · X 00 s N
3|X 0 |2 s + T · X 00 2 T.
@t
We see from these that the expressions in the lemma are rather coarse as
not all of the allowed terms arise, but they are sufficient for our purposes.
In fact, inspecting the proof below we could have used a much weaker form:
@X (k)
(k) +Q, where P =  N 2 T and Q is a polynomial
s
@t has the form P ·X
in T · X (j) for j < k and @sj  for j  k.
Proof. The proof is a straightforward induction on k: In the case k = 1
both expressions are clearly true (we verified these in the remark above).
Now suppose we have established both expressions for k = 1, . . . , m. Then
we write
N · X (m+2) = (N · X (m+1) )0
N0 · X (m+1) = (N · X (m+1) )0
|X 0 |T · X m+1 ,
where we used the relation f 0 = |X 0 |fs for any function f , and the SerretFrenet equation. In the first term we substitute the expression for N·X (m+1)
coming from the inductive hypothesis. Di↵erentiating a factor T · X (i) gives
T · X (i+1) |X 0 |N · X (i) . In the second of these terms we substitute the
expression for N · X (i) coming from the inductive hypothesis. We leave it to
the reader to check that all of the resulting terms are of the form allowed in
the claimed expression for N · X (m+2) . Similarly, di↵erentiating a term @sj 
gives |X 0 |@sj+1 , which again is of the form allowed, and we have completed
the induction. The argument for the evolution equation is similar.
⇤
We will now prove bounds on X (k) :
Lemma 2.18. For each k there exist constants Ak and Bk such that |X (k) | 
@
Ak and @t
X (k)  Bk .
One can compute these exactly, obtaining:
N · X 000 =
2. CURVE SHORTENING AND GRAYSON’S THEOREM
Computing these exactly we find
1 we have
X
26
4|X 0 |T · X 000 .
Similarly in the evolution equations we get the expressions
@X 0
1
1
1
1
= (C2,0
|X 0 |2 + C0,1
|X 0 |s )N + (D2,0
|X 0 |2 + D0,1
|X 0 |s )T
@t
Proof. We prove the the two statements simultaneously by induction
on k. The statement is clearly true for k = 0 by the curvature bound and
equation (2.38). We suppose the bounds on X (k) hold for k = 1, . . . , m, and
consider the evolution equation for X (m+1) from Lemma 2.16: This gives
(using the bounds from the inductive hypothesis)
@ (m+1)
X
 C1 X (m+1) + C2 ,
@t
14. THE AVOIDANCE PRINCIPLE
27
|X (m+1) |
from which bounds on
(and hence also the time derivative) follow
since the time interval is finite. This completes the induction and proves the
lemma.
⇤
We now complete the proof of convergence as t approaches T : We use
the fact that C k (R/(AZ), R2 ) is complete. The bound on the time derivative
gives
kX(., t2 ) X(., t1 )kC k  C|t2 t1 |
for any t1 and t2 in [0, t0 ]. In particular, for any sequence tn approaching
T , the sequence X(., tn ) is Cauchy in C k , and hence converges to a limit
X(., T ). But then we also have
kX(., t)
X(., T )kC k  C(T
t),
proving that X(., t) converges to X(., T ) in C k .
Finally, we apply the short time existence result (Theorem 2.13) to prove
the theorem. We leave it to the reader to check that if we extend the solution
to the time interval [T, T + ) in this way then the resulting map X is smooth
and is a solution of curve shortening flow on M ⇥ [0, T + ).
⇤
14. The avoidance principle
There is a very nice geometric principle which holds for solutions of curve
shortening flow:
Theorem 2.19. Let Xi : M1i ⇥ [0, t0 ] ! R2 be smooth solutions of CSF
such that X1 (x, 0) 6= X2 (y, 0) for all x 2 M1 and y 2 M2 . Then X1 (x, t) 6=
X2 (y, t) for all x 2 M1 , y 2 M2 , and t 2 [0, t0 ].
Proof. In fact we will prove something stronger: The length of the
shortest line segment joining the two curves is not decreasing in time. The
simple geometric reason behind this is as follows: If we take the shortest
such line segment and rotate the picture to make this vertical, then the
curvature at the ‘top’ point must be more in the upward direction than the
curvature at the ‘bottom’ point (otherwise nearby vertical segments have
shorter length), and so the curves are moving apart at the endpoints (see
the inequality (2.42) below).
To demonstrate this we will apply a maximum principle argument, but
unlike our previous maximum principle arguments we will work with a function defined not just on an evolving curve but on a product of two curves:
That is, a function of a pair of points (x, y) 2 M1 ⇥ M2 . This is a useful
technique which we will exploit below in other situations.
Define d : M1 ⇥ M2 ⇥ [0, t0 ] ! R as follows:
d(x, y, t) = |X2 (x, t)
X1 (y, t)| .
Since M1 and M2 are compact, we have d0 = inf{d(x, y, 0) : (x, y) 2
M1 ⇥ M2 } > 0. We will prove that d(x, y, t) d0 for all (x, y, t) 2 M1 ⇥
M2 ⇥ [0, t0 ], by proving that de"(1+t) > d0 for any " > 0.
28
2. CURVE SHORTENING AND GRAYSON’S THEOREM
At t = 0 we have de"(1+t)
d0 e" > d0 . Suppose it is not true that
d(x, y, t)e"(1+t) > d0 for all (x, y, t) 2 M1 ⇥ M2 ⇥ [0, t0 ]. Then since
inf{d(x, y, t)e"(1+t) : (x, y) 2 M1 ⇥ M2 } is continuous in t, there is a first
time t1 2 (0, t0 ] such that inf{d(x, y, t1 )e"(1+t1 ) : (x, y) 2 M1 ⇥ M2 } = d0 .
Since M1 ⇥ M2 is compact, there exists (x1 , y1 ) 2 M1 ⇥ M2 such that
d(x1 , y1 , t1 ) = d0 . We will derive a contradiction by considering derivatives of d at the point (x1 , y1 , t1 ). Specifically, we have the inequalities
@
"(1+t)
 0, the first derivatives on M1 ⇥ M2 vanish, and
@t de
(x1 ,y1 ,t1 )
the matrix of second derivatives is non-negative definite at this point. Our
computations are made simpler by exploiting the parametrisation-invariance
of the curve shortening flow: We choose our parametrisation of Mi to be
an arc length parametrisation induced by the immersion X1 (., t1 ) (this still
leaves us freedom to reverse the direction of parametrisation, a fact which
we will exploit below).
The vanishing of the first spatial derivatives give the following identities:
(2.40)
0=
@d
=
@x
hw, T1 i ;
0=
@d
= hw, T2 i ,
@y
X2 X1
where w = |X
is the unit vector pointing from X1 (x1 , t1 ) to X2 (y1 , t1 ),
2 X1 |
and Ti is the unit tangent vector to the parametrised curve Xi (., t1 ) at xi ,
for i = 1, 2. It follows that both T1 and T2 are orthogonal to the unit vector
w. By reversing the direction of parametrisation of Mi if necessary, we can
ensure that ⌫2 = ⌫1 = w (hence also T1 (x1 , t1 ) = T2 (y1 , t1 )). Finally, we
compute the second derivatives
(2.41)
@2d
hT1 w · T1 w, T1 i
=
hw, 1 N1 i ;
@x2
|X1 X2 |
2
@ d
hT2 w · T2 w, T2 i
=
+ hw, 2 N2 i ;
@y 2
|X1 X2 |
@2d
hT1 w · T1 w, T2 i
=
.
@x@y
|X1 X2 |
Here we used the Serret-Frenet equation (2.2) to di↵erentiate the unit tangent vectors arising in (2.40). Using the identities T1 = T2 and Ni = w,
these become
@2d
1
=
@x2
d
1 ;
@2d
1
= + 2 ;
@y 2
d
@2d
=
@x@y
1
.
d
In particular the non-negativity of the Hessian implies the following:
(2.42)
0
✓
@
@
+
@x @y
◆2
d = 2
1 .
14. THE AVOIDANCE PRINCIPLE
29
The inequality from the time derivative is as follows:
@ ⇣ "(1+t) ⌘
0 e "(1+t)
de
@t
⌧
X2 X1
= "d +
, 1 N 1 + 2 N 2
|X2 X1 |
> hw, 1 N1 + 2 N2 i
= 2  1
0,
a contradiction, where in the last step we applied the inequality (2.42). This
contradicts our assumption that de"(1+t) decreases to d0 at some time, so
we have proved that de"(1+t) > d0 for each " > 0, and hence d
d0 on
M1 ⇥ M2 ⇥ [0, t0 ].
⇤
Remark 2.20. In the proof there is an important point which will reappear
several times in the discussion below: Since X2 and X1 both evolve according
to the curve shortening flow it is not surprising that the quantity d we construct from them also satisfies a heat-type equation on M1 ⇥ M2 . However
it is important that we do not try to make it too much like the usual heat
equation: Essentially we showed in the proof above that
✓
◆
@d
@
@ 2
=
+
d (modulo gradient terms),
@t
@s1 @s2
(see also Exercise 2.21 below) where s1 and s2 are the arc length parameters
on M1 and M2 . This is not the heat equation on M1 ⇥ M2 in the induced
metric, but rather a degenerate heat equation with non-trivial di↵usion coefficient in only one tangent direction @s@ 1 + @s@ 2 . We can of course write d
as a solution of something more like a standard heat equation:
✓ 2
◆
@d
@
@2
2
=
+ 2 d
(modulo gradient terms),
2
@t
d
@s1 @s2
but then the maximum principle cannot be applied to keep d greater than d0 .
To put this another way: The non-negativity of the Hessian of d as a function
on M1 ⇥ M2 contains more information than just the non-negativity of the
Laplacian, and it is crucial in the argument that this be exploited.
Exercise 2.21. Show that d satisfies the following degenerate heat-type
equation on M1 ⇥ M2 :
✓ 2
◆
@d
@
@2
@2
=
+
+ 2T1 · T2
d
@t
@s1 @s2
@s21 @s22
!
✓
◆2 ✓
◆2
1
@d
@d
@d @d
+
2T1 · T2
,
d
@s1
@s2
@s1 @s2
where s1 and s2 are arc length parameters on M1 and M2 respectively.
Theorem 2.19 then follows by applying the maximum principle.
30
2. CURVE SHORTENING AND GRAYSON’S THEOREM
Corollary 2.22. If X0 : R/(AZ) ! R2 is an immersion, then the maximal
time of existence for the curve shortening flow with initial data X0 is finite.
Proof. Let X : R/(AZ) ! R2 be the solution of curve shortening
flow with initial data X0 . Since R/(AZ) is compact, the image of X0 is
contained in some large ball BR (0) in R2 . In particular X0 disjoint from
the circle X̃0 (✓) = (R cos ✓, R sin ✓). By the avoidance principle this remains
true for positive times, and so p
X(M, t) is enclosed by the circle given by
the image of the map X̃(✓, t) = R2 2t (cos ✓, sin ✓). The enclosed region
disappears at time R2 /2, so the time of existence can be no greater than
this.
⇤
Exercise 2.23. Define f : M ⇥ [0, T ) ! R by f (x, t) = |X(x, t)|2 . Derive
the evolution equation
@f
@2f
=
2.
@t
@s2
Use this to give another proof of Corollary 2.22.
15. Preserving embeddedness
Next we prove a result which is closely related to the avoidance principle:
The preservation of embeddedness under the curve shortening flow. While
the avoidance principle states that two curves which are initially disjoint
cannot cross each other, the preservation of embeddedness says that a curve
which does not initially cross itself will not develop self-intersections.
Theorem 2.24. Let X : M⇥[0, t0 ] ! R2 be a smooth solution to the curve
shortening flow, and suppose that X(., 0) is an embedding. Then X(., t) is
an embedding for each t 2 [0, t0 ].
Proof. Since M is compact, X(., t) is an embedding if and only if X is
injective and has injective derivative. The fact that the derivative remains
injective if the curvature is bounded was established in the proof of the
global existence theorem (see Lemma 2.15). We prove here that X(., t) is
injective for t 2 [0, t0 ]: For all x, y 2 M with y 6= x, we must prove that
X(y, t) 6= X(x, t).
The proof is very similar to that for Theorem 2.19, but there is a complication since the function d can no longer be expected to be positive on a
compact manifold without boundary: d : M ⇥ M ⇥ [0, t0 ] ! R is given by
d(x, y, t) = |X(y, t)
X(x, t)|.
However d is zero on the ‘diagonal’ subset {(x, x) : x 2 M}, so we cannot
hope to show that d remains positive on M ⇥ M. Instead we will first use
the fact that the curvature is bounded to control d on a neighbourhood of
the diagonal, and then apply the maximum principle to show that d remains
positive on the remaining region.
15. PRESERVING EMBEDDEDNESS
Lemma 2.25. If X :
x 2 M, then
M1
|X(y)
for all x, y with `(x, y) 
!
R2
is an immersion with |(x)|  K for all
2
sin
K
X(x)|
⇡
K,
31
✓
K`(x, y)
2
◆
where `(x, y) is the arc length from x to y.
Proof. Let x and y be two points on the curve joined by a curve seg⇡
ment of length `  K
. Let s be an arc length parameter with s(x)
R s = `/2
and s(y) = `/2. For any s 2 [ `/2, `/2] we have |✓(s) ✓(0)|  0  ds 
⇡
K|s|  K`
2  2 . We compute
Z s/2
(X(y) X(x)) · T(0) =
T(s) · T(0) ds
=
Z
Z
=
s/2
`/2
cos(|✓(s)
✓(0)|) ds
`/2
`/2
cos(K|s|) ds
`/2
2
sin
K
✓
K`
2
◆
since cos(x) is decreasing on [0, ⇡/2]. Since |X(y)
T(0) the result follows.
,
X(x)|
(X(y)
X(x)) ·
⇤
Now we can complete the proof that X(., t) is injective: Since M ⇥ [0, t0 ]
is compact, there exists a constant K such that |(x,
⇣ t)|  K⌘ for all x 2 M
and t 2 [0, t0 ]. We have |X(y, t) X(x, t)| K2 sin K`(x,y,t)
> 0 whenever
2
⇡
0 < `(x, y, t)  K
by Lemma 2.25. Now we apply the maximum principle
⇡
to d on the set S = {(x, y, t) 2 M ⇥ M ⇥ [0, t0 ] : `(x, y, t)
K }. On
⇡
2
the spatial boundary of S we have inf{d(x, y, t) : `(x, y, t) = K
}
K by
Lemma 2.25. The argument of Exercise 2.21 shows that on S we have
✓ 2
◆
@d
@
@2
@2
=
+ 2 + 2Tx · Ty
d
@t
@s2y
@sx
@sy @sx
!
✓
◆2 ✓
◆2
1
@d
@d
@d @d
+
2Tx · Ty
,
d
@sy
@sx
@sy @sx
and the maximum principle applies to give
⇢ n
d(x, y, t) min inf d(x, y, 0) : `(x, y, 0)
⇡o 2
,
K
K
.
⇡
Since X(., 0) is injective we have inf{d(x, y, 0) : `(x, y, 0)
K } > 0. This
establishes that d(x, y, t) > 0 for all y 6= x and t 2 [0, t0 ], so X(., t) is
injective for each t 2 [0, t0 ] and the theorem is proved.
⇤
32
2. CURVE SHORTENING AND GRAYSON’S THEOREM
16. Huisken’s distance comparison estimate
Next we will consider a refinement of the above argument due to Gerhard
Huisken [124]. As in the argument above we will derive a positive lower
bound on d(x, y, t) = |X(y, t) X(x, t)| for y 6= x, but this time the lower
bound will not depend explicitly on a bound on curvature as in the argument
above. Instead we will show that d can be bounded from below in terms of
the arc length `(x, y, t) and the total length of the curve.
The motivation for the precise estimate proved by Huisken comes from
a consideration of the shrinking circle solution. Why is this important? If
an estimate is to be useful for analysing singularities, it should be scalinginvariant. But the shrinking circle solution is itself scaling-invariant (selfsimilar), so any scaling-invariant quantity is constant in time. If the maximum principle applies to preserve an inequality, then the strong maximum
principle says that the inequality becomes strict unless the quantity is constant along the directions where the di↵usion coefficient is non-trivial. For
this reason it makes sense to look for quantities that are constant on circles.
There is essentially a unique identity relating the distance d to the arc
length ` and the total length L for a pair of points on a circle: If ✓ is the
angle subtended at the centre by the two points, then we have `/L = ✓/2⇡
and d/L = 2 sin(✓/2)/2⇡. Rearranging this we find
✓ ◆
L
⇡`
(2.43)
d = sin
.
⇡
L
Huisken proved the following:
Theorem 2.26. Let X : M2 ⇥ [0, T ) ! R2 be a smooth solution of
the curve shortening flow, such that X(., 0) is an embedding. Define Z :
M1 ⇥ M1 \ {(x, x) : x 2 M} ⇥ [0, T ) ! R by
✓
◆
L(t)
⇡`(x, y, t)
Z(x, y, t) =
sin
,
d(x, y, t)
L(t)
where d(x, y, t) = |X(y, t) X(x, t)|, `(x, y, t) is the arc length from x to y at
time t, and L is the total length of the curve X(M, t). Then sup{Z(x, y, t) :
x, y 2 M} is non-increasing in t, strictly unless the curve is a circle.
Note that this result implies in particular that embeddedness is preserved, with estimates that do not depend on any explicit curvature bound.
Proof. We apply the maximum principle to Z as a function on M1 ⇥
M1 ⇥ [0, T ). To understand the behaviour of Z near the diagonal {(x, x) :
x 2 M} we begin with a simple observation:
Lemma 2.27. If X(., t) is embedded then Z is smooth, and extends to a
continuous function on M1 ⇥ M1 ⇥ [0, T ) with Z(x, x, t) = ⇡.
Proof. Embeddedness implies that d(x, y, t) 6= 0 when x =
6 y in M.
Since d2 is smooth and d is positive, d is also, and Z is smooth on M1 ⇥ M1 \
16. HUISKEN’S DISTANCE COMPARISON ESTIMATE
33
{(x, x) : x 2 M}. We proceed to show continuity on the diagonal: Clearly
the arc length cannot be less than the chord
so we have d  `.
⇣ distance,
⌘
K`(x,y)
2
By Lemma 2.25 we also have d
for `  ⇡/K, where
K sin
2
1 3
6x
K = supM ||. Since x
✓ ◆
L
⇡`
Z = sin

d
L
for ` < ⇡/K, while
L
K
 sin(x)  x, we have
L
2
K
⇣
It follows that |Z(x, y, t)
only on K and L.
sin
⇡`
L
⇡`
L
K`
2
1
6

⇡` 3
L
`
2
K
⌘
⇣
⇡`
K`
2
✓
=⇡ 1
1
6
K` 3
2
⌘=
⇡
1
K2 2
24 `
◆
⇡2 2
` .
6L2
⇡|  C`(x, y, t)2 for ` < r, where C and r depend
⇤
This is complemented by the following:
Lemma 2.28. sup{Z(x, y, t) : x, y 2 M} > ⇡ if X(M, t) is not a circle.
Proof. Let K be the maximum absolute value of the curvature  at
time t. Note that we have
Z
1
2⇡
(2.44)
K>
 ds =
,
L M
L
where the strict inequality holds because the curve is not circular and the
curvature is therefore not constant, and the second equality is the theorem
of turning tangents.
Choose an arc length parameter s such that s = 0 is a point where the
maximum curvature is attained. We use the following Taylor expansion for
X, which can be computed using the definition of T and the Serret-Frenet
formulae (2.2):
(2.45)
X(s, t) = X(0, t) + sT
s2
N
2
s3
(s N + 2 T)
6
s4
(3 ss )N 3s T + O(s5 )
24
as s ! 0, where T, N, , s and ss are all evaluated at (0, t). From this we
find
⇣s ⌘
⇣ s ⌘
s3
(2.46)
X
,t
X
, t = sT
s N + 2 T + O(s5 )
2
2
24
as s ! 0. From this we find
⇣s ⌘
⇣ s ⌘2
s4 2
,t
X
,t
= s2
 + O(s6 ),
(2.47)
X
2
2
12
and therefore
⇣s ⌘
⇣ s ⌘
s3 2
(2.48)
X
,t
X
,t = s
 + O(s5 ).
2
2
24
+
34
2. CURVE SHORTENING AND GRAYSON’S THEOREM
3
⇡s
1 ⇡s
We also have sin ⇡s
+ O(s5 ) from the Taylor expansion of
L = L
6 L
the sine function. Combining these we find
⇣
⌘
2 2
⇡ 1 ⇡Ls2 + O(s4 )
(2.49)
Z(s/2, s/2, t) =
2
1 s 2 + O(s4 )
✓ 24 ✓
◆
◆
1
4⇡ 2
=⇡ 1+
2
s2 + O(s4 ) .
2
24
L
Since we are working at the point of maximum absolute value of curvature
2
we have 2 = K 2 > 4⇡
by equation (2.44), and it follows that the right
L2
hand side of (2.49) is greater than ⇡ for sufficiently small values of s.
⇤
Of course if the initial curve is a circle, then the solution is the shrinking
circle solution and Z remains constant. So we proceed by assuming that the
curve is not circular, and prove that Z is strictly decreasing. Precisely, we
prove the equivalent statement that for any t0 2 [0, T ) there does not exist
t1 > t0 such that sup{Z(x, y, t1 ) : x, y 2 M} sup{Z(x, y, t0 ) : x, y 2 M}.
For this it suffices to prove that there does not exist t1 > t0 such that
sup{Z(x, y, t1 ) : x, y 2 M} = sup{Z(x, y, t) : x, y 2 M, t 2 [t0 , t1 ]}.
Suppose such times t0 and t1 exist. Then there exist x0 and y0 in M such
that Z(x0 , y0 , t1 ) = sup{Z(x, y, t) : x, y 2 M, t 2 [t0 , t1 ]}. By Lemma 2.27
and Lemma 2.28 we must have x0 6= y0 . At this point we have @Z
0
@t
and the matrix of second derivatives of Z is non-positive. We will derive a
contradiction from these two inequalities.
R
d
2
We first compute the time derivative: Observing that dt
L=
M  ds
Ry 2
@
and @t
`(x, y, t) =

ds,
we
find
x
✓ ✓ ◆
✓ ◆◆ Z
@Z
⇡`
⇡`
⇡`
d
= Z hw, y Ny x Nx i
sin
cos
2 ds
@t
L
L
L
M
✓ ◆Z y
⇡`
(2.50)
⇡ cos
2 ds.
L
x
Since we can assume without loss of generality that 0  `  L/2 we have
⇡`
⇡`
sin ⇡`
0 and cos ⇡`
0, so the terms after the first
L
L cos L
L
are non-positive. To understand the first term we use the spatial second
variation. First, we compute the first derivatives, working at a point (x, y, t)
with respect to arc length parametrisation at time t. We orient the curve so
that the parameter increases from x to y along the shorter of the two arcs
@`
@`
of the curve, so that @x
= 1 and @y
= 1. This gives
✓ ◆
@Z
⇡`
(2.51)
d
= Zhw, Tx i ⇡ cos
;
@x
L
✓ ◆
@Z
⇡`
d
= Zhw, Ty i + ⇡ cos
.
@y
L
16. HUISKEN’S DISTANCE COMPARISON ESTIMATE
35
From this we find the following expressions for the second derivatives (at an
extremum where the first derivatives vanish):
✓ ◆
@2Z
Z
⇡2
⇡`
(2.52)
d 2 = Zhw, x Nx i
1 hw, Tx i2
sin
@x
d
L
L
✓
◆
@2Z
Z
⇡2
⇡`
2
d 2 = Zhw, y Ny i
1 hw, Ty i
sin
@y
d
L
L
✓ ◆
@2Z
Z
⇡2
⇡`
d
= (hTy , Tx i hw, Ty ihw, Tx i) +
sin
.
@x@y
d
L
L
The vanishing of the first derivatives in identities (2.51) tell us that hw, Tx i =
hw, Ty i = Z⇡ cos ⇡`
L . This leads to two possibilities: Either the two tangent
directions Tx and Ty are parallel, or they are bisected by the direction w.
We consider the first case first: Then we compute:
✓
◆ !
✓ ✓ ◆
✓ ◆◆ Z
@
@
@ 2
⇡`
⇡`
⇡`
d
Z
+
Z =
sin
cos
2 ds
@t
@x @y
L
L
L
M
✓ ◆Z y
⇡`
(2.53)
⇡ cos
2 ds
L
x
< 0.
This contradicts the fact that we are at a point where a new supremum of
Z is attained. Next consider the second possibility: Then let ✓ be the angle
from w to Tx , so that in this case the angle from Tx to Ty is 2✓. Then we
compute
✓
◆ !
@
@
@ 2
Z
d
Z
Z =
2 sin2 ✓ + 2(cos(2✓) cos2 ✓)
@t
@x @y
d
✓ ◆
✓ ◆Z y
4⇡ 2
⇡`
⇡`
(2.54)
+
sin
⇡ cos
2 ds
L
L
L
✓ ✓ ◆
✓ ◆◆ Zx
⇡`
⇡`
⇡`
sin
cos
2 ds
L
L
L
M
The terms in the first bracket simplify to zero. The remainder we estimate as
follows: The coefficients of each of the two integrals of 2 are non-negative,
so we can estimate each of the integrals using the Cauchy-Schwarz inequality
and the turning-angle identity ??: The integral over the whole curve gives
✓Z
◆2
Z
1
4⇡ 2
(2.55)
2 ds >
 ds =
,
L
L
M
M
where the strict inequality holds because  is not constant if the curve is
not a circle. The integral over the segment from x to y gives
✓Z y
◆2
Z y
1
4✓2
(2.56)
2 ds
 ds =
.
`
`
x
x
36
2. CURVE SHORTENING AND GRAYSON’S THEOREM
Substituting these we arrive at the following:
✓
◆ !
✓ ◆
@
@
@ 2
4⇡
⇡`
(2.57) d
Z
Z <
cos
✓2
@t
@x @y
`
L
✓
⇡`
L
◆2 !
.
To complete the argument we need to compare ✓ to the length fraction ⇡`/L.
The required estimate comes from first derivative identity (2.51), which says
that Z cos(✓) = ⇡ cos ⇡`
L , and the fact that Z > ⇡ at the supremum by
Lemma 2.28. These imply that 0  cos ✓  cos ⇡`
L , and since the cosine
⇡`
function is decreasing on the interval [0, ⇡] we deduce that ✓
L . The
last bracket is therefore non-negative and the time derivative of Z is strictly
negative at the maximum point. This is a contradiction, so we have ruled
out both cases and the theorem is proved.
⇤
Remark 2.29. The reader may have noted that the ‘heat equation’ we derive
for Z di↵ers in the two cases. There is a simple geometric reason for this,
which we will try to explain here: Our choice of the second order operator is
governed by the need to get the best inequality possible. We want to show that
the maximum of Z is decreasing, which we do by showing that @Z
LZ  0
@t
for a suitable second order operator L. To give ourselves the best chance of
proving this we choose L to maximize LZ at the maximum point, and this
corresponds to choosing L to minimise Ld.
Note that d is a convex function, and the second derivatives are strictly
positive except in radial directions. This means that the second derivative of
d is minimized when we keep the vector w parallel. In the first case of the
proof the tangent lines are parallel and traversed in the same direction (see
Figure 2.29), so w is constant if we move x and y in direction @x + @y (that
is, x moves in direction @x and y moves in direction @y ). Accordingly we
choose L = (@x + @y )2 .
In the second case of the proof the corresponding tangent lines meet the w
direction at equal angles, as in Figure 2.29. In this case w remains constant
if we move x and y in direction @y @x , so the optimal choice of L is
(@y @x )2 .
17. A curvature bound by distance comparison
Now we come to one final refinement of the argument: In our first proof
that embeddedness is preserved, we assumed a curvature bound and deduced
‘injectivity’ bounds depending on a bound for curvature. Huisken’s estimate
improves this by removing the dependence on curvature, producing lower
bounds for d in terms of ` which depend only on initial lower bounds and
the total length of the curve. We want to prove Grayson’s theorem, which
says that if the initial curve is embedded, then the evolving curve shrinks to a
point, becoming circular in the process. In particular the solution continues
to exist as long as the length remains positive.
Huisken’s estimate is not enough by itself to deduce that the solution
continues to exist while the length remains positive: To apply the global
17. A CURVATURE BOUND BY DISTANCE COMPARISON
37
38
2. CURVE SHORTENING AND GRAYSON’S THEOREM
The idea is as follows: If the curvature  is large at a point x on a smooth
curve, then it looks locally like a part of circle of radius 1/. In particular
2 `3
we would expect that the distance d is close to 2 sin `
2 '`
24 for pairs
of points close to x. Conversely, if we can prove that d(x, y)
'(`(x, y))
3
3
where '(x) x Cx + o(x ) as x ! 0, then we must have 2  24C at
every point of the curve. So the challenge is to prove a lower bound on d of
this kind. Note that the bound coming from Huisken’s estimate is not strong
⇡`
enough: That estimate gives d '(`) = cL
⇡ sin L for some c 2 (0, 1), so
we have '(x) ' cx as x ! 0.
First let us establish the implication more carefully:
Definition 2.30. Let X : M1 ! R2 be a smooth embedding. The chord-arc
profile of X is the function X : [0, 1) ! R defined by
X (z)
Figure 3. Case 1: Tx = Ty . Moving (x, y) in direction
@y + @x keeps w constant, while moving in direction @y @x
rotates w and gives a positive second derivative of d.
= inf{|X(x)
Proof. By Lemma 2.25 we have d(x, y) K2 sin K`
for all (x, y) with
2
2
⇡
`(x, y)  K
, so X (z) K2 sin Kx
= z K24 z 3 +O(z 5 ) as z ! 0. The proof
2
of Lemma 2.28 gives the reverse inequality: We showed there (in equation
(2.48)) that for any small z there exist points (x, y) with `(x, y) = z and
d(x, y)  z
Therefore by definition we have also
Figure 4. Case 2: Tx 6= Ty . Moving (x, y) in direction
@y @x keeps w constant, while moving in direction @y + @x
rotates w and gives a positive second derivative of d.
existence theorem to deduce this we would need to prove that a curvature
bound holds as long as the length remains positive. We will later see how
an analysis of blow-up limits would allow us to derive Grayson’s theorem
from Huisken’s estimate. But first we will see how to improve Huisken’s
argument to derive a curvature bound directly, giving a rather easy proof of
Grayson’s theorem.
X(y)| : `(x, y) = z}.
Proposition 2.31. Let X : M1 ! R2 be a smooth embedding, and let
2
K = supM ||. Then X (z) = z K24 z 3 + O(z 5 ) as z ! 0.
K2 3
z + Cz 5 .
24
X (z)
z
K2 3
5
24 z +O(z )
as z ! 0.
⇤
It follows that we can read o↵ the maximum curvature from the chordarc profile, and that a strong enough lower bound for the chord-arc profile (of
the form X (z) z Cz 3 for small z) implies a curvature bound (precisely,
2  24C).
Then next problem is: How can we prove good lower bounds for the
chord-arc profile? Let us see what conditions are required for a lower bound
on the chord-arc profile to be preserved. To take into account the fact
that the curve is expected to shrink to a point, it makes sense to consider a
relative chord-arc profile, which compares the chord distance as a proportion
of total length to the arc length as a proportion of total length (the relative
chord-arc profile of a curve X is therefore just the chord-arc profile of the
curve obtained by rescaling X to have total length 1): Define
✓
◆
`(x, y, t)
(2.58)
Z(x, y, t) = d(x, y, t) L(t)'
,t ,
L(t)
where ' is a smooth function yet to be determined. The idea is to find
conditions on ' which will imply that positivity of Z is preserved. We
will begin somewhat formally, by considering what the implications are for
' when a new interior zero minimum point (x, y, t) of Z is attained with
17. A CURVATURE BOUND BY DISTANCE COMPARISON
39
40
2. CURVE SHORTENING AND GRAYSON’S THEOREM
y 6= x. At such a point we compute the following (working in arc length
parametrisation):
@Z
@Z
(2.59)
= hw, Tx i + '0 ;
= hw, Ty i '0
@x
@y
X(y,t) X(x,t)
Thus as before the cosine of the angle between Tx and w = |X(y,t)
X(x,t)| ,
0
and that between Ty and w, both equal ' . The second derivatives give
@2Z
1
'00
= hw, x Nx i + hTx w · Tx , Tx i
@x2
d
L
@2Z
1
'00
=
hw,

N
i
+
hT
w
·
T
,
T
i
y
y
y
y
y
@y 2
d
L
@2Z
1
'00
=
hTx w · Tx , Ty i +
@x@y
d
L
The time derivative of Z is as follows:
✓
◆Z
Z y
@Z
` 0
@'
(2.61)
= hw, x Nx y Ny i+ '
'
2 ds+'0
2 ds L .
@t
L
@t
M
x
(2.60)
The vanishing of the first derivatives implies as in Huisken’s result (Theorem 2.26) that there are two possible cases: Either Tx = Ty , or the direction
w bisects Tx and Ty . We can eliminate one of these possibilities as long as
' is concave:
Lemma 2.32. Suppose '(., t) is strictly concave with |'0 | < 1 on (0, 1),
and let (x, y) 2 (M ⇥ M) \ {(x, x) : x 2 M} be a critical point of Z with
Tx = Ty and Z(x, y, t) = 0. Then there exists (u, v) 2 M ⇥ M such that
Z(u, v, t) < 0.
Proof. By the Jordan curve theorem, the curve X(., t) separates R2
into precisely two open connected regions. Note that by equation (2.59)
we have hw, Tx i = '0 2 ( 1, 1), so Tx and w are linearly independent. It
follows that for small a > 0, the point X(x, t) + aw lies in one of the two
connected components of R2 \ X(M). Call this component ⌦. We can
assume (by reversing parametrisation if necessary) that ⌦ also contains the
points X(x, t) + aN(x, t) for small a > 0. But then by connectedness ⌦ also
contains the points X(z, t) + aN(z, t) for any z 2 M, and in particular the
points X(y, t) + aN(y, t). But since Ty = Tx we have Ny = Nx , so ⌦ contains
the points X(y, t) + aw for small a > 0, hence not the points X(y, t) aw
for small a > 0.
Write d = d(x, y, t), and let s⇤ = sup{s 2 (0, d) : X(x, t) + sw 2
⌦}. Then the above argument shows that 0 < s⇤ < d, and X(x, t) +
s⇤ w 2 @⌦ = X(M). Therefore there exists u 2 M such that X(u, t) =
X(x, t) + s⇤ w. In particular we have d(x, u, t) + d(u, y, t) = d(x, y, t), and
we also have `(x, u, t) + `(u, y, t) = `(x, y, t). Therefore by strict concavity
`(x,y,t)
we have '( `(x,u,t)
, t) + '( `(u,y,t)
L
L , t) > '( L ). By assumption we have
d(x,y,t)
'( `(x,y,t)
= d(x,u,t)
+ d(u,y,t)
, so we deduce that '( `(x,u,t)
, t) +
L , t) =
L
L
L
L
Figure 5. If Tx = Ty there must be a point X(u, t) on the
curve along the line joining X(x, t) to X(y, t), and it follows
(if '(., t) is concave) that (x, y) does not minimize Z(., ., t).
d(x,u,t)
'( `(u,y,t)
+ d(u,y,t)
and hence either '( `(x,u,t)
, t) > d(x,u,t)
or
L , t) >
L
L
L
L
d(u,y,t)
'( `(u,y,t)
,
t)
>
.
That
is,
either
Z(x,
u,
t)
<
0
or
Z(u,
y, t) < 0.
⇤
L
L
Since by construction we have Z(., ., t)
0, it follows that the only
possibility is the case where Tx and Ty are bisected by w. As before, we let
✓ be the angle between Tx and w, which then equals the angle between w
and Ty .
Now we compute
✓
◆ !
Z y
@
@
@ 2
@'
'00
(2.62)
Z= L
+4
+ '0
2 ds
@t
@x @y
@t
L
x
✓
◆Z
` 0
+ '
'
2 ds
L
M
To proceed as we did in the proof of Theorem 2.26 we must assume that
the coefficients of the two integrals of 2 are non-positive, so that ' must be
17. A CURVATURE BOUND BY DISTANCE COMPARISON
1
2]
41
42
z'0 (z, t)
Proof. If y = (arccos(x))2 , a direct computation gives that
p
2 y
y0 =
p
sin( y)
increasing on [0, and we must have '(z, t)
0. These follow
from the assumption of concavity that we made previously:
Lemma 2.33. If '(., t) is strictly concave and positive on [0, 1], and '(z, t) =
'(1 z), then for all z 2 [0, 1/2],
'0 > 0
and
'
z'0 > 0.
Proof. By symmetry ' has a positive critical point at 1/2, and hence
by strict convexity ' > 0 for z < 1/2. The function u(z) = ' z'0 is zero
at the origin, and u0 (z) = z'00 > 0, so u > 0 on (0, 1].
⇤
We can then use the Cauchy-Schwarz inequality to estimate the integrals,
yielding
✓
◆ !
✓
◆
@
@
@ 2
@'
'00 4✓2 0 4⇡ 2
` 0
(2.63)
Z> L
+4 +
'+
'
'
@t
@x @y
@t
L
`
L
L
The strict inequality holds because we have a strict inequality in the CauchySchwarz inequality if  is not constant (that is, if the curve is not a circle).
Finally, the first derivative condition allows us to relate ✓ to ': This says
that cos ✓ = hw, Tx i = '0 , or ✓ = arccos ('0 ) (note that are necessarily
working at a point where '0 is less than one). This gives
✓
◆ !
@
@
@ 2
@'
'00 4 (arccos('0 ))2 0
(2.64)
Z> L
+4
+
'
@t
@x @y
@t
L
`
✓
◆
2
4⇡
` 0
+
'
' .
L
L
The important thing to note is that the right hand side of this equation
depends only on ' (and ` and L — note that ' is itself a function of `/L
and t). To put this another way, we can rule out the possibility of an interior
maximum of Z if ' satisfies the di↵erential inequality
(2.65)
L2
@'
4(arccos('0 ))2 '0
(z, t)  4'00 +
+ 4⇡ 2 ('
@t
z
z'0 ).
The appearance of the total length L at first sight looks like a problem, since
this depends on time itself. However we can remove this factor simply by
redefining our ‘time
R t variable’: Precisely, we define a new parameter ⌧ by the
equation ⌧ = 0 L12 dt. Then equation (2.65) becomes simply a parabolic
di↵erential inequality for a function of one spatial variable and one time
variable. The inverse trigonometric function can also be removed, using the
following:
Lemma 2.34. The function x 7! (arccos(x))2 is convex on [0, 1], and
⇡z
(arccos(x))2 ⇡ 2 z 2 + 2 sin(⇡z)
(cos(⇡z) x) for x, z 2 [0, 1].
2. CURVE SHORTENING AND GRAYSON’S THEOREM
and
p
2 cos( y)
p
p
(tan( y)
y) > 0,
3 p
sin ( y)
so y is convex. Therefore y(x) y(cos(⇡z)) + y 0 (cos ⇡z)(x cos(⇡z)), and
2⇡z
the result follows since y(cos(⇡z)) = ⇡ 2 z 2 and y 0 (cos(⇡z)) = sin(⇡z)
.
⇤
y 00 =
A comment on the form of the result above, which might otherwise look
rather mysterious: In seeking to estimate the terms which arise, we must
keep an important principle in mind: If we want to prove a result which is
strong enough to imply that the curves become circular, then in particular
we must be able to prove that all inequalities we use hold exactly in the
case of the shrinking circle. In the case of a circle the chord-arc profile is
exactly ' = ⇡1 sin(⇡x), as derived in equation (2.43). In this case we have
'0 = cos(⇡z). For this reason, it is natural to look for an estimate which
holds exactly in the case x = cos(⇡z), which we obtain in the lemma by
using the linear approximation about this point. Now we can put everything
together to get the following result:
Proposition 2.35. Suppose ' : [0, 1] ⇥ [0, 1) ! R is a smooth function
which is non-negative and concave in the first argument, with the symmetry
'(z, ⌧ ) = '(1 z, ⌧ ), and satisfying the inequality
(2.66)
@'
8⇡'0
 4 '00 + ⇡ 2 ' +
@⌧
tan(⇡z)
8⇡('0 )2
sin(⇡x)
2
at any point with |'0 |  1. Let X : M⇥[0,
T) !
⇣
⌘ R be an embedded solution
of CSF such that d(x, y, 0) L(0)' `(x,y,0)
,
0
for
all x, y 2 M, and define
L(0)
⇣
⌘
Rt 1
`(x,y,t)
⌧ = 0 L(t)2 dt. Then d(x, y, t) L(t)' L(t) , ⌧ (t) for all x, y 2 M and
all t 2 [0, T ].
This leaves us with the problem of how to find a function ' satisfying
the conditions of the Proposition. Equation (2.66) is not a simple equation,
so it is quite remarkable that we can find a simple, explicit function for
which equality holds. To find this we seek a ‘similarity’ solution of a quite
general form: We try the ansatz '(z, ⌧ ) = A(t) (B(⌧ )C(z)), where A,
B, C and
are smooth functions. We want a solution which always has
'0 (0, ⌧ ) = 1 in order to derive a curvature bound using Proposition 2.31.
0
Since ' (0, ⌧ ) = A(⌧ )B(⌧ ) 0 (0)C 0 (0), we look for a solution with 0 (0) = 1,
C 0 (0) = 1, and A(⌧ )B(⌧ ) = 1. Substituting this form in equation (2.66), we
obtain
(2.67)
✓
◆
CA0 0
8⇡ 0 C 0
C0 0
A0
= 4 00 (C 0 )2 /A + 4 0 C 00 + 4⇡ 2 A +
1
.
A
tan(⇡z)
cos(⇡z)
17. A CURVATURE BOUND BY DISTANCE COMPARISON
43
44
2. CURVE SHORTENING AND GRAYSON’S THEOREM
2
e4⇡ ⌧ .
We cancel the
terms by choosing A =
We also simplify the last
bracket by making the choice C = ⇡1 sin(⇡z), so that the factor C 0 / cos(⇡z)
disappears. Miraculously the second term on the left then cancels exactly
with the second term on the right, and the equation becomes
✓
◆
0 (1
0)
2
00
(2.68)
0 = 4e 4⇡ ⌧ cos2 (⇡z)
+2
,
⇠
2
where ⇠ = ⇡1 e 4⇡ ⌧ sin(⇡z) is the argument of . Thus the problem has
0
0
reduced to finding the solutions of the first order ODE 00 + 2 (1⇠ ) = 0
with the initial conditions (0) = 0 and 0 (0) = 1, which are (z) = z,
(z) = a1 arctan(az), or (z) = a1 arctanh(az). The first of these yields
exactly the circle chord-arc profile ' = ⇡1 sin(⇡z). The third has '0 > 1 and
so is not useful for our purposes, but the second family is exactly what we
need:
Corollary 2.36. If X : M ⇥ [0, T ) ! R2 is a smooth solution of curveshortening flow with
✓
✓
◆◆
L(0)
a
⇡`(x, y, 0)
(2.69)
d(x, y, 0)
arctan
sin
a
⇡
L(0)
for some a > 0 and all x, y 2 M, then
✓
✓
◆◆
2
L(t)e4⇡ ⌧ (t)
a
⇡`(x, y, t)
(2.70)
d(x, y, t)
arctan
2 ⌧ (t) sin
4⇡
a
L(t)
⇡e
Rt 1
for all x, y 2 M and t 2 [0, T ), where ⌧ (t) = 0 L(t)
2 dt.
We leave it to the reader to check that this function is concave and has
gradient less than 1 on (0, 1) as required in our argument:
Importantly, this lower bound on the chord-arc profile is indeed strong
enough to deduce a curvature bound using Proposition 2.31: The first few
terms in the Taylor expansion about zero give the following:
✓ 2
◆
⇣a
⌘
1
a
⇡2
(2.71)
arctan
sin (⇡z) = z
+
z 3 + O(z 5 ).
a
⇡
3
6
Comparing with Proposition 2.31 we see that a bound of the form d
L
a arctan(a/⇡ sin(⇡`/L)) implies a curvature bound
2  (4⇡ 2 + 8a2 )/L2 .
In particular the bound (2.70) implies a curvature bound
✓ ◆2
2
2⇡
8a2 e 8⇡ ⌧
2 
+
.
L
L2
Corollary 2.37. If X : M ⇥ [0, T ) ! R2 is a smooth solution of curveshortening flow satisfying (2.69) for some a > 0 and all x, y 2 M, then
✓
◆ ✓
◆
2⇡ 2
2a2
2
(2.72)
2 (x, t) 
1 + 2 e 8⇡ ⌧ (t)
L(t)
⇡
Figure
6. Plots
of
the
chord-arc
comparisons
1
a
for varying values of a. As a ina arctan ⇡ sin(⇡x)
creases to infinity the function approaches zero, while as
a approaches zero the function approaches the chord-arc
profile of the circle ⇡1 sin(⇡x).
for all x 2 M and all t 2 [0, T ).
We need one more observation to make this useful, which we leave to
the reader in the following exercise:
Exercise 2.38. Show that for any smooth initial embedding X0 : M ! R2
from a compact 1-manifold to R2 , there exists a constant a > 0 such that
the estimate (2.69) holds.
Exercise 2.39. Why is it necessary to rule out the situation with Tx = Ty
using Lemma 2.32, rather than proceeding as we did in the Huisken estimate
in Section 16?
18. Grayson’s theorem
We are now in a position to prove the most famous result concerning
curve-shortening flow: The curve shortening flow starting from any simple
closed curve shrinks to a point and becomes circular in shape:
18. GRAYSON’S THEOREM
45
R2
Theorem 2.40 (Grayson’s theorem). Let X0 : M !
be a smooth
embedding of a compact connected 1-manifold M. Then the solution of
curve-shortening flow with initial data X0 exists on a maximal time interval
[0, T ), and there exists p 2 R2 such that the rescaled embeddings
X(., t)
X̃(., t) = p
2(T
p
t)
are smooth embeddings for each t 2 [0, T ), and converge in C 1 to a limit
X̃1 , which is an embedding with image equal to the unit circle about the
origin.
Proof. The idea is to show that as the final time is approached, the
length approaches zero and ⌧ approaches infinity. The curvature bound then
shows the maximum curvature approaches 2⇡/L(t) near the final time. Since
the theorem of turning tangents implies the average curvature is exactly
2⇡/L(t), this means the curvature must be approaching a constant and
the curve is becoming circular. As we will see, there is some further work
required to prove the strong convergence statement of the Theorem.
By Exercise 2.38 there exists a > 0 such that (2.69) holds for all x, y 2
M. By Corollary 2.37 we have the curvature bound
✓
◆ ✓
◆
2⇡ 2
2a2
2
(2.73)
2 (x, t) 
1 + 2 e 8⇡ ⌧ (t)
L(t)
⇡
where ⌧ (t)
0 for each t 2 [0, T ).
Lemma 2.41. limt!T L(t) = 0.
@
Proof. We know L(t) is decreasing (since @t
ds = 2 ds by Exercise
??. If L(t) does not approach zero then we have L(t)
L0 > 0 for all
t 2 [0, T ) and hence 2  C/L20 remains bounded. This contradicts the
global existence theorem (Theorem 2.14).
⇤
p
Lemma 2.42. L(t) is comparable to T t:
r
p
2a2
2⇡ 2(T t)  L(t)  2⇡ 2(1 + 2 )(T t)
⇡
for all t 2 [0, T ).
R 2
d
Proof. These follow from the evolution equation dt
L=
 ds and
the inequalities
R
Z
2
 ds
4⇡ 2
2 ds
=
L
L
and
Z
2a2
2
4⇡ (1 + ⇡2 )
2 ds 
.
L(t)
⇤
46
2. CURVE SHORTENING AND GRAYSON’S THEOREM
Substituting these inequalities in the identity ⌧ =
the following:
R
1
L2
dt we conclude
Lemma 2.43. ⌧ (t) approaches infinity as t ! T , and
✓
◆
✓
1
1
t
log 1
 ⌧ (t) 
log 1
8⇡ 2 + 16a2
T
8⇡ 2
t
T
◆
.
Now the curvature bound (2.73) becomes
✓
◆
✓
◆ 1 !
2⇡ 2
2a2
t 1+2a2
2 
1+ 2 1
.
L(t)
⇡
T
Using this in the evolution equation for L(t) yields an improvement of
Lemma 2.42:
Lemma 2.44. There exists C such that
q
p
2⇡ 2(T t)  L(t)  2⇡ 2(T
t)(1 + C(T
1
t) 1+2a2
2
L
This implies in particular that the isoperimetric ratio 4⇡A
approaches 1
R
d
as the final time is approached, since we have dt
A=
 ds = 2⇡ and
A(t) ! 0 as t ! T , so A(t) = 2⇡(T t).
Exercise 2.45. By substituting the result of Lemmap
2.44 into the definition
2
of ⌧ , show that 2  2⇡
(1+C(T t)) and L  2⇡ 2(T t)(1+C(T t))
L
for some C.
In order to extract a limit of the rescaled embeddings X̃t as t approaches
T , we need to establish uniform bounds on X̃t and all of its derivatives. The
first step is to control the derivatives of curvature:
Lemma 2.46. There exist constants C̃k for each k > 0 such that
@k
@sk
2

(T
C̃k
.
t)1+k
Proof. We apply the smoothing estimates from Theorem 2.12. The
curvature bound (2.73) gives that || is bounded by a multiple of L 1 , which
by Lemma 2.42 is comparable to (T t)1/2 , so there exists C0 such that
||  C0 (T
t)
1/2
.
C2
a
In particular, on the time interval [T a, T 1+C0 2 a] of length 1+C
2 we have
0
0
q
1+C02
|| 
a . Therefore we can apply Theorem 2.73, to deduce
r
@k
1 + C02

C
(t T + a) k/2 .
k
a
@sk
18. GRAYSON’S THEOREM
n
1+C02
(T
C02
For given t, choosing a = min T,
(r
@k
1 + C02
(2.74)

C
max
t
k
T
@sk
t)
k/2
o
,
47
C02
T
t
)
◆ k+1
2
.
We also have estimates for small t from the short time existence theorem,
and the result follows.
⇤
k+1
k
Note that for each k, (T t) 2 @@sk is just the kth derivative of curvature
with respect to arc length on the rescaled curve, so Lemma 2.46 simply states
that all derivatives of curvature remain bounded on the rescaled curves.
Next we deduce that the curvature of the rescaled curves approaches 1:
Lemma 2.47. There exists C > 0 such that
p
p
 2(T t) 1  C T
t
Proof. First we prove convergence in an integral sense: We have
p
Z
Z
⇣ p
⌘2
2 2(T t)
1
2(T t)
 2(T t) 1 ds =
2 ds
 ds + 1
L M
L
L
M
M
8⇡ 2
4⇡ p
 2 (T t)(1 + C(T t))
2(T t) + 1
L
L
(2.75)
 C(T t),
Z
where we used the curvature bound (from Exercise 2.45) in the first term
on the right , the theorem of turning tangents in the second, and the length
estimate (also from Exercise 2.45) in both terms on the second line.
Since we have bounds on the derivatives of curvature from Lemma 2.46
we can convert the integral bounds into pointwise
bounds: The idea is that
p
if there is some point where the di↵erence | 2(T t) 1| is large, then the
derivative estimates imply that the di↵erence is large on a neighbourhood,
so that the integral must also be large. This is a simple example of an
interpolation inequality, which is a common tool in the analysis of partial
di↵erential equations.
Exercise 2.48. Make this argument precise: Let f be a smooth function on a
R
1/2
simple closed curve of length L, with M1 = supM |fs |, kf k2 = M f 2 ds
,
and M0 = supM |f |. Using the gradient bound find an interval on which
|f | 2M0 , and use this to obtain a lower bound on kf k2 . Deduce that
2/3
2. CURVE SHORTENING AND GRAYSON’S THEOREM
By a similar argument we can show that all of the derivatives of the
curvature of the rescaled curves decay to zero:
gives
✓
48
1/3
M0  max{22/3 kf k2 M1 , 2kf k2 /L}.
p
In our case if f =  2(T t) 1 then we have M1 bounded. Using this
we deduce that
p
(2.76)
 2(T t) 1  C(T t)1/2 .
We remark that by using interpolation inequalities involving higher derivatives one can improve this estimate to C (T t) for any < 1.
⇤
Lemma 2.49. For each k > 0 there exist C̄k such that
2
@k
 C̄k (T t)1/4 .
@sk
Proof. This is again an exercise in interpolation: For a smooth function
f , if the second derivative is bounded, and the absolute value of f is very
small, then the first derivative must also be quite small. The reason is that
if the derivative were large somewhere, then the second derivative bound
would imply that the derivative was large on a whole neighbourhood, and
so the function itself could not be small everywhere. A simple proof can be
given using the Taylor approximation of degree 2: Choosing x to be a point
where the maximum derivative is attained, we have
t)1+k
(T
f (x + h) = f (x) + f 0 (x)h +
h2 00
f (z)
2
for some z. Rearranging this gives
f (x + h) f (x) h 00
f 0 (x) =
f (z),
h
2
from which we deduce
2
h
M1  M0 + M 2 ,
h
2
k
where Mk = supM @s f for each k. Optimizing over h gives
1/2
1/2
M1  2M0 M2 .
(2.77)
Applying this with f replaced by the kth derivative of f gives
1/2
1/2
Mk+1  2Mk Mk+1 .
(2.78)
Using this we can control the size of any derivative of f if we have control
on a higher derivative and a lower derivative.
Exercise 2.50. Using the inequality (2.78), show that for any integers 0 
i < j < k we have
Mj  2
(2.79)
(k j)(j
2
i)
k j
j
i
Mik i Mkk i .
p
Now for given k we use equation (2.79) applied to f =  2(T t) 1
to interpolate the kth derivative of  between the result of Lemma 2.47 and
the bound on a very high derivative of  from Lemma 2.46. For concreteness
we can interpolate using the 2kth derivative:
1
p
k2
1
1
2
(2.80)
|@sk |  2 2 sup  2(T t) 1 sup |@s2k | 2 (2(T t)) 4
M
M
1
1
 C(T
2
t) 4 C̃2k
(T
= C(T
t)
k
2
1
4
.
t)
1+2k
4
(T
t)
1
4
18. GRAYSON’S THEOREM
49
(2.81)
(T
t)
|@sk |
1
4
 C(T
t) ,
as required. We remark that by interpolating with higher derivatives both
here and in Lemma 2.47 we could improve the bound on the right to C(T t)
for any < 1.
⇤
Finally, we control all of the derivatives of the rescaled embedding X̃:
Lemma 2.51. There exists C > 0 such that
X̃ 0
and for each k
C ,
1 there exists Ck such that
X̃ (k)  Ck
and
@ (n+1)
X̃
 C(T
@t
(2.85)
t)
3/4
3/4
is integrable on [0, T ).
.
⇤
This completes the induction and the proof.
We can now complete the proof: The last set of estimates in Lemma
2.51 show that the rescaled embeddings X̃t are Cauchy in C k for every k:
We have for any t1 < t2 < T
Z t2
@ (k)
(2.86)
X̃ (k) (t2 ) X̃ (k) (t1 ) 
X̃
dt
@t
t1
Z t2
 Ck
(T t) 3/4 dt
t1
@ (k)
X̃
 Ck (T
@t
t)
3/4
 4Ck (T
.
Proof. This is a straightforward adaptation of the proofs of Lemmas
0
2.15 and 2.18. We have X̃ 0 = p X
, so
2(T t)
(2.82)
✓
✓
@ 0
X0
X̃ = |X̃ 0 |@s (N)+
= |X̃ 0 | s N + 2
3/2
@t
(2(T t))
From this we find
✓
◆
@
1
log |X̃ 0 | =
2
.
@t
2(T t)
1
2(T
t)
◆ ◆
T .
The right-hand side is bounded by a constant by Lemma 2.47, so log |X̃ 0 | is
uniformly bounded (since the time of existence is finite).
To control the higher derivatives we adapt the result of Lemma 2.16:
First we rewrite equation (2.83) as follows (noting that @s̃ = p 1
@s
2(T t)
where s̃ is the arc length parameter on X̃):
(2.83)
2. CURVE SHORTENING AND GRAYSON’S THEOREM
It follows that X̃ (n+1) is bounded, since (T t)
Substituting this bound in equation (2.84) gives
This implies
1+k
2
50
@ 0
X̃ =
@t
|X̃ 0 |
̃s̃ N + ̃2
2(T t)
On di↵erentiating this we find that
@ X̃ (k)
@t
1 T .
is equal to
1
2(T t)
times a polyno-
mial in ̃, . . . , @s̃ ̃, . . . , @s̃k ̃, and T· X̃ (i) for i  k, with every term containing
either a factor @s̃i ̃ for some i > 1 or a factor ̃2 1, and every term containing at most one factor T · X (k) . We proceed by induction on k: Assuming
we have already
p obtained the result for k = 1, . . . , n, we observe that since
|̃2 1|  C T t and |@s̃i ̃|  C(T t)1/4 for each i 1, we obtain the
estimate
⇣
⌘
@ (n+1)
(2.84)
X̃
 C(T t) 3/4 X̃ (n+1) + 1 .
@t
t1 )1/4
which approaches zero as t1 and t2 approach T . It follows that X̃t0 converges
in Ck for every k (that is, in C 1 ) to a limit. Therefore for each t there exists
p(t) 2 R2 (for concreteness, the centre of a ball of largest radius enclosed
by Xt (M)) such that pXt pt converges in C 1 to a limiting map X̃1 . The
2(T t)
first estimate of Lemma 2.51 implies that X̃1 has nonvanishing derivative,
hence is an immersion. The estimate of Lemma 2.47 implies that X̃1 has
curvature equal to 1 everywhere, and Lemma 2.44 implies that X̃1 has
length equal to 2⇡. Therefore by the choice of pt the image X̃1 (M) is a
circle of radius 1 with centre at the origin.
It remains only to show the convergence of the points pt . The curvature
of Xt0 is no less than (2(T t0 )(1 + C(T pt0 ))) 1/2 . It follows that Xt is
contained in Br+ (t0 ) (pt0 ), where r+ (t0 ) = 2(T t0 )(1 + C(T t0 )). By
the avoidance principle, for t > t0 , Xt is contained in the ball of radius
p
2
2(t t0 ) about pt0 . This implies that |X(x, t) X(y, t0 )| <
pr+ (t0 )
2 2(T t0 )(1 + C(T t0 )) and all x, y 2 M and t > t0 , so X(., t) is
Cauchy in sup norm at t ! T , and converges
to a constant p. Since X(., t)
p
converges to p and |X(x,
t) pt0 | < r+ (t0p
)2 2(t t0 ) for each x 2 M,
p
we have |p pt0 |  r+ (t0 )2 2(T t0 ) = 2C(T t0 ). Finally, we have
that X̃t = pXt p = pXt pt + ppt p converges in C 1 to X̃1 , since
2(T t)
ppt
p
2(T t)
Ck

2(T t)
p
2C(T
2(T t)
t0 ) for any k.
⇤
19. Notes and commentary
There is a book on the CSF by K.-S. Chou and X.-P. Zhu [45]. A book
on the 1-dimensional heat equation is Cannon [34]. There is a rich literature
on the CSF and related flows, we shall only cite a few.
19. NOTES AND COMMENTARY
51
Mullins [157] discovered the grim reaper solution (a terminology later
coined by Matt Grayson).
That the CSF evolves closed convex embedded curves to round points
was proved by Gage and Hamilton (see [87], [88], and [90]). The space of
convex curves (or convex bodies) in R2 has some interesting structure which
can be exploited to understand curve flows, and the argument of Gage and
Hamilton made use of this structure in several ways: Gage [87] proved that
the isoperimetric ratio of the curves improves with time in the convex case.
Gage and Hamilton proved a Li-Yau type di↵erential Harnack estimate, and
also found a monotone ‘Entropy’ quantity. These were both used in the proof
of convergence, and similar ideas play a central role in our understanding
of higher dimensional mean curvature flow and Ricci flow. We will discuss
some of these techniques for curve shortening flow in the next Chapter, and
expand on them later in a more general context.
Grayson [96] extended the Gage-Hamilton theorem to general closed embedded curves, proving that in this situation the evolving curves eventually
become convex. Grayson’s original proof made strong use of ‘zero-counting’
arguments, which in this case imply that the number of inflexion points of
the curve cannot increase with time. Grayson made use of this to prove a
key geometric lemma (the ‘ -whisker lemma’) which can be seen as a precursor of the distance comparison estimates we discussed above. The argument
involves a careful accounting of the behaviour and interaction between ‘nice’
subarcs of the curve, and requires analysis of various possible cases.
There are now various approaches to Grayson’s theorem using the idea
of controlling or proving monotonicity of an isoperimetric ratio, all of which
greatly simplify Grayson’s original argument. We presented in section 16
the approach using chord-arc distance comparison due to Huisken [124].
A similar argument was provided by Hamilton [116], who showed how to
control the geometry by bounding the isoperimetric profile of the region
enclosed by the evolving curve. Both of these methods can be combined
with a blow-up argument and some classification of singularities to deduce
Grayson’s theorem. We will discuss the techniques required for this in the
next chapter. In the presentation we adopted here, based on a refinement
of Huisken’s distance comparison due to Andrews and Bryan [13], no separate analysis of the convex case is needed and no analysis of singularities
is required. A similar extension of Hamilton’s isoperimetric profile estimate
was found later [?], with the additional feature that the isoperimetric profile
can be compared to that of a suitable model solution of curve shortening
flow (such as the paperclip solution), so that one does not have to construct
‘by hand’ a solution of the di↵erential inequality as we did here in equations
(2.67) and (2.68): An exact solution can be constructed from the model
solution.
While these approaches to the proof allow for a streamlined exposition
and a quick route to a famous theorem, the drawback is that many beautiful
and useful techniques have not been discussed along the way. Accordingly
52
2. CURVE SHORTENING AND GRAYSON’S THEOREM
we will devote the next chapter to the elaboration of some of the other
results and methods which have been developed for curve shortening flow,
many of which will later appear and play a crucial role in the analysis of
other geometric evolution equations.
We mention some extensions of Grayson’s theorem (see the next chapter
for further discussion of the convex case): The CSF of curves on surfaces was
studied by Grayson [97], Angenent [15], [16], and Oaks [166]. Applications
of the CSF to closed geodesics on surfaces was given by Angenent [17]. Oaks
[166] and Chou and Zhu [?] gave some extensions of Grayson’s theorem
to anisotropic analogues of curve shortening flow. Angenent, Sapiro and
Tannenbaum [18] proved a result similar to Grayson’s theorem for an affinegeometric curvature flow of non-convex curves in the plane (in which the
speed is proportional to the cube root of the curvature at each point), and
Chou and Zhu [45] discussed flows where the speed is proportional to other
powers of curvature.
Lauer [?] considered the curve shortening flow with highly irregular initial data (essentially an arbitrary connected, locally-connected compact subset of the plane), proving in particular the existence of a unique smooth
solution starting from any Jordan curve with finite length.
Huisken and Polden [] also considered the curve shortening flow in noncompact situations, assuming some control on the curves near infinity. [did
this ever appear? maybe other non-compact refs]