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arXiv:1106.4583v2 [math.DG] 4 Aug 2011
HELICOIDAL SURFACES ROTATING/TRANSLATING UNDER
THE MEAN CURVATURE FLOW
HOESKULDUR P. HALLDORSSON
Abstract. We describe all possible self-similar motions of immersed hyper-
surfaces in Euclidean space under the mean curvature flow and derive the cor-
responding surface equations. Then we present a new two-dimensional family
of immersed helicoidal surfaces that rotate/translate with constant velocity un-
der the flow. We look at their limiting behaviour as the pitch of the helicoidal
motion goes to 0 and compare it with the limiting behaviour of the classical
helicoidal minimal surfaces. Finally, we give a classification of the immersed
cylinders in the family of constant mean curvature helicoidal surfaces.
1. Introduction
The mean curvature flow (MCF) of immersed hypersurfaces in Euclidean space
is defined as follows. Let Mnbe an n-dimensional manifold and consider a family
of smooth immersions Ft= F(·,t) : Mn→ Rn+1, t ∈ I, with Mt= Ft(Mn). The
family of surfaces (Mt)t∈I is said to move by mean curvature if
∂F
∂t(p,t) = H(F(p,t))
for p ∈ Mnand t ∈ I. Here H is the mean curvature vector of Mt, given by
H = −Hn, where n is a choice of unit normal field and H = divMtn is the mean
curvature of Mt.
Surfaces which move in a self-similar manner under the MCF play an important
role in the singularity theory of the flow. The most important ones are the shrinking
surfaces and the translating surfaces, which model the so-called type 1 and type 2
singularities respectively (see [12]). Examples of shrinking surfaces can for example
be found in [1], [3], [5], [14] and [15], and examples of translating surfaces in [6],
[16] and [18]. Other types of self-similar motions under the flow have been studied
less thoroughly. Some examples of expanding surfaces can be found in [2] and [9].
In [10], the author gave a complete classification of all self-similar solutions to the
flow in the case n = 1, the so-called curve shortening flow in the plane. However,
the classification problem is significantly harder in higher dimensions.
In this paper, we present a new two-dimensional family of complete immersed
hypersurfaces which rotate under the MCF. The surfaces belong to the family
of so-called helicoidal surfaces, i.e., surfaces invariant under a helicoidal motion.
Since the surfaces rotate around their helicoidal axis, they can also be viewed as
translating solutions to the flow. However, they are not convex. The idea to look
for these surfaces came from [13].
The paper is structured as follows. In Section 2, we describe all possible self-
similar motions of immersed hypersurfaces under the MCF and derive the equations
2010 Mathematics Subject Classification. Primary 53C44. Secondary 53A10, 53C42.
1
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2HOESKULDUR P. HALLDORSSON
the surfaces have to satisfy. It turns out that the only possible self-similar motions
are the following:
• dilation with scaling function√2bt + 1, i.e., shrinking with increasing speed
if b < 0 and expanding with decreasing speed if b > 0,
• translation with constant velocity,
• rotation with constant angular velocity,
• dilation and rotation together, with scaling function
lar velocity in a fixed direction but with angular speed proportional to
log(2bt+1)
2b
,
• translation and rotation together but in orthogonal directions, with con-
stant velocity and angular velocity.
√2bt + 1 and angu-
In Section 3 we introduce helicoidal surfaces. In this paper, we focus on the pla-
nar curves given as the intersection of the surfaces with a plane orthogonal to their
helicoidal axis. These curves generate the surfaces and give rise to a parametriza-
tion different from the one most common in the literature. Using this approach,
we derive a new method for constructing helicoidal surfaces with prescribed mean
curvature. This method is used in the sections that follow.
Section 4 contains the heart of the paper. There we present the two-dimensional
family of helicoidal surfaces that rotate with unit speed under the mean curvature
flow. We derive some basic properties of these surfaces and investigate the limiting
behaviour of the generating curves as the pitch of the helicoidal motion goes to 0.
A specific choice of initial values gives convergence to a circle, so the corresponding
surfaces in some sense converge to a cylinder.
In Section 5 we take a look at the classical helicoidal minimal surfaces described
in [19]. We derive their parametrization and investigate their limiting behavior
as the pitch goes to 0. Finally, in Section 6 we look at the classical constant
mean curvature helicoidal surfaces, studied in [8], [11] and [17]. We derive their
parametrization and in Theorem 6.5, we classify the immersed cylinders in this
family of surfaces, in the spirit of the classification of the self-shrinking solutions to
the curve shortening flow in the plane given by Abresch and Langer in [1].
2. Self-similar motions under the mean curvature flow
Let Σnbe a complete n-dimensional surface immersed in Rn+1. We call the
immersion F and to simplify notation we identify Σ with F(Σ). A self-similar
motion of Σ is a family (Σt)t∈I of immersed surfaces where the immersions are of
the form
F(p,t) = g(t)Q(t)F(p) + v(t),p ∈ Σ,t ∈ I.
Here I is an interval containing 0 and g : I → R, Q : I → SO(n + 1) and
v : I → Rn+1are differentiable functions s.t. g(0) = 1, Q(0) = I and v(0) = 0,
and hence F(p,0) = F(p), i.e., Σ0= Σ.
This motion is the mean curvature flow of Σ (up to tangential diffeomorphisms)
if and only if the equation
?∂F
∂t(p,t),n(p,t)
?
= −H(p,t)
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HELICOIDAL SURFACES ROTATING/TRANSLATING UNDER THE MCF3
holds for all p ∈ Σ, t ∈ I. Simple calculations yield that this equation is equivalent
to
g(t)g′(t)?F(p),n(p)? + g2(t)?QT(t)Q′(t)F(p),n(p)?
+ g(t)?QT(t)v′(t),n(p)? = −H(p).
By looking at this equation at time t = 0, we see that Σ has to satisfy
(2.1)
(2.2)
where b = g′(0), A = Q′(0) ∈ so(n + 1) and c = v′(0). It turns out that satisfying
an equation of this form is also a sufficient condition for Σ to move in a self-similar
manner under the MCF. To see that, we look separately at two cases.
b?F,n? + ?AF,n? + ?c,n? = −H
Dilation and rotation: First let’s look at the case where there is no transla-
tion term. Assume the surface Σ satisfies
(2.3)
b?F,n? + ?AF,n? = −H.
Now, if the functions g and Q satisfy g(t)g′(t) = b and g2(t)QT(t)Q′(t) = A for all
t ∈ I, then Equation (2.1) is satisfied for all p ∈ Σ, t ∈ I. Solving these differential
equations with our initial values gives
g(t) =
√2bt + 1and
Q(t) =
?
exp(log(2bt+1)
exp(tA)
2b
A)if b ?= 0,
if b = 0.
Therefore, under the MCF the surface Σ either expands and rotates forever with
decreasing speed (b > 0), rotates forever with constant speed (b = 0) or shrinks and
rotates with increasing speed until a singularity forms at time t = −1
course, there is no rotation if A = 0. Note that when b = −1 and A = 0, Equation
(2.3) reduces to the famous self-shrinker equation. In [10], the author showed the
existence of all these motions in the case of the curve shortening flow in the plane.
2b(b < 0). Of
Translation and rotation: Now let’s look at the case where there is no dila-
tion. Assume the surface Σ satisfies
(2.4)?AF,n? + ?c,n? = −H
and furthermore impose the condition Ac = 0, i.e., the translation and rotation
are in orthogonal directions. If the functions Q and v satisfy QT(t)Q′(t) = A and
QT(t)v′(t) = c for all t ∈ I, then Equation (2.1) is satisfied for all p ∈ Σ, t ∈ I.
Solving these differential equations with our initial values yields
Q(t) = exp(tA)andv(t) = tc.
Therefore, under the MCF the surface translates and rotates forever with constant
velocity and angular velocity, and these motions are orthogonal.
But what happens if we include all three terms in the left hand side of Equation
(2.2)? Assume the surface Σ satisfies (2.2). By translating Σ by a vector w, we get
a surfaceˆΣ satisfying the equation
b?ˆF, ˆ n? + ?AˆF, ˆ n? + ?c − (A + bI)w, ˆ n? = −ˆH.
Note that since A ∈ so(n + 1), A is skew-symmetric, i.e., AT= −A. Therefore, A
has no non-zero real eigenvalues and ker(A) = (im(A))⊥. Thus, we can make the
following choice of w:
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4 HOESKULDUR P. HALLDORSSON
• If b ?= 0, we let w be the unique solution to the equation (A + bI)w = c.
ThenˆΣ satisfies an equation of the form (2.3), so Σ dilates and rotates
around the point w.
• If b = 0, we let w be such that c = Aw + c0 where Ac0 = 0. ThenˆΣ
satisfies an equation of the form (2.4), so Σ translates and rotates around
the point w.
In short, the general case can always be reduced to one of the two cases already cov-
ered. Hence, they give all possible self-similar motions of immersed hypersurfaces
under the MCF.
3. Helicoidal surfaces
For h ∈ R, let γh
rigid motions of R3given by
t: R3→ R3be the one-parameter subgroup of the group of
γh
t(x,y,z) = (xcost − ysint,xsint + y cost,z + ht),
This motion is called a helicoidal motion with axis the z-axis and pitch h. A
helicoidal surface with axis the z-axis and pitch h is a surface that is invariant
under γh
tfor all t. When h = 0 it reduces to a rotationally symmetric surface, but
in this paper we will focus on the case h ?= 0. By reflecting across the xy-plane
if necessary, we can assume h > 0. Now, the parameter
speed of the rotation as we move along the z-axis with unit speed. By setting that
parameter equal to 0 (corresponding to the limit case h → ∞) we get surfaces
invariant under translation along the z-axis. Most surface equations in this paper
extend smoothly to that case so we will often use the notation h = ∞.
A helicoidal surface with h > 0 can be parametrized in the following way. Let
X : R → R2be an immersed curve in the xy-plane, parametrized by arc length.
We translate the curve X along the z-axis with speed h and at the same time rotate
it counterclockwise around with z-axis with unit angular speed. Then the curve
traces an immersed helicoidal surface Σ with axis the z-axis and pitch h, whose
parametrization is given by
t ∈ R.
1
hrepresents the angular
F(s,t) = (eitX(s),ht),s ∈ R,t ∈ R.
Note that in the literature, it is more common to parametrize the surface so
that the generating curve lies in the xz-plane. There is also the so-called natu-
ral parametrization used in [8].
The following notation will be used. Let T =dX
N = iT its leftward pointing normal. The signed curvature of X is k = ?d2X
turns out to be convenient to work with the functions τ = ?X,T? and ν = ?X,N?.
They satisfy
dsbe the unit tangent along X and
ds2,N?. It
(3.1)
d
dsτ = 1 + kν
d
dsν = −kτ
and of course we have
(3.2)
X = (τ + iν)T,r2:= |X|2= τ2+ ν2.
An interesting geometric interpretation of (τ,ν) is given in [17].
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HELICOIDAL SURFACES ROTATING/TRANSLATING UNDER THE MCF5
Now, the tangent space of Σ is spanned by the coordinate tangent vectors
∂F
∂s= (eitT,0),
and one easily verifies that a unit normal is given by
n =(heitN,−τ)
√τ2+ h2.
The metric is given by the matrix
∂F
∂t
= (ieitX,h),
gij=
?
1−ν
−νr2+ h2
?
and the inverse metric by
gij=
1
τ2+ h2
?
r2+ h2
ν
ν
1
?
.
The second derivatives are
∂2F
∂s2= (eitkN,0),
so the second fundamental form is given by the matrix
∂2F
∂s∂t= (ieitT,0),
∂2F
∂t2= (−eitX,0),
Aij= −?Fij,n? = −
h
√τ2+ h2
?
k
1
1
−ν
?
.
Finally, the mean curvature of Σ is
(3.3)
H = gijAij= −h(k(r2+ h2) + ν)
(τ2+ h2)
3
2
.
The following theorem shows how to get a helicoidal surface of prescribed mean
curvature.
Theorem 3.1. For every smooth function Ψ : R2→ R and h > 0, there exists a
complete immersed helicoidal surface of pitch h satisfying the equation H = Ψ(τ,ν).
Proof. By solving Equation (3.3) for k, we have k written as a smooth function of
τ and ν, so the proof is reduced to the following lemma.
?
Lemma 3.2. For every smooth function Φ : R2→ R, point z0 ∈ R2and angle
θ0∈ [0,2π), there is a unique immersed curve X : R → R2satisfying the equation
k = Φ(τ,ν) and going through z0with angle θ0.
Proof. Keeping in mind (3.1) and (3.2), we let τ,ν,θ be the unique solution to the
ODE system
τ′= 1 + νΦ(τ,ν)
ν′= −τΦ(τ,ν)
θ′= Φ(τ,ν)
(3.4)
with initial values θ(0) = θ0, τ(0) + iν(0) = e−iθ0z0, and then define the curve as
X = (τ + iν)eiθ.
Since
(3.5)
d
ds
?
τ2+ ν2=ττ′+ νν′
√τ2+ ν2=
τ
√τ2+ ν2≤ 1,
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6 HOESKULDUR P. HALLDORSSON
the solution cannot blow up in finite time and hence is defined on all of R. Note
that
X′= (τ′+ iν′+ iθ′(τ + iν))eiθ= eiθ,
so X is parametrized by arc length with tangent T = eiθand hence the curvature
k is equal to θ′= Φ(τ,ν). Finally,
?X,T? = Re(Xe−iθ) = τ,
?X,N? = Re(X(−i)e−iθ) = ν,
finishing the proof.
?
Theorem 3.1 generalizes in some sense the results in [4], which treats the case
when H is a function of r =
complete surface. However, [4] gives an explicit integral formula.
Typically the system of ODEs (3.4) for τ and ν has a one-dimensional trajectory
space, so we get a one-dimensional set of surfaces for each h. Varying h gives
another dimension.
In the remaining three sections, we look separately at three types of helicoidal
surfaces, namely surfaces rotating under the MCF and then the classical minimal
and constant mean curvature helicoidal surfaces.
√τ2+ ν2. Moreover, our method always gives a
4. Helicoidal surfaces rotating/translating under the MCF
In this section, we describe a two-dimensional family of immersed helicoidal
surfaces that rotate around the z-axis with unit speed under the MCF. Since the
surfaces are invariant under the helicoidal motion γh
as a translation with speed h in the negative z-direction.
The matrix R =
?0 −1 0
surface rotating with unit speed if and only if
t, this motion can also be seen
1 0 0
0 0 0
?
generates the rotation, so by Equation (2.4) we get a
−H = ?RF,n? =
hτ
√τ2+ h2.
Since H is given as a smooth function of τ, the existence of these surfaces is guar-
anteed by Theorem 3.1. The corresponding equation for the curve X is
(4.1)
k = ττ2+ h2
r2+ h2−
ν
r2+ h2
so the two-dimensional system of ODEs for τ and ν becomes
(4.2)
τ′=τ2+ h2
r2+ h2(1 + τν)
ν′=τν − τ2(τ2+ h2)
r2+ h2
.
Note that the right hand side of (4.2) remains the same when (τ,ν) is replaced by
(−τ,−ν). Therefore, s ?→ −(τ(−s),ν(−s)) is also a solution to the system, which
of course just corresponds to the curve X parametrized backwards. This symmetry
will simplify some of our arguments. Also note that the system has no equilibrium
points.
The limit case h = ∞ yields k = τ, the rotation equation for the curve shortening
flow in the plane, and the rotating surface is simply the product X × R. These
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HELICOIDAL SURFACES ROTATING/TRANSLATING UNDER THE MCF7
curves were described by the author in [10]. It turns out that when h is finite, X
still has most of the same properties although it no longer is always embedded.
Theorem 4.1. For each h > 0 there exists a one-parameter family of helicoidal
surfaces with pitch h, that rotate with unit speed around their helicoidal axis under
the MCF. The corresponding generating curves have one point closest to the origin
and consist of two properly embedded arms coming from this point which strictly go
away to infinity. Each arm has infinite total curvature and spirals infinitely many
circles around the origin. The curvature goes to 0 along each arm and the limiting
angle between the tangent and the location isπ
2.
Examples of the generating curves appear in Figures 1-3.
The proof will be given through a series of lemmas. We start by investigating
the limit behavior of τ and ν.
Lemma 4.2. Both τ and ν have (possibly infinite) limits as s → ±∞.
Proof. Note that if τ(s) = 0, then τ′(s) =
and is negative before it and positive after it.
If k(s) = 0, then it can be shown that k′(s) > 0, by differentiating (4.1) directly
and using the fact that τ′(s) = 1, ν′(s) = 0 and τ(s)ν(s) ≥ 0. Therefore, k also
has at most one zero, and is positive before it and negative after it.
Since ν′= −kτ, ν has at most two extrema and therefore has a (possibly infinite)
limit in each direction.
If τ′(s) = 0, then by differentiating τ′= 1+kν and using (4.1) and the fact that
τ(s)ν(s) = k(s)ν(s) = −1, we get that τ′′(s) = −k′(s)+τ4(s)
Hence τ has a local minimum at s if τ(s) < 0 and a local maximum if τ(s) > 0.
But that means τ has at most two extrema and hence has a (possibly infinite) limit
in each direction.
h2
ν2(s)+h2 > 0 so τ has at most one zero,
τ(s)
and k′(s) + τ4(s) > 0.
?
Lemma 4.3. The ratioν
τis decreasing wherever both τ and ν are positive.
Proof. Direct calculations yield
d
ds
ν
τ= −τ5+ τ3ν2+ h2τ3+ h2τν2+ h2ν
τ2(r2+ h2)
,
making the statement obvious.
?
Lemma 4.4. lims→±∞ν = ∓∞.
Proof. Assume lims→∞ν is finite. Since there are no equilibrium points, lims→∞τ
cannot also be finite so it has to be either ∞ or −∞. But by (4.1), that implies k
goes to ∞ or −∞ (respectively) and hence ν′= −kτ goes to −∞, a contradiction.
Assume lims→∞ν = ∞. If lims→∞τ = −∞, then by (4.1), eventually k <
0 and thus ν′= −kτ < 0, a contradiction. Assume τ has a finite limit.
(4.1), eventually k < 0 and since −kτ = ν′> 0, τ eventually becomes positive.
But Lemma 4.3 then gives us an estimate ν ≤ Cτ, so τ is forced to go to ∞,
a contradiction. Finally, assume τ goes to ∞. Then eventually both τ and ν are
positive so by Lemma 4.3 we get an estimate ν ≤ Cτ. By (4.1), that implies k → ∞
and thus ν′= −kτ → −∞, a contradiction.
The limits in the other direction follow by symmetry.
By
?
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8 HOESKULDUR P. HALLDORSSON
?4
?224
?4
?2
2
4
(a) The symmetric curve (h = 1)
?4
?224
?4
?2
2
4
(b) Curve at distance 1 to
the origin (h = 1)
Figure 1. The symmetric curve is always embedded but when
the distance to the origin has reached a certain value (depending
on h) the curve stops being embedded.
Lemma 4.5. τ has a finite limit in each direction.
Proof. If lims→∞τ = ∞, then, by Lemma 4.4, eventually 1 + τν < 0 so τ′< 0 by
(4.2), a contradiction. Similarly, if lims→∞τ = −∞ then eventually 1 + τν > 0
so τ′> 0 by (4.2), a contradiction. The limits in the other direction follow by
symmetry.
?
Lemma 4.6. lims→±∞k = 0±.
Proof. This follows directly from (4.1) and Lemmas 4.4 and 4.5.
?
Lemma 4.7. lims→±∞r = ∞ and r has exactly one extremum, a global mininum.
Proof. The first statement follows from Lemma 4.4, since r2= τ2+ν2. The second
one follows from the fact that
4.2 that τ has at most one zero.
d
dsr2= 2τ and the observation in the proof of Lemma
?
A consequence of the previous two lemmas is that each of τ and k does indeed
have a zero, and is negative before it and positive after it.
Lemma 4.7 implies that X has a unique point closest to the origin and consists
of two arms coming out from this point and strictly going away to infinity. Hence,
each of these arms is properly embedded but they can cross each other. The limiting
growing direction of the arms is given by the following lemma.
Lemma 4.8. lims→±∞
Proof. This follows directly from Lemmas 4.4 and 4.5, sincerT
rT
X= ±i.
X=τ−iν
r
.
?
Now, let φ be the angle of X. Then we have the following.
Lemma 4.9. lims→±∞φ = +∞.
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HELICOIDAL SURFACES ROTATING/TRANSLATING UNDER THE MCF9
?4
?224
?4
?2
2
4
(a) Curve at distance 1 to
the origin (h = 5)
?4
?224
?4
?2
2
4
(b) Curve passing close to
the origin (h =
1
16)
Figure 2. As h goes to ∞, the curves converge to the rotating
curves of the curve shortening flow. When h is close to 0, the
curvature can increase dramatically where the curve passes close
to the origin.
Proof. Since
Lemma 4.7.
dφ
dlog(r)= −ν
τgoes to ∞ in each direction, the result follows from
?
This means each arm spirals infinitely many times around the origin. Moreover,
we have the following.
Lemma 4.10. lims→±∞θ = +∞. In other words,?∞
Proof. Follows from the identity φ = θ +arg(τ +iν), Lemma 4.9 and the fact that
arg(τ + iν) has finite limit in each direction.
s0kds = +∞ and?s0
−∞kds =
−∞, so each arm has infinite total curvature.
?
This concludes the proof of Theorem 4.1.
Our parametrization for the helicoidal surfaces does not remain valid in the case
h = 0, but we can nevertheless investigate what happens to the generating curves
in the limit h → 0. When h = 0, Equations (4.1) and (4.2) become
k =τ3− ν
r2
and
τ′=τ2
r2(1 + τν)
ν′=τν − τ4
r2
.
In this ODE system, every point on the ν-axis except for (0,0) is a fixed point.
These fixed points correspond to solutions in which the curve X is a circle around
the origin. The other trajectories are found by noticing that the functionν
constant. Therefore, the trajectories are the algebraic curves τ3+ τν2+ 2ν = 2aτ
where a is any real constant, representing the slope of the trajectory as it goes
τ+1
2r2is
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10 HOESKULDUR P. HALLDORSSON
through the origin. Unless a = 0, the curvature k blows up as the trajectory goes
through the origin. Therefore, we can only take half of the trajectory to create the
smooth curve X. That curve X is embedded, has one end spiraling out to infinity
and the other one spiraling into the origin in finite time where its curvature blows
up. When a = 0, we get a complete embedded curve X.
These curves give us an idea about the limiting behaviour of our generating
curves as h → 0. In the case of the circular curves corresponding to the fixed points
on the ν-axis, it is easy to make a precise convergence statement.
Theorem 4.11. Assume Φ is a smooth function on R3\ {(0,0,0)} and A ?= 0.
For each h ≥ 0, let (τh,νh) be the solution to the initial value problem
If (τ0,ν0) is the constant solution (0,A), then as h → 0, (τh,νh) → (τ0,ν0) in the
sense of uniform Ck-convergence on compact subsets of R for each k.
τ′
ν′
h= 1 + νhΦ(τh,νh,h)
h= −τhΦ(τh,νh,h)
τh(0) = 0
νh(0) = A.
Proof. Here is a sketch of the proof. Let K be the annulus in the τν-plane defined
by
2|A| ≤ r ≤
ing the fundamental estimate (4.5) below, we get uniform Ck-convergence to the
constant solution (0,A) on any interval where (τh,νh) ∈ K. In particular, (τh,νh)
goes uniformly into L. Then, since |r′
fixed amount the interval where (τh,νh) ∈ K. Repeating this process, we manage
to cover all of R.
1
3
2|A| and let L be the ball or radius
1
4|A| around (0,A). By us-
h| ≤ 1 by (3.5), we can extend uniformly by a
To simplify notation, let
F0(τ,ν,h) = 1 + νΦ(τ,ν,h),G0(τ,ν,h) = −τΦ(τ,ν,h)
and for k ≥ 0
Fk+1=∂Fk
∂τF +∂Fk
∂νG,
Gk+1=∂Gk
∂τ
F +∂Gk
∂νG.
The functions Fkand Gkare smooth on R3\ {(0,0,0)} and
?τ(k+1)
ν(k+1)
(4.3)
h
= Fk(τh,νh,h)
h
= Gk(τh,νh,h).
Define the constants D0= 0 and for k ≥ 0,
Dk+1=sup
K×[0,1](|∇Fk| + |∇Gk|).
Now, imagine we fix some h ∈ [0,1] and let I be an interval containing 0 such
that (τh(s),νh(s)) ∈ K for all s ∈ I. Then, by the fundamental theorem of calculus
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HELICOIDAL SURFACES ROTATING/TRANSLATING UNDER THE MCF11
?4
?224
?4
?2
2
4
(a) Curve at distance 1 to
the origin (h =1
8)
?4
?224
?4
?2
2
4
(b) Curve at distance 1 to
the origin (h =
1
16)
Figure 3. As h decreases to 0, the curve at distance 1 to the
origin converges to the unit cirlce.
and (4.3), we have for s ∈ I
|τ(k)
h(s) − τ(k)
0 (s)| ≤ |τ(k)
h(0) − τ(k)
?s
≤ hDk+
?s
0 (0)| +
?s
0
|τ(k+1)
h
(t) − τ(k+1)
0
(t)|dt
≤ hDk+
0
|Fk(τh(t),νh(t),h) − Fk(τ0(t),ν0(t),0)|dt
?s
0
|Fk(τh(t),νh(t),h) − Fk(τh(t),νh(t),0)|dt
+
0
|Fk(τh(t),νh(t),0) − Fk(τ0(t),ν0(t),0)|dt.
Here and below, the limits of integration should be swapped if s < 0.
the first integrand is bounded by hsupK×[0,1]|∇Fk|. The second is bounded by
|(τh(t),νh(t))−(τ0(t),ν0(t))|π
be joined by a piecewise smooth path in K of length less than or equal toπ
Therefore, if we put Ck= πDk+1, the estimate above and the corresponding one
for ν yield for s ∈ I
???(τ(k)
≤ 2hDk+ Ck
0
Now,
2supK×{0}|∇Fk|, since every two points p,q ∈ K can
2|p−q|.
(4.4)
h(s),ν(k)
h(s)) − (τ(k)
?s
0 (s),ν(k)
0(s))
???
h + |(τh(t),νh(t)) − (τ0(t),ν0(t))|dt.
In the case k = 0, we let u(s) = h + |(τh(s),νh(s)) − (τ0(s),ν0(s))| and then (4.4)
becomes
?s
u(s) ≤ h + C0
0
u(t)dt,s ∈ I.
Page 12
12 HOESKULDUR P. HALLDORSSON
By Gr¨ onwall’s inequality, this yields
u(s) ≤ heC0|s|,s ∈ I
or equivalently
|(τh(s),νh(s)) − (τ0(s),ν0(s))| ≤ h(eC0|s|− 1),
Putting this into (4.4) yields our fundamental estimate, valid for all k
s ∈ I.
(4.5)
???(τ(k)
h(s),ν(k)
h(s)) − (τ(k)
0 (s),ν(k)
0(s))
??? ≤ h
?
2Dk+Ck
C0(eC0|s|− 1)
?
2|A|]. Since
, s ∈ I.
We are now ready to finish the proof. Let I1be the interval [−1
h| ≤ 1 it is clear that (τh(s),νh(s)) ∈ K for all s ∈ I1and h ∈ [0,1]. Therefore,
as h → 0, we get by (4.5) that (τ(k)
Pick δ1so small that (τh(s),νh(s)) ∈ L for all s ∈ I1and h ∈ [0,δ1].
We continue by induction. Assume that for some n ≥ 1 we have (τh(s),νh(s)) ∈
L for all s ∈ Inand h ∈ [0,δn]. Let In+1be the closed interval obtained from Inby
extending it by
for all s ∈ In+1if h ∈ [0,δn]. Then as h → 0, by (4.5), (τ(k)
uniformly on In+1for each k. Finally, we pick δn+1so small that (τh(s),νh(s)) ∈ L
for all s ∈ In+1if h ∈ [0,δn+1].
Corollary 4.12. Under the assumptions of Theorem 4.11, let Xh be the curve
corresponding to (τh,νh) for some fixed choice of θ0. Then, as h → 0, Xhconverges
to the circle s ?→ iAe−is
Proof. Since Xh(s) = (τh(s) + iνh(s))eiθh(s), where θh(s) =
kh(s) = Φ(τh(s),νh(s),h), the result follows directly from Theorem 4.11.
2|A|,1
|r′
h,ν(k)
h) → (τ(k)
0 ,ν(k)
0) uniformly on I1for each k.
1
4|A| in each direction. Since |r′
h| ≤ 1 we have (τh(s),νh(s)) ∈ K
h,ν(k)
h) → (τ(k)
0,ν(k)
0)
?
A+iθ0, Ck-uniformly on compact subsets of R for each k.
?s
0kh(t)dt + θ0 and
?
For h > 0, circular generating curves around (0,0) correspond to cylinders so
this result can in some sense be interpreted as our surfaces converging to a cylinder.
5. Helicoidal minimal surfaces
Wunderlich studied the helicoidal minimal surfaces in [19]. We include them
here to investigate their limit behaviour as h → 0.
From Equation (3.3) we see that the curves that generate the helicoidal minimal
surfaces are given by
(5.1)
k = −
ν
r2+ h2.
Hence, the two-dimensional system of ODEs for τ and ν becomes
(5.2)
τ′=τ2+ h2
r2+ h2
ν′=
r2+ h2.
τν
The limit case h = ∞ yields the equation k = 0. That means X is just a straight
line and the surface Σ becomes a plane, an embedded minimal surface.
In the general case, direct calculations yield that the function
h2s
A2+h2 and by solving for ν, k and θ in terms of τ we get that the
curve is given by
ν
√τ2+h2is constant.
That gives τ =
(5.3)
X = (τ + iA
h
?
τ2+ h2)e−iA
harsinhτ
h+iθ0,τ ∈ R,
Page 13
HELICOIDAL SURFACES ROTATING/TRANSLATING UNDER THE MCF13
?10
?55 10
?10
?5
5
10
(a) h =1
2, A = 1
?10
?55 10
?10
?5
5
10
(b) h =1
8, A = 1
Figure 4. As h decreases to 0, the curve at distance 1 to the
origin converges to the unit cirlce.
where A in any real constant. Letting u =1
tion
X = (hsinhhu + iAcoshhu)e−iAu+iθ0,
harsinhτ
hyields Wunderlich’s parametriza-
u ∈ R.
These curves appear in Figure 4.
Note that |A| gives the distance of the curve from the origin. When A = 0, X
is just a straight line through the origin so the surface Σ is the helicoid, the well-
known embedded minimal surface (see [7]). In the other cases, the curve spirals
infinitely many times around the origin, intersecting itself infinitely many times. It
is symmetric with respect to reflections across the line through the origin and the
point ieiθ0. The curvature never changes sign, reaches its maximum absolute value
at the point closest to the origin and goes to zero in each direction. The limiting
growing angle is given by
lim
s→±∞
rT
X
=
±h − iA
√h2+ A2.
In the limit h → 0, Equations (5.1) and (5.2) become
k = −ν
r2
and
τ′=τ2
r2
ν′=τν
r2.
Like in the rotating surface case, every nonzero point on the ν-axis is a fixed
point of this system. The other trajectories are found by noticing that the function
ν
τis constant so they are straight lines of the form ν = aτ for some real constant
a. This gives τ′=
so θ = −alog|s|. Unless a = 0, the curvature k blows up at the origin so the
corresponding curve X is only defined for s > 0 (or s < 0) where it is given by the
1
a2+1so τ =
1
a2+1s and thus ν =
a
a2+1s. Moreover, k = −a
s
Page 14
14 HOESKULDUR P. HALLDORSSON
simple formula
(5.4)
X =1 + ia
a2+ 1se−ia log s=s1−ia
1 − ia,
s > 0.
These curves of course have a constant growing anglerT
Just as in the rotating surface case, we can apply Theorem 4.11 to get the same
type of convergence of certain generating curves to the circular curves corresponding
to the fixed points on the ν-axis. However, here this convergence can also be seen
directly from (5.3).
The curve given by (5.4) can also be realized as a limit.
θ0= alog
X=
1−ia
√1+a2.
Put A = ah and
2
h(a2+1)into (5.3) to get
X =s + ia?s2+ h2(a2+ 1)2
When h → 0, this curve converges to the curve given by (5.4), Ck-uniformly on
compact subets of (0,∞) for each k.
a2+ 1
e−ialogs+√
s2+h2(a2+1)2
2
,s ∈ R.
6. Helicoidal surfaces of constant mean curvature
The constant mean curvature helicoidal surfaces were studied by Do Carmo and
Dajczer in [8] and further by Hitt and Roussos in [11] and Perdomo in [17]. We
include them here to give a classification of the immersed cylinders in this family
of surfaces.
From Equation (3.3) we see that the helicoidal surfaces of constant mean curva-
ture H are given by the equation
k = −H(τ2+ h2)
h(r2+ h2)
3
2
−
ν
r2+ h2.
In the special case h = ∞, the equation reduces to k = −H. Therefore, X is a circle
and the surface Σ is a cylinder, a well-known embedded constant mean curvature
surface.
Now, by rescaling the surface and parametrizing X backwards if necessary, we
can assume H = −1. Then we have the following two-dimensional system of ODEs
for τ and ν, which also appeared in [17]
(6.1)
τ′=
τ2+ h2
h(r2+ h2)(h + ν
ν′=hτν − τ(τ2+ h2)
h(r2+ h2)
?
3
2
τ2+ h2)
The system has a unique fixed point (0,−1). To find the other trajectories we make
a change of variables. First, define the norm-preserving involution
?√r2+ h2
Φh: R2→ R2,
(x1,x2) ?→
?x2
2+ h2x2,
h
?x2
2+ h2x1
?
,
where r2= x2
let x and y be the functions given by (x,y) = Φh(ν,τ). Direct calculations using
1+ x2
2. Note that Φ∞is just reflection across the line x1= x2. Then
Page 15
HELICOIDAL SURFACES ROTATING/TRANSLATING UNDER THE MCF15
(6.1) yield that the functions x and y are solutions to the system of ODEs
x′=
h
?y2+ h2(y + 1)
−hx
?y2+ h2.
ds=
y′=
Now, introduce a new variable u such thatdu
h
√
y2+h2. Then we get the simpler
system
dx
du= y + 1
dy
du= −x,
whose trajectory space consists of concentric circles around the fixed point (0,−1).
The solutions are parametrized by (x,y) = (Rcosu,−1−Rsinu), where R ∈ [0,∞)
is any constant. Now, the solutions to the original system are given by (ν,τ) =
Φh(x,y), so they are also periodic. Since Φh is norm-preserving, each solution is
contained in an annulus with center at the origin, inner radius |R − 1| and outer
radius R + 1. It touches each boundary once in every period. The corresponding
curve X is therefore also contained in the same annulus and consists of repeated
identical excursions between the two boundaries of the annulus. Note that X passes
through the origin if and only if R = 1.
Now, let’s examine a whole excursion, i.e., we go from the outer boundary, touch
the inner boundary and go back out to the outer boundary. This corresponds to
the (τ,ν) trajectory going through one period. Let ∆θ denote the difference be-
tween the values of the tangent angle before and after the excursion. There are two
possibilities:
1) The angle difference ∆θ is of the formp2π
integers p and q. Then after q excursions the curve X closes up. Thus, X is a
closed curve, an immersed S1, with q-fold rotational symmetry and the surface is
an immersed cylinder. The number p is the rotation number of X.
q
for some relatively prime, positive
2) The angle difference ∆θ is not of this form. In this cases the curve X never
closes up and takes infinitely many excursions. It is dense in the annulus.
In the rest of this section, we find all possible values of ∆θ in order to obtain a
classification of the closed immersed curves mentioned above, in the spirit of the
classification of the immersed self-shrinkers of the curve shortening flow in the plane
given by Abresch and Langer in [1].
Since dθ = kds, ∆θ is given by integrating k over one period:
h√r2+ h2
y2+ h2
h√R2+ 1 + 2Rsinu + h2
(1 + Rsinu)2+ h2
(6.2)
∆θ =
?
?π
kds =
?π
−π
?
−
y
h√r2+ h2
?
du
=
−π
?
+
1 + Rsinu
h√R2+ 1 + 2Rsinu + h2
?
du.
Page 16
16 HOESKULDUR P. HALLDORSSON
Clearly, ∆θ is smooth in R and h on [0,∞) × (0,∞]. Note that when h = ∞,
k = 1 so ∆θ = 2π for all R and therefore
It turns out to be convenient to work with ∆φ, where φ is the angle of the curve
X, i.e., X = reiφ. Since X = (τ +iν)eiθ, we get the following relation between ∆φ
and ∆θ, by looking at the angle difference of the (τ,ν) trajectory:
∂
∂R∆θ = 0.
(6.3)∆φ =
∆θ
∆θ − π
∆θ − 2π
if 0 ≤ R < 1,
if R = 1,
if R > 1.
Since dφ = −ν
r2ds, ∆φ is given by integrating over one period:
(6.4)
∆φ = −
?
ν
r2ds = −
√R2+ 1 + 2Rsinu + h2
h(R2+ 1 + 2Rsinu)
?π
π
√r2+ h2
hr2
ydu
=
?π
−π
(1 + Rsinu)du.
We should mention that the integral (6.2) also appears in [17], whereas the integral
(6.4) is used in [11].
Lemma 6.1. When R=0, ∆φ takes the value
2π√h2+1
h
.
Proof. This follows directly from setting R = 0 in Equation (6.4).
?
The complete elliptic integral of the second kind shows up in our next lemma.
It is defined as
?
and gives1
4of the circumference of an ellipse with semi-major axis 1 and eccentricity
k.
Lemma 6.2. When R=1, ∆φ takes the value2√h2+4
E(k) =
π
2
0
?
1 − k2sin2θdθ.
h
E(
2
√h2+4).
Proof. Set R = 1 in Equation (6.4) to get
∆φ =
?π
√h2+ 4
h
−π
√h2+ 2 + 2sinu
2h
du
=
?
π
2
−π
2
?
1 −
4
h2+ 4sin2θdθ,
where in the second equation we replaced sinu with cosu, set u = 2θ and used
cos2θ = 1 − 2sin2θ.
The value in the lemma is half the circumference of an ellipse with semi-minor
axis 1 and semi-major axis
h
.
?
√h2+4
Lemma 6.3. For each h, limR→∞∆φ = 0.
Proof. If R ≥ 2, then r ≥ 1 so the integrand in (6.4) is bounded by
dominated convergence, limR→∞∆φ =?π
Lemma 6.4. For each h, ∆φ is strictly decreasing in R.
√h2+1
h
. By
−π
sinu
h
= 0.
?
Page 17
HELICOIDAL SURFACES ROTATING/TRANSLATING UNDER THE MCF17
(a) h = 1, p = 4, q = 3 (b) h = 1, p = 6, q = 5
Figure 5
Proof. We need to show that
case h = ∞, it is enough to show that
∂
∂h
∂
∂R∆φ < 0 when R ?= 1. Since
∂
∂h
∂
∂R
∂
∂R
−π
?π
∂
∂R∆φ = 0 in the limit
∂
∂R∆φ > 0. Now,
∂
∂R∆φ =
∂
∂h∆φ
?π
=
y
h2√r2+ h2du
Rcos2u − h2sinu
h2(R2+ 1 + 2Rsinu + h2)
=
−π
3
2du.
The first term is clearly positive. However, so is the integral of the second term,
since the denominator is smaller when sinu takes its negative values.
?
Now, from the previous lemmas and Equation (6.3) we conclude that for each h,
∆θ is a smooth, decreasing function of R satisfying
• ∆θ =2π√h2+1
• ∆θ =2√h2+4
• limR→∞∆θ = 2π.
Therefore, we have the following theorem. The trichotomy corresponds to R = 1,
R < 1 and R > 1 respectively.
h
, when R = 0,
E(
√h2+4) + π, when R = 1,
h
2
Theorem 6.5. Let p and q be relatively prime positive integers, such that 1 <
p
q<
h
. Then there exists a unique (up to rotation) closed immersed curve
X, with rotation number p and q-fold rotational symmetry, such that it generates
a helicoidal surface with pitch h and constant mean curvature 1. Moreover, if
αh=
hπ
E(
• if
• if
• if
A few of these generating curves appear in Figures 5 - 11, where they are rescaled
to have the same outer radius.
√h2+1
√h2+4
2
√h2+4) +1
q= αh, X goes through the origin,
p
q> αh, X has winding number p around the origin,
p
q< αh, X has winding number p − q around the origin.
2then the following holds:
p
Page 18
18 HOESKULDUR P. HALLDORSSON
(a) h = 1, p = 11, q = 8(b) h = 1, p = 19, q = 15
Figure 6
(a) h =1
2, p = 2, q = 1
(b) h =1
2, p = 3, q = 2
Figure 7
(a) h =1
2, p = 5, q = 3
(b) h =1
2, p = 11, q = 6
Figure 8
Page 19
HELICOIDAL SURFACES ROTATING/TRANSLATING UNDER THE MCF19
(a) h =1
5, p = 15, q = 4
(b) h =1
5, p = 13, q = 5
Figure 9
(a) h = 2, p = 11, q = 10 (b) h = 2, p = 29, q = 26
Figure 10
(a) h = 5, p = 52, q = 51(b) h = 5, p = 55, q = 54
Figure 11
Page 20
20 HOESKULDUR P. HALLDORSSON
Acknowledgements
The author would like to thank Eric Marberg for reading over the draft and
providing valuable feedback. He would also like to thank Oscar Perdomo for sharing
with him his recent paper on the CMC helicoidal surfaces. Finally, he would like
to thank his advisor, Tobias Colding, for guidance and support.
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MIT, Department of Mathematics, 77 Massachusetts Avenue, Cambridge, MA 02139-
4307.
E-mail address: hph@math.mit.edu