Helicoidal surfaces rotating/translating under the mean curvature flow

Abstract

We describe all possible self-similar motions of immersed hypersurfaces in
Euclidean space under the mean curvature flow and derive the corresponding
hypersurface equations. Then we present a new two-parameter family of immersed
helicoidal surfaces that rotate/translate with constant velocity under 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.

Full text

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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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2 HOESKULDUR 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,