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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.

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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,