Helicoidal minimal surfaces in R3

Illinois Journal of Mathematics

Volume 57, Number 1, Spring 2013, Pages 87–104

S 0019-2082

HELICOIDAL MINIMAL SURFACES IN R

OSCAR M. PERDOMO

Abstract. In 1841, Delaunay (J. Math. Pures Appl. 6 (1841)

309–320) showed that the intersection of a constant mean cur-

vature surface of revolution in R

and a plane Π that contains

itsaxisofsymmetryl can be described as the trace of the focus

of a conic when this conic rolls without slipping in the plane Π

along the line l. In the same way surfaces of revolution are foli-

ated by circles perpendicular to the axis of symmetry, helicoidal

surfaces are foliated by helices, all of them symmetric to a line l.

Roughly speaking, helicoidal surfaces are surfaces invariant un-

der a screw-motion. In this paper, we show that the intersection

of a helicoidal minimal surface S in R

and a plane π perpen-

dicular to line l—where l is the axis of symmetry of the screw

motion—is characterized by the property that if we roll the curve

C = S ∩ π on a ﬂat treadmill located on another plane Π, then,

the point P = π ∩ l describes a hyperbola on the plane Π centered

at the ﬁxed point of contact of the treadmill with the curve C.

This way of generating a curve using another curve, similar to the

well known “Roulette,” was introduced by the author in (Paciﬁc

J. Math. 258 (2012) 459–485) and it was called the “Treadmill-

Sled.” We will also prove several properties of the TreadmillSled,

in particular we will classify all curves that are the TreadmillSled

of another curve.

1. Introduction

The surface of revolution generated by the regular curve α =(y(s),z(s))

with z(s) =0is given by

φ(s, t)=



z(s)sin(t),y(s),z(s) cos(t)



Received March 7, 2012; received in ﬁnal form June 26, 2012.

2010 Mathematics Subject Classiﬁcation. 53C42, 53A10.

2014 University of Illinois

88 O. M. PERDOMO

Figure 1. An unduloid and the construction of its proﬁle curve.

The curve α is called the proﬁle curve. In [2], Delaunay proved that a

surface of revolution has constant mean curvature, CMC, if and only if, it is

a sphere, a cylinder or if its proﬁle curve lies in the trace made by the focus

of a conic, when this conic rolls along the y-axis. When the conic used is a

parabola, the surface is minimal and it is called catenoid; if the conic used is a

hyperbola, the surface is called a nodoid and if the conic used is an ellipse, the

surface is called an unduloid. Since an ellipse has two foci, the trace of each

one of them generates an undoloid. It is not diﬃcult to see that these two

unduloids are essentially the same, one is a translation of the other. Figure 1

shows how the proﬁle curve of an unduloid is constructed using an ellipse.

When we roll the parabola its focus traces a curve of inﬁnite length. Fig-

ure 2 shows a catenoid and its proﬁle curve.

When we roll a set of hyperbolas, their foci trace two curves of ﬁnite length,

each curve generates a CMC surface. It can be proven that we can translate

one of these surfaces to obtain a smooth connected surface with constant

mean curvature. If we repeat this connected piece over and over, we obtain

a complete CMC surface. In Figure 3 we show, the trace of the foci of the

two-branch hyperbola when one of its branches is rolled on a line, the two-

piece CMC surface and the connected piece made by gluing the translation of

one of the connected components of the initial two-piece surface to the other

connected component.

In order to compare the main result in this paper with Delaunay’s result,

let us think of the operator Roll that takes regular curves in R

into curves

in R

given in the following way: For a regular curve α :[a, b] →R

,lets(t)

HELICOIDAL MINIMAL SURFACES IN R

Figure 2. A catenoid and the construction of its proﬁle curve.

Figure 3. A nodoid and the construction of its proﬁle curve.

denote the length of the curve from α(a)toα(t) and let us deﬁne

Roll(α)=







: T

is an oriented isometry in R



α(t)





s(t)



and dT





(t)

|α



(t)|







Notice that Roll(α) is the “Roulette” of the curve α when the rolling occurs

over the x-axis and the tracing point in the plane that contains α is the origin.

90 O. M. PERDOMO

With this operator, Delaunay’s theorem implies that if α :[0,l] → R

is a

piece of a conic with focus at the origin, then Roll(α) is the proﬁle curve of a

surface of revolution with constant mean curvature. Notice how the origin of

the curve α plays an important role in the deﬁnition of Roll(α).

The helicoidal surface generated by the regular curve α =(x(s),z(s)) is

given by

φ(s, t)=



x(s) cos(wt)+z(s)sin(wt),t,−x(s)sin(wt)+z(s) cos(wt)



where w>0 is ﬁxed. The curve α is called the proﬁle curve. Before explaining

our interpretation for the proﬁle curve of helicoidal minimal surface we need to

explain the notion of “TreadmillSled” introduced by the author in [7]. Think

of the operator TSS that takes regular curves in R

into curves in R

given

in the following way: for a regular curve α :[a, b] → R

, let us deﬁne

TSS(α)=







: T

is an oriented isometry in R



α(t)







and dT





(t)

|α



(t)|







The letters TSS stand for TreadmillSled Set. Notice that the only diﬀerence

in the geometrical interpretation for Roll(α) and TSS(α)isthatRoll(α)isthe

curve that we obtain by rolling the curve α without slipping along a line, and

TSS(α) is the curve that we obtain by rolling the curve α with full slipping so

that at every time the point of α that is making contact with the line is not

moving forward but staying in the same spot. Therefore, the curve α will not

look like rolling anymore but it will look like moving on a treadmill. Notice

how the origin of curve α plays an important role in the deﬁnition of TSS(α).

Figure 4 shows the TreadmillSled of the graph of a polynomial of degree 3.

In this particular example, the graph of the polynomial does not contain the

origin. We can easily see that, α contains the origin if and only if TSS(α)

contains the origin. We will use x and z for the coordinates of the plane that

Figure 4. The TreadmillSled of the graph of a cubic polynomial.

HELICOIDAL MINIMAL SURFACES IN R

contains the curve α(s)andu and v for the coordinates of the plane that

contains TTS(α).

We are ready to state our main theorem:

Theorem 1.1. A complete helicoidal surface

φ(s, t)=



x(s) cos(wt)+z(s)sin(wt),t,−x(s)sin(wt)+z(s) cos(wt)



is minimal if and only if the TreadmillSled of its proﬁle curve either is the

u-axis and φ is a helicoid or it is one of the branches of the hyperbola

−

=1 for some nonzero M .

Besides a detailed proof of the theorem above, in this paper we will show

several properties for the TreadmillSled Operator. For a motivation of the

study of this operator, we refer to the following papers [3], [5], [6], [7].

As more applications of the TreadmillSled, in [7], the author showed that

a helicoidal surface has zero Gauss curvature if and only if the TreadmillSled

of its proﬁle curve lies in a vertical semi-line contained in the upper or lower

plane (see Figure 5).

Also in the same paper the author showed that a helicoidal surface has

constant mean curvature 1 if and only if the TreadmillSled of its proﬁle satisﬁes

the following equation

(1.1) u

+ v

−

√

1+w

= M for some M>−

Also, Khuns and Palmer in [6] found a dynamical interpretation for heli-

coidal surfaces with constant anisotropic mean curvature.

Besides the nice dynamical interpretation of the TreadmillSled, this oper-

ator essentially represents a change of coordinates. Actually, one of the main

Figure 5. The TreadmillSled of the proﬁle curve of a ﬂat

helicoidal surface lies in a vertical semiline.

92 O. M. PERDOMO

aspects in the papers [6]and[7] is the simpliﬁcation of an ordinary diﬀerential

equation, ODE, after changing to “TreadmillSled coordinates,” that is, after

considering the ODE for the TS(α) instead of the ODE for α.

For purposes of a better understanding, we will ﬁnd an explicit parametriza-

tion for the TreadmillSled of a curve and we will be referring to this parame-

trization of the TreadmillSled as just TS; in this way, TS becomes an operator

that takes a parametrized regular curve into a parametrized curve. We will

show that this operator acts like the derivative operator for functions. For

example,

• Given α :[a, b] → R

,TS(α) is an expression of α and α



• If TS(α)=TS(β), then α and β diﬀer by a “constant.” This time the

constant does not represent a translation on the graph like in the case of

the derivative operator but it represents an oriented rotation that ﬁxes

the origin. More precisely, identifying R

with the complex numbers, if

TS(α)=TS(β)thenβ =e

α for some constant c.

• When γ is in the image of the operator TS, there is a formula for TS

−1

(γ)

that depends on γ, γ



and only one antiderivative. The ambiguity of this

antiderivative is responsible for the existence of the whole 1 parametric

family of curves with the same TreadmillSled.

• If we change the orientation of α, that is, if we consider the curve β(t)=

α(−t), then TS(β)(t)=−TS(α)(−t).

If we look at Figure 4, we notice in this example that, the curve that is a

TreadmillSled have the property that its velocity vector is horizontal where

the curve intercepts the v axis. This is not a coincidence; actually we will

show that a curve γ(t)=(u(t),v(t)) is the TreadmillSled of a regular curve α

if and only if

• v



(t)=−f(t)u(t) for some continuous function f and

• v(t)f (t) −u



(t) is a positive function.

From the ﬁrst property, we see that if u(t

) = 0, then v



) = 0 and there-

fore the velocity vector γ



) is horizontal on points along the v-axis. The

second property is the reason why a whole vertical line cannot be the Tread-

millSled of a regular curve. For a vertical line, u



(t) always vanishes and there-

fore when the line touches the u-axis, the function v(t)f (t) − u



(t)vanishes

making the second property fail at this point. Also notice that if the vertical

semi-line is contained in the v-axis, then by the relation v



(t)=−f(t)u(t), we

must have that v



(t) vanishes and therefore γ reduces to just a point. It is

easy to see that the TreadmillSled of a circle centered at the origin is just a

point in the v-axis. Notice that if the proﬁle curve of a helicoidal surface is a

circle centered at the origin then the surface is a cylinder.

HELICOIDAL MINIMAL SURFACES IN R

2. The φ-TreadmillSled of a curve

Given a curve α,wehavethatRoll(α)andTSS(α) are curves constructed

using a motion of the plane containing the curve α and a pencil placed at the

origin of this plane. It is clear that the curve obtained using the curve α and a

tracing point diﬀerent from the origin can be obtained by using a translation

of α and the origin as tracing point. Therefore, there is not loss of generality

by assuming that the point describing Roll(α)orTSS(α) is the origin. The

roulette, which is related with the rolling of a curve α along another curve β,

can be deﬁned as

Roulette(α)=







: T

is an oriented isometry in R



α(s)



= β(s)anddT



(s)



= β



(s)



In the previous expression we are assuming that both curves, α and β are

parametrized by arc-length, and again, this is the roulette described when the

tracing point is located at the origin.

2.1. Deﬁnition and interpretation. Let us start this section with a deﬁ-

nition that extends the notion of TreadmillSled. This extension is similar to

the one that we obtain when we generalize the notion Roll(α), which is the

Roulette over a line, to the notion of Roulette over any curve. In this gener-

alization from Roll(α) to Roulette(α), we are allowing the point of contact to

move along a general curve instead of just a line, and we are also adjusting or

moving the plane containing α to force the velocity vector α



(s) to be aligned

with the velocity vector β



(s) instead of just the vector (1, 0) (the velocity

vector of the horizontal line). Since the TreadmillSled of a curve α is a rolling

with full slipping, then, the point of contact is ﬁxed and we cannot generalize

this notion by moving the curve α along a curve β, nevertheless, we can force

the velocity vector of α to be aligned with any direction we wish.

Here is the deﬁnition of this generalization of the notion of TreadmillSled.

Definition 2.1. Given a regular curve α :[a, b] → R

and a function φ :

[a, b] →R, we deﬁne the φ-TreadmillSled of α as the set of points







: T

is an oriented isometry in R



α(s)







and dT





(s)

|α



(s)|





cos(φ(s))

sin(φ(s))



This set of points will be denoted by φ-TS(α).

Remark 2.2. Notice that the deﬁnition of the φ-TS(α) is independent of

the parametrization, it only depends on the orientation of the curve. That

94 O. M. PERDOMO

is, if h :[c, d] → [a, b] is a function with positive derivative and ˜α(t)=α(h(t))

and

φ(t)=φ(h(t)), then

φ-TS(˜α)=φ-TS(α).

It is not diﬃcult to see that the φ-TreadmillSled of α can be viewed as the

curve generated by doing the following steps:

• Imagine that the curve α is in a plane which can freely move. Moreover,

let us assume that there is a hole in the origin of this plane and also let us

assume that we have placed a pencil in this hole.

• Imagine that another plane, this one ﬁxed, contains a treadmill based at

the origin with a device that allows the treadmill to incline at any angle.

• The curve α in the moving plane will generate another curve in the ﬁxed

plane, the φ-TreadmillSled of α.

• The φ-TreadmillSled of α is the curve drawn on the ﬁxed plane by the

pencil located at the origin of the moving plane, when the curve α passes

on the treadmill with the property that, anytime the point α(s)isonthe

treadmill, the treadmill is aligned in the direction (cos(φ(s)), sin(φ(s))).

2.1.1. Generalization of the Roulette. We can extend the φ-TreadmillSled and

the Roulette in the following way.

Definition 2.3. Given two curves α :[0,l

] → R

and β :[0,l

] → R

parametrized by arc-length, and two functions ρ :[0,l

] → [0,l

]and

φ :[0,l

] →R, we can deﬁne the Generalized Roulette of α over β with func-

tions ρ and φ as

GR(α)=







: T

is an oriented isometry in R



α(s)



= β



ρ(s)



and dT



(s)





cos(φ(s))

sin(φ(s))



Remark 2.4. With this deﬁnition we have that if ρ(s)=s and φ(s) satisﬁes

the equation β



(s) = (cos φ(s), sin φ(s)), then GR(α) agrees with the Roulette

of α.Ifρ(s) = 0 for all s and β is any curve that satisﬁes β(0) = (0, 0), then

GR(α) agrees with the φ-TreadmillSled of α.

2.2. Parametrization of the φ-TreadmillSled. The following proposition

will provide a formula to ﬁnd the φ-TreadmillSled of a curve α.

Proposition 2.5. Let α :[a, b] → R

be a regular curve in R

, if α(s)=

(x(s),y(s))



x(s)

y(s)



, then,

(2.1) β(s)=A



θ(s)



α(s)=−A



−φ(s)





ρ(s)



α(s)

is a parametrization of the φ-TreadmillSled of α. Here

A(τ)=



cos(τ)sin(τ )

−sin(τ ) cos(τ)



HELICOIDAL MINIMAL SURFACES IN R

and

θ(s)=ρ(s) −φ(s)+π and



cos(ρ(s))

sin(ρ(s))



|α



(s)|



(s).

Proof. We will use the parameter s to describe points in the set φ-TS(α).

For a ﬁxed s ∈ [a, b], let us ﬁnd an oriented isometry of R

such that T

(α(s)) =





and dT

(



(s)

|α



(s)|



cos(φ(s))

sin(φ(s))



. We know that





= A



θ(s)









(s)



Notice that once we ﬁnd

θ(s), c

(s)andc

(s), using the Deﬁnition 2.1,we

get that β(s)=(c

(s),c

(s))

is a point in φ-TS(α); and therefore,

when we vary s in the interval [a, b], we obtain that β(s)=(c

(s),c

(s)) is

a parametrization of φ-TS(α).

Since





= A



θ(s)







and dT





(s)

|α



(s)|





cos(φ(s))

sin(φ(s))



we have that



θ(s)





cos(ρ(s))

sin(ρ(s))





cos(φ(s))

sin(φ(s))



and therefore,



θ(s)





−ρ(s)







= A



−φ(s)







Since A(τ

+ τ

)=A(τ

)A(τ

), the last equation implies that A(

θ(s) −

ρ(s)+φ(s))









, which implies that

θ(s)=ρ(s) −φ(s).

Now, using the equation T

(α(s)) =





we get that



(s)



= −A



θ(s)



α(s)=A



θ(s)



α(s).

Since β(s)=(c

(s),c

(s))

, then the proposition follows. 

2.3. TreadmillSled as a parametrized curve. The deﬁnition of Tread-

millSled of a curve given in the Introduction corresponds with the φ-Tread-

millSled when φ is the zero function. Sometimes we will view φ-TS(α) not as a

set but as the parametrized curve described in (2.1). In particular, we have the

following way to deﬁne the TreadmillSled not as a set but as a parametrized

curve.

Definition 2.6. Let α :[a, b] →R



x(s)

y(s)



bearegularcurve. Wedeﬁne

the TreadmillSled of α as the parametric curve TS(α):[a, b] → R

given by

TS(α)(s)=





(s)

+ y



(s)



−x



(s)x(s) −y



(s)y(s)

x(s)y



(s) −y(s)x



(s)



96 O. M. PERDOMO

The following remark gives us some insight about the nature of the operator

φ-TreadmillSled deﬁned in the set of regular curves. As we already noticed,

the φ-TreadmillSled of a curve is independent of the parametrization as long

as the orientation is preserved. Therefore, there is not loss of generality if

we assume that the curves in the domain of the operator φ-TreadmillSled are

parametrized by arc-length.

Remark 2.7. Let us deﬁne J =



0 −1



.Ifα is an arc-length parametrized

curve and



u(s)

v(s)



=TS(α), then,

u = −



α, α





and v =





,Jα



where ·, · is the Euclidean inner product.

With this deﬁnition of J we have that the curvature of α is k(s)=α



(s),

J(α



(s)). Notice that if α



(s)=



cos(ρ(s))

sin(ρ(s))



,thenk(s)=ρ



(s). Therefore, if we

know the curvature k(s) of a curve parametrized by arc-Length and a given

angle for the velocity vector, let’s say α



(a)=



cos(ρ

)

sin(ρ

)



,then

TS(α)=−A



ρ(s)



α(s)whereρ(s)=



k(u) du + ρ

2.4. The φ-TreadmillSled and complex numbers. The parametrization

given in Proposition 2.5 can be viewed in the following way.

Corollary 2.8. If we identify each point





∈ R

with the complex

number x

+ ix

, then

φ-TS(α)=e

iφ

TS(α).

For any curve α(s)=x

(s)+ix

(s). Moreover, if the function φ is ﬁxed,

then, the φ-TreadmillSled of two curves is the same, if and only if the Tread-

millSled of the curves is the same.

One of the reasons we introduce this extension to the notion of Tread-

millSled is because it provides an interpretation for the curve h(t)α(t)when

h :[a, b] → C, α :[a, b] → C are curves in the complex plane with |h(t)| =1.

The following corollary provides a way to program the inclination on a tread-

mill (ﬁnd the function φ) if we want to get the curve e

ig(t)

α(t)astheφ-

TreadmillSled of α.

Corollary 2.9. If α :[a, b] −→ C

∼

is a regular curve with curvature

function κ and g :[a, b] →R isafunction, then

ig(t)

α(t)=φ-TS(α),

where φ(t)=



κ(τ)|α



(τ)|dτ + ρ

+ g(t)+π and α



(a)=



cos(ρ

)

sin(ρ

)



HELICOIDAL MINIMAL SURFACES IN R

2.5. The inverse image of the TreadmillSled operator. In this section

we will point out that even though the TreadmillSled operator is not one to

one, given a curve β in the range of the operator TreadmillSled, there is only

a one-parametric family of curves whose image is the curve β, moreover, all

these curves in the inverse image of the curve β diﬀer only by an oriented

rotation about the origin. More precisely, we have the following proposition.

Proposition 2.10. Let α

:[a, b] → R

and α

:[a, b] → R

be two curves

parametrized by arc-length.TS(α

)=TS(α

) if and only i f α

(s)=A(τ)α

(s)

for some constant τ .

Proof. Let ρ

(s)andρ

(s) be functions such that



cos(ρ

(s))

sin(ρ

(s))



= α



(s). If

(s)=A(τ)α

(s), then



(s)=A(τ)



cos(ρ

(s))

sin(ρ

(s))





cos(ρ

(s) −τ )

sin(ρ

(s) −τ )



Therefore, we may assume that ρ

(s)=ρ

(s) − τ . Using Proposition 2.5,we

obtain that

TS(α

)=A(ρ

+ π)α

= A(ρ

−τ + π)A(τ )α

= A(ρ

+ π)α

=TS(α

Therefore, we have proven that if α

= A(τ )α

,thenTS(α

)=TS(α

Now let us assume that TS(α

)=TS(α

). Let us ﬁx an s

∈ [a, b]such

that |α

)| =0. Since TS(α

)(s

)=TS(α

)(s

)then|α

)| = |α

)|.

Let τ be a real number such that A(τ )α

)=α

) and let us consider

(s)=A(τ)α

(s). We have, TS(α

)=TS(α

), and moreover, we have that

)=α

). Using Deﬁnition 2.6 we get that if α(s)=



x(s)

y(s)



is a curve

parametrized by arc-length and TS(α)(s)=



u(s)

v(s)



,then

u(s)=−x



(s)x(s) −y



(s)y(s),

v(s)=x(s)y



(s) −y(s)x



(s).

For values of s such that x(s)

+ y(s)

> 0, we get that



(s)=−

x(s)

+ y(s)



x(s)u(s)+y(s)v(s)





(s)=

x(s)

+ y(s)



x(s)v(s) −y(s)u(s)



By the Existence and Uniqueness theorem of ordinary diﬀerential equations

we get that the conditions α

)=α

)andTS(α

)=TS(α

) imply that

(s)=α

(s) for all s near s

. Since both curves are regular, by a continu-

ity argument we conclude that the real number τ is independent of s

and

therefore α

(s)=α

(s) for all s. We then get α

= A(τ)α

for some τ .This

ﬁnishes the proof of the proposition. 

98 O. M. PERDOMO

2.6. The range of the TreadmillSled operator and a formu la for

the inverse of the TreadmillSled. In this section, we will characterize the

range of the operator TreadmillSled. Moreover, we will provide two easy-

to-check properties such that, any curve that satisﬁes them, must be the

TreadmillSled of another curve. Moreover, for any curve γ that satisfy these

two easy-to-check properties we will ﬁnd an explicit formula for a curve α

whose TreadmillSled is the curve γ. Under the assumption that a curve γ is in

the range of the operator TS, the formula for the inverse of the TreadmillSled

provided below was found in [6].

Proposition 2.11. Let γ(s)=



u(s)

v(s)



be a regular curve. γ is the Tread-

millsled of a regular curve α if and only if v



(s)=−f(s)u(s) for some contin-

uous function f and vf −u



> 0. More precisely, if f,v and u satisfy the two

previous conditions, and F (s) is an antiderivative of f (s), then,

TS(α)=γ where α(t)=−A



−F (t)



γ(t).

Proof. Let us assume that γ(s) is the TreadmillSled of a curve α.Letus

ﬁrst consider the case when α is parametrized by arc-Length. If we denote by

the curvature of α, then, using Remark 2.7 we obtain,

u = −



α, α





and v =





,Jα



Therefore,







,Jα







,Jα





= k



Jα



,Jα



= k





,α



= −k

and



= −1 −



α, α





= −1 −k



α, Jα





= −1+k



Jα,α





= −1+k

Taking f = k

, we conclude that v



= −fu and fv − u



=1. If we now

consider a regular curve ˜α,thenwehavethat˜α(t)=α(h(t)) where α is

parametrized by arc-length and h(t) is a function with h



(t) > 0. Therefore,

by either Remark 2.2 or by Deﬁnition 2.6, we get that if ˜γ =



˜u

˜v



is the Tread-

millSled of ˜α,then˜γ(t)=γ(h(t)) where γ =





is the TreadmillSled of α.

Since α is parametrized by arc-length, then v



(h(t)) = −f(h(t))u(h(t)) for

some continuous function f and fv −u



= 1. A direct veriﬁcation shows that

f(t)=h



(t)f(h(t)) is a continuous function that satisﬁes ˜v



(t)=−

f(t)˜u(t),

and

˜v(t)

f(t) − ˜u



(t)=h



(t)v



h(t)





h(t)



−h



(t)u



h(t)



= h



(t) > 0.

This inequality ﬁnishes the proof of one of the implications of the proposi-

tion. Let us assume now that the functions u, v are given and that v



= −fu

for some continuous function f , and that fv−u



> 0. We need to prove that

HELICOIDAL MINIMAL SURFACES IN R

if F



= f then

α(t)=−A



−F (t)



γ(t)

satisﬁes that TS(α)=γ. Using the fact that

dA(τ)

dτ

= A



τ +



= A(τ)A





= −A(τ)J

we get that



= −fA(−F )Jγ −A(−F )γ



= −A(−F )



fJγ + γ





Therefore,





,α







fJγ + γ



,fJγ + γ





= f

γ,γ+2f



Jγ,γ









,γ





Before we continue with the proof, we point out that the set {s : u(s)=0}

cannot contain an open set because of the regularity of the curve γ. Indeed, if

(a, b) is an open interval contained in {s : u(s)=0} and s

∈(a, b), then clearly



) = 0 and, using the equation v



)=f(s

)u(s

) we conclude that v



)

is also zero, which is impossible because we are assuming that γ



(s)doesnot

vanish for any s. As a consequence of this observation, we have that if two

continuous functions agree in the complement of the set {s : u(s)=0}, then,

they agree in the whole domain of the function γ.

For every point where the function u does not vanish, we have that f = −



Moreover, we get that





,α







)



+ v



−2



−vu





















fv −u





Since we have that fv−u



> 0, then we conclude that |α



|= fv−u



anytime

u does not vanish.

Since we have that {s : u(s)=0} does not contains an open interval, then,

by the continuity of the functions |α



,α



| and fv − u



we conclude that

|α



,α



| = fv −u



> 0 everywhere, and therefore α is a regular curve. More-

over, we have

TS(α)=

|α





−α



,α

α



,Jα



|α





−γ,fJγ + γ





fJγ + γ



,Jγ



|α





−γ,γ





fγ,γ+ γ



,Jγ



Since,

−



γ,γ





= −vv



−uu



= vfu −uu



= u



vf −u





= u|α



100 O. M. PERDOMO

and,

fγ,γ+





,Jγ



= −



+ v



+ uv



−vu



= −



−vu



= v



fv −u





= v|α



we conclude that TS(α)=γ. This completes the proof of the proposition. 

Remark 2.12. The regularity condition for the curve γ in the previous

proposition can be replaced by the weaker condition that γ



does not vanish

on an open set, that is, it can be replaced by the condition that the curve γ

is not constant on a open interval.

3. A dynamical interpretation for helicoidal minimal surfaces

Helicoidal minimal hypersurfaces have been understood for a long time. For

a detailed study, we refer to the last section of the last chapter of the book of

Diﬀerential Geometry by Graustein [4]. We have that all the isometric surfaces

(except for the catenoid) from the well-known family of surfaces that starts

with a helicoid and ends with a catenoid are helicoidal minimal surfaces. See

Figure 6. Actually, every helicoidal minimal surface belongs to one of these

families.

Figure 6. A helicoid (left), a helicoidal minimal surface

(center) and a catenoid (right) are part of a family of iso-

metric minimal surfaces.

HELICOIDAL MINIMAL SURFACES IN R

101

Figure 7. The TreadmillSled of the proﬁle curve of a heli-

coidal minimal surface is either the u-axis or a hyperbola. In

the notation of Theorem 3.1,inthisﬁgure,wehavew =1

and M =1.

A similar result for helicoidal CMC surfaces was proven in [1]byDoCarmo

and Dajczer. They proved that every helicoidal surface belongs to a family

of isometric surfaces that continuously move from an unduloid to a nodoid.

In this section, we provide a dynamical interpretation for the proﬁle curve of

a helicoidal minimal surface (see Figure 7). Let us state and prove the main

theorem in this section.

Theorem 3.1. A complete helicoidal surface

φ(s, t)=



x(s) cos(wt)+z(s)sin(wt),t,−x(s)sin(wt)+z(s) cos(wt)



is minimal if and only if the TreadmillSled of its proﬁle either is the u-axis

and φ is a helicoid or it is one of the branches of the hyperbola

−w

for some nonzero M .

Proof. Let us assume that the proﬁle curve α(s)=(x(s),z(s)) is parame-

trized by arc-length. If TS(α)(s)=(ξ

(s),ξ

(s)) then by Deﬁnition 2.6 we

have

(s)=−x



(s)x(s) −z



(s)z(s)andξ

(s)=x(s)z



(s) −z(s)x



(s).

Since we are assuming that α is parametrized by arc-length, there exists a

function θ such that α



(s) = (cos(θ(s)), sin(θ(s)). From the previous equation,

we get that



(s)=x



(s)z



(s) −z



(s)x



(s).

102 O. M. PERDOMO

With this deﬁnition of θ(s) and the deﬁnition of the function ξ

(s)and

(s) given above, we obtain that

x(s)=−ξ

(s) cos



θ(s)



+ ξ

(s)sin



θ(s)



and

z(s)=−ξ

(s)sin



θ(s)



−ξ

(s) cos



θ(s)



A direct computation shows that

ν =



sin(wt −θ)



1+w

, −

wξ



1+w

cos(wt −θ)



1+w



is a Gauss map of the immersion φ and, with respect to this Gauss map, the

ﬁrst and second fundamental forms are given by

E =1,F= −wξ

,G=1+w



+ ξ



and

e =





1+w

,f=

−w



1+w

,g=



1+w

Using the values above we get that the mean curvature H of the φ is given

H =

−w

+ θ



(1 + w

(ξ

+ ξ

))

2(1 + w

)

Therefore, the equation H = 0, that is, the minimality of the immersion φ,

implies



1+w

(ξ

+ ξ

)

From the deﬁnition of ξ

and ξ

we get that



= −x



x −





−z



z −





= θ



x sin(θ) − θ



z cos(θ) −1

= θ



−1.

Likewise we obtain that ξ



= −θ



. Therefore if φ is minimal, replacing

the expression for θ



above, we get that



1+w

(ξ

+ ξ

)

−1,



= −

1+w

(ξ

+ ξ

)

A direct veriﬁcation shows that if (ξ

(s),ξ

(s)) satisﬁes the diﬀerential

equation above then,

(s)



1+w

(s)

= M for some constant M.

If M =0, then ξ

(s) = 0. That is, in this case the TreadmillSled of α

is the u-axis. If M is not zero, we get by squaring the centered equation

HELICOIDAL MINIMAL SURFACES IN R

103

above, that the TreadmillSled lies in one of the branches of the hyperbola

−w

= 1. Therefore, one implication of the theorem follows. We will use

the inverse formula for the TreadmillSled, see Proposition 2.11,toprovethe

other implication. Let us assume that the TreadmillSled of the proﬁle curve

α is the horizontal line γ(s)=(γ

,γ

)=(−s, 0). In this case the function f

from Proposition 2.11 is given by f = −



(s)

=0,sincef is continuous we have

that the ﬁrst of the two easy-to-check properties holds. Since γ

f −γ



=1> 0,

then the second of the two easy-to-check properties holds too. In this case

F (s)=c where c is any constant. Therefore, it follows that an inverse of the

horizontal line is given by

α(s)=−



cos(−c)sin(−c)

−sin(−c) cos(−c)



−s





s cos(c)

s sin(c)



That is, α is a line through the origin and therefore the surface φ is a

helicoid. Now, let us assume that the TreadmillSled of the proﬁle curve sat-

isﬁes the equation

− w

= 1. We can assume that this TreadmillSled

is parametrized as γ(s)=(γ

,γ

)=(−

sinh(s),Mcosh(s)). In this case, the

function f from Proposition 2.11 is given by f = −



(s)

= Mw,sincef is

continuous we have that the ﬁrst of the two easy-to-check properties holds.

Since γ

f − γ



1+M

cosh(s), then the second of the two easy-to-check

properties holds too. In this case, we can take the function F (s)=Mws.

Therefore, using Proposition 2.11, we get that an inverse of the branch of the

hyperbola is

α(s)=−



cos(−Mws)sin(−Mws)

−sin(−Mws) cos(−Mws)



−

sinh(s)

M cosh(s)





M cosh(s)sin(Mws)+

sinh(s) cos(Mws)

−M cosh(s) cos(Mws)+

sinh(s)sin(Mws)



A direct veriﬁcation shows that if we deﬁne the functions x(s)andz(s)by

the equation α(s)=(x(s),z(s)), then, φ(s, t)=(x(s) cos(wt)+z(s)sin(wt),t,

−x(s)sin(wt)+z(s) cos(wt)) is minimal. This ﬁnishes the proof of the theo-

rem. 

Acknowledgment. I would like to thank the referee for valuables comments

and corrections that helped improve this paper.

References

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in R

, S

and H

, available at arXiv:1110.1068v1.

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Oscar M. Perdomo, Department of Mathematics, Central Connecticut State

University, New Britain, CT 06050, USA

E-mail address: [email protected]