EUCLIDEAN GAUSS MAP

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EUCLIDEAN GAUSS MAP

RICARDO SA EARP AND ERIC TOUBIANA

Received 6 July 2004 and in revised form 19 November 2004

We present another proof of a theorem due to Hoﬀman and Osserman in Euclidean space concerning the determination of a conformal immersion by its Gauss map. Our approach depends on geometric quantities, that is, the hyperbolic Gauss mapGand formulae obtained in hyperbolic space. We use the idea that the Euclidean Gauss map and the hyperbolic Gauss map with some compatibility relation determine a conformal immersion, proved in a previous paper.

1. Introduction

Throughout this paper we will consider surfaces immersed in Euclidean space thinking that,locally, the surface is immersed into the upper half-space R³+= {(u,v,w), w >0}. Of course there are two important metrics inR³+: the standard Euclidean metric and the hyperbolic metric given by ds²=(1/w²)·(du²+ dv²+ dw²). Notice thatR³+ endowed with the hyperbolic metric is the upper half-space model of hyperbolic spaceH³.

Our main goal is to show how certain geometric quantities relating to the oriented Euclidean Gauss map, the hyperbolic Gauss map, and the coordinate functions for conformal immersions of a planar domain into the upper half-space model of hyperbolic space, can be used to infer that the oriented Euclidean Gauss map determines locally a conformal immersion in Euclidean space with nonvanishing mean curvature, up to a homothety and Euclidean translation. This is a theorem due to Hoﬀman and Osserman [8]

on account of a theorem by Kenmotsu [9]. Their proof specializes to dimension-3 results for the Euclidean Gauss map of conformal immersions inRⁿ. The main idea of our proof is that both Euclidean Gauss mapEand hyperbolic Gauss mapGtogether with some compatibility relation determine a conformal immersion into upper half-space. Thus, we use in the proof the relation between Euclidean and hyperbolic geometry. We hope that this approach gives a significant insight into the theory.

LetX:U⊂CR³+,z=x+iy→X(z) be an oriented conformal immersion of a simply connected domainU into the upper half-spaceR³+. Let N be the Euclidean Gauss map of X such that (Xx,Xy,N)(z) is a positively oriented basis of R³ for eachz∈U,

International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences 2005:00 (2005) 1–7 DOI:10.1155/IJMMS.2005.1

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whereXx=∂X/∂xandXy=∂X/∂y. That is, N= Xx∧Xy

Xx∧Xy, (1.1)

where| · |stands for the Euclidean norm and∧for the Euclidean vector product. We call N=(N1,N2,N3) theoriented Euclidean Gauss mapof X, or more briefly theEuclidean Gauss mapofX.

LetΠ:S²→C∪ {∞}be the standard stereographic projection. We set E=Π◦N=N1+iN2

1−N3 , (1.2)

so that

N=(2 E, 2E,EE−1)

EE+ 1 . (1.3)

We callEtheoriented Euclidean Gauss mapofX.

Given a point p=X(z), the geodesic ray lying in hyperbolic spaceH³, issuing from X(z) in the direction ofN, fits the asymptotic boundaryC∪ {∞}at a pointG(z). This defines an applicationG:U→C∪ {∞}, called the hyperbolic Gauss map.

For everyC¹-function f :U→C∪ {∞}, the notation fz(resp., fz) stands for the de- rivative of f with respect toz(resp.,z), that is,

fz=1 2

fx−i fy

, fz=1 2

fx+i fy

. (1.4)

We will prove the following theorem.

Theorem1.1 (Hoﬀman and Osserman). Let Ebe aC³ function in a simply connected domainΩ. AssumeE_z=0everywhere. IfEsatisfies

E_zz

E_z ⁻2E E_z EE+ 1

z= (E)_zz

(E)z −2E (E)_z EE+ 1

z. (1.5)

then there is a conformal immersionX:ΩR³ ofΩintoR³ with nonvanishing mean curvatureH. Furthermore,Eis the oriented Euclidean Gauss map and any geometric quan- tity related to the immersionXcan be determined in terms ofE. The mean curvatureH is determined up to a positive multiplicative constant and is given by

HEzz−2E E_zE_z EE+ 1

=HzEz (Kenmotsu’s equation). (1.6) IfXandXare two conformal immersions of a simply connected domainUintoR³with the same oriented Euclidean Gauss mapEand nonvanishing mean curvature, thenX=Xup to a homothety and Euclidean translation.

Remark 1.2. The authors have deduced a Kenmotsu-type theorem in hyperbolic space [15]. In fact, given aC²functionEand aC¹functionᏴon a simply connected domain

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Ω, we have stated a necessary and suﬃcient compatibility equation to obtain a conformal (possibly branched) immersion ofΩinto hyperbolic space with oriented Euclidean Gauss mapEand hyperbolic mean curvatureᏴ. Namely, the equation in hyperbolic space cor- responding to (1.6) is the following:

2 + (Ᏼ−1)(1 +EE)(1 +EE)Ezz−21 + (Ᏼ−1)(1 +EE)EEzEz−(1 +EE)²EzᏴz=0.

(1.7) Notice that Aiyama and Akutagawa proved a related result in the 3-sphere, see [1].

2. Formulae for conformal immersions inR³+

We now recall formulae about the immersionX(z)=(u(z) +iv(z),w(z)), established by the authors in a previous work [17].

Proposition2.1. LetX:U⊂CR³+,z=x+iy→X(z)be a conformal immersion of a simply connected domainUinto the upper half-spaceR³+. LetH be the Euclidean mean curvature. AssumeE(z)= ∞, then

We removed “We have”

fromProposition 2.1 and replaced it with

“then” to avoid using personal pronouns in propositions. Please check similar cases throughout.

u+iv=G−wE, (2.1)

Gz=wEz, (2.2)

ds²=Gz−wE_z²|dz|² (induced Euclidean metric), (2.3) wz= E

EE+ 1

Gz−wEz

, (2.4)

HGz−wEz

= −2 E_z

EE+ 1. (2.5)

Remark 2.2. (1) Notice that from (2.5), it follows thatX is a minimal conformal immersion into Euclidean space if and only if the Euclidean Gauss map is a meromorphic function.

(2) The following formula has important significance for surfaces theory in hyperbolic space. LetᏴbe the mean curvature ofXin hyperbolic space

(1−Ᏼ)Gz−wEz

=2 Gz

EE+ 1. (2.6)

Thus,X has mean curvature 1 in hyperbolic space if and only if the hyperbolic Gauss map is a meromorphic function. This is a result proved by Bryant [4] and seems to be known by Bianchi, see [3,6]. Mean curvature one surfaces in hyperbolic space has been intensively studied after Bryant’s work. See, for example, [2,5,7,11,12,13,14,16,19].

(3) We remark that in the context where we allow branch point, that is, the metric ds²=0 at some points, then from (2.3) and (2.5), we conclude thatz0is a branch point of an immersion with nonvanishing mean curvatureHif and only ifEz(z0)=0.

(4) When the mean curvatureHis a real constant, we sayH=cte, we infer from (1.6), Kenmotsu’s equation,

Ezz=2E E_zE_z

EE+ 1. (2.7)

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The above equation shows thatE:Ω⊂C→C∪ {∞}, with the sphereC∪ {∞}equip- ped with the standard metric dσ²=(4/(1 +ζζ)²)|dζ|², is a harmonic map [18]. We observe that ifH=cte=0, then a pointzsuch thatE_z=0 is a zero of the Hopf functionφ related to the harmonic mapE, sinceφis holomorphic, we deduce that the set of points {E_z=0}is discrete.

(5) An equation relating the mean curvatureHin Euclidean space and the mean cur- vatureHH³in hyperbolic space can also be deduced, that is,

HH³−1E_z=HGz. (2.8)

(6) An important equation in hyperbolic space is given by E_zz=E E_zE_z

EE+ 1. (2.9)

Indeed, this equation is satisfied for mean curvature one surfaces in hyperbolic space, taking into account (1.7). Any solution of this equation gives rise to mean curvature one conformal immersion into hyperbolic space [17]. All solutions of this equation can be explicitly given by meromorphic data (h,T) according to a quite simple formula [17].

It can be considered as a harmonic map, taking a complete metric inCgiven by dσ²= (2/(1 +ζζ))|dζ|². As far as we know, the solutions of (2.7) cannot be expressed in a simple way. However, there are results of Ritor´e [10].

3. Proof of the theorem

The idea of the proof to obtain existence is the following. We outline it locally. Using (1.5), it is straightforward to infer a candidateH for the mean curvature, up to a multiplicative positive constant. Then, using the equations established for hyperbolic space, seeProposition 2.1, we obtain a candidate for the heightw >0. Now we recall that the Euclidean Gauss mapEand hyperbolic Gauss mapG, with some compatibility relation, determine a conformal immersion into upper half-space [13,16]. Thus we expect to ob- tainGwith the aid of (1.5). This happens to be true, and usingE,w, andG, we are able to find the horizontal coordinatesu,vand to finally get the desired immersion.

We will now proceed with the details of the proof. Note first that from integrability condition (1.5), we deduce the existence of a real functionFdefined in the whole simply connected domainΩsuch that

E_zz−2E E_zE_z

EE+ 1⁼FzE_z. (3.1)

We define a realC¹positive functionHonΩsettingH:=e^F. Notice thatHis defined up to a multiplicative positive constant, sinceFis well defined up to additive constant.

We infer therefore Kenmotsu’s equation (1.6). We fix now the (height)won the wholeΩ by choosing a solution of the following equation:

wz= E EE+ 1^·

−2E_z

HEE+ 1. (3.2)

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This can be done since a computation shows that (1.6) is the compatibility equation for (3.2).

Let∪n≥1Un=Ωbe an exhaustion ofΩby open simply connected subdomainsUn⊂⊂

Ω, that is,Un⊂Un+1⊂ΩandUnis compact.

We will now proceed with the construction of the coordinate functions (u+iv,w) of the desired immersion by constructing an auxiliary conformal immersionX⁽ⁿ⁾ofUninto R³+by recurrence such that the height ofX⁽ⁿ⁾is equal tow+αn>0.

We will see that X⁽ⁿ⁺¹⁾−(0, 0,αn+1)=X⁽ⁿ⁾−(0, 0,αn) on Un, n=1, 2,.... This will provide the desired immersion. Notice that (3.2) for a conformal immersionXofΩinto R³+can be deduced with the aid of (2.4) and (2.5). Of course this choice is done up to an additive constant.

We will first work inU1. We choose a positive constantα1such thatw1:=w+α1>0.

We defineG=G1onU1by Gz=

w1− 2 H(EE+ 1)

E_z, Gz=w1E_z. (3.3) With a bit of surprise, we find that the compatibility of (3.3) is still Kenmotsu’s equation (1.6). We observe that (3.3) for a given conformal immersion follows again from Proposition 2.1. Of course this choice is done up to an additive complex constant. Now using (2.1), we obtain the horizontal coordinatesu1+iv1, that is,u1+iv1:=G1−w1E.

Hence we get aC² mapX⁽¹⁾:U1→R³+, byX⁽¹⁾(z) :=(u1(z) +iv1(z),w1(z)), z=x+iy. On account of the above construction, we get (we writew=w1,G=G1for simplicity)

wz= E 1 +EE

Gz−wEz+Gz−wEz

= E 1 +EE

Gx−wEx

= E 1 +EE

Gz−wE_z−

Gz−wE_z= E

1 +EEiGy−wE_y.

(3.4)

Thus

wx=2

EGx−wEx EE+ 1

, wy=2

EGy−wEy EE+ 1

. (3.5)

We infer from (3.5) that Xx⁽¹⁾·Xx⁽¹⁾= |Gx−wE_x|²= X⁽¹⁾y ·X⁽¹⁾y = |Gy−wE_y|²=

|Gz−wEz|²andXx⁽¹⁾·Xy⁽¹⁾=0 inU1, where·stands for the inner product in Euclidean space. We conclude therefore thatX⁽¹⁾is indeed a conformal immersion. Now computa- tions show that the unit normalN toX⁽¹⁾ is given byN=(2 E, 2E,EE−1)/(EE+ 1), henceEis the oriented Euclidean Gauss map andH (nonvanishing) is the mean curvature of the immersionX⁽¹⁾, see [17] for further details. We point out that fixingwand H, the immersionX⁽¹⁾is uniquely defined up to a horizontal translation. We define the immersionXonU1settingX:=X⁽¹⁾−(0, 0,α1). Of course the functionwis the height, as desired. Now we will work in U2. Ifw+α1>0 on U2 we are done, if not we first choose a positive constantα2> α1such thatw2:=w+α2is positive onU2. Now making again the same construction as before, we can find a conformal immersionX⁽²⁾ ofU2

intoR³+whose oriented Gauss map isEand a mean curvature isHsuch thatu2+iv2=

We added the highlighted “a”. Please check.

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u1+iv1 onU1. In fact, on account of (3.3), we deduce (G2)_z=(G1+ (α2−α1)E)_z and (G2)_z=(G1+ (α2−α1)E)_z. Hence, there is a complex constanta+ib such that G2= G1+ (α2−α1)E+a+ibinU1. Thus doing a horizontal translation, if necessary, we may assume thatu2+iv2=u1+iv1 onU1. This impliesX⁽²⁾−(0, 0,α2)|U1=X⁽¹⁾−(0, 0,α1).

We then defineX:=X⁽²⁾−(0, 0,α2) onU2. Now we can infer by recurrence that there ex- ist positive constantsαn,n∈N^∗, and conformal immersionsX⁽ⁿ⁾ofUnintoR³+such that X⁽ⁿ⁺¹⁾−(0, 0,αn+1)=X⁽ⁿ⁾−(0, 0,αn) onUn,n=1, 2,....We therefore inductively define a conformal immersionX ofΩintoR³, settingX:=X⁽ⁿ⁾−(0, 0,αn) onUnwith nonvanishing mean curvature, whose Euclidean Gauss map isE. The uniqueness part of the statement uses the same ideas and we will outline it to the reader. Firstly, notice that from (1.6), it follows that the mean curvature of a given conformal immersion with Euclidean Gauss mapEis determined up to a positive multiplicative constant. Secondly, using (3.2), we see that the height (3.2) is determined up to an additive real constant and the same multiplicative constant. Finally, we conclude therefore that two given immersions with nonvanishing mean curvature are the same up to a homothety and Euclidean translation.

Acknowledgments

The first author likes to thank theInstitut de Mathématiques de Jussieufor their hospitality and support during his visit in the first semester of 2004 asProfesseur Invité. Also thanks to Conselho Nacional de Desenvolvimento Cientifico e Tecnol ógico (CNPq) and PRONEX of Brazil for partial financial support.

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Ricardo Sa Earp:Departamento de Matemática, Pontif´ıcia Universidade Cat ólica do Rio de Janeiro, Rua Marquês de São Vicente, 225 -Gávea , Rio de Janeiro, Brazil

We added the highlighted parts in the address of the first author and removed

“24 453-900” from the address. Please check.

E-mail address:earp@mat.puc-rio.br

Eric Toubiana: Centre de Math´ematiques de Jussieu, Universit´e Paris 7-Denis Diderot, 2 Place

We changed

“Universit´e Paris VII, Denis Diderot” to

“Universit´e Paris 7-Denis Diderot”.

Please check.

Jussieu, F-75251 Paris Cedex 05, France E-mail address:toubiana@math.jussieu.fr