A relation between the right triangle and circular tori with constant mean curvature in the unit 3-sphere

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A relation between the right triangle and circular tori with constant mean curvature in the unit3-sphere

ABDÊNAGO BARROS

Departamento de Matemática-UFC, Bl 914, Campus do Pici, 60455-760 Fortaleza, CE, Brasil Manuscript received on May 30, 2003; accepted for publication on June 14, 2004;

presented byManfredo do Carmo*

ABSTRACT

In this note we will show that the inverse image under the stereographic projection of a circular torus of revolution in the 3-dimensional euclidean space has constant mean curvature in the unit 3-sphere if and only if their radii are the catet and the hypotenuse of an appropriate right triangle.

Key words:Flat torus, constant mean curvature, circular tori, stereographic projection.

1 INTRODUCTION

We will denote byT (r, a)the standard circular torus of revolution inR3_{obtained from the circle}

Ŵin thexz₋planecentered at(r,0,0)with radiusa < r, i.e.

T (r, a)₌(x, y, z)_∈R3_:(x2_+y2_−r)2

+z2₌a2.

Now let ρ_: S3

\ {n_{} →} R3_{be the stereographic projection of the Euclidean sphere S}3

= {x _∈

R4

:| x _|2

= 1_}, where n ₌ (0,0,0,1) is its north pole. The inverse image of a circular torus inR3 _{under the stereographic projection will be called a circular torus in}S3_{. We would like to}

know when circular tori inR3_{comes from constant mean curvature circular tori inS}3 _{under the} stereographic projection. A circular torus inS3_{meant that it is obtained from a revolution of a} circle inS3 _{under a rigid motion. A generalT (r, a)will not satisfy the above requirement. For} instance, it was proved by Montiel and Ros (Montiel and Ros 1981) that a compact embedded surfaceS with constant mean curvature contained in an open hemisphere ofS3_{must be a round}

sphere. Hence forT (r, a)contained inside or outside of the unit ballB(1)_⊂R3_,ρ−1_{(T (r, a))will} be contained in an open hemisphere ofS3_{and can not have constant mean curvature. Then among}

*Member Academia Brasileira de Ciências E-mail: abbarros@mat.ufc.br

all tori T (r, a)which intercept the inside and the outside of the unit ballB(1)we will describe those which have the desired property. We will show that to construct such a torus we take an arbitrary pointP (α)₌(cosα,0,sinα)on the unit circle of thexz₋plane,0< α < π/2,draw its tangent until it meets thexaxis at the pointQ(α)₌(secα,0,0)which will be the center of the circleŴwhereas its radius will bea ₌tanα, i.e. the torusT (secα,tanα)will satisfy the previous requirement. We note ifOdenotes the origin ofR3_{then the triangleOP Qis a right triangle. This} description will yield that the Clifford torus is associated to a right triangle with two equal sides. More precisely, our aim in this note is to present a proof of the following fact:

Theorem_{1. LetT}2 _⊂S3_{be a circular torus of constant mean curvature. Then}

T2₌ρ−1(T (secα,tanα))₌S1(cosα)_×S1(sinα).

Moreover, the mean curvature ofT2is given byH ₌ (tan

2_α −1) 2 tanα .

2 PRELIMINARIES

For an immersionf_:M _→Mbetween Riemannian manifolds we will denote byds2

f the induced metric onM byf. Now letMn,M₁m andM₂mbe Riemannian manifolds, where the superscript denote the dimension of the manifold. Considerψ_:Mn _→M₁mbe an immersion,ρ_:M₁m _→M₂m a conformal mapping and setϕ ₌ρ_◦ψ. Letφ_:M _→R_{be a function verifyingds}2

ϕ =e2φdsψ2. Ifki¯ andki denote the principal curvatures ofψandϕ ₌ρ_◦ψ, respectively, then we get

ki ₌e−φ

¯

ki ₋∂φ ∂ξ

, (1)

whereξis a unit normal vector field toψ (M), see for instance (Abe 1982) or (Willmore 1982). At first we will recall the following known lemma of which we sketch the proof.

Lemma _{1. Let ψ = (ψ1, ψ2, ψ3, ψ4):M}2 _→ S3

\ {n_} be an immersion of a surface M2, set

ϕ ₌ρ_◦ψand supposedsϕ2=e2φdsψ2. Then we get

ki ₌e−φ(ki¯ ₋g), (2)

whereg₌ ν, ϕdenotes the support function onM2 ⊂R3_.

Proof. _{If we putψ =ψ (u1, u2)then a direct computation gives}

_∂ui∂ϕ, ∂ϕ ∂uj =λ

2

_∂ui∂ψ, ∂ψ

∂uj, (3)

whereλ ₌ (1₋ψ4)−1 = 1+|ϕ|

2

2 . So we can writeds 2

ϕ = e2φdsψ2 with e φ

= 1+|2ϕ|2. Thus ifν denotes a unit normal vector field toϕ(M2_)thenν

=e−φ_{ξ, whereξstands for a unit normal vector} field toψ (M2_{). Hence we have from (1)}

ki ₌e−φki¯ ₋∂φ ∂ν =e

−φ_(ki¯

− ν, ϕ)₌e−φ(ki¯ ₋g),

3 PROOF OF THE THEOREM

Proof. _{First we note that the circleŴ=(x,0, z)∈}R3_:(x−r)2_+z2_=a2_{can be parametrized}

by the mapγ_: [0,2π]_→R3_{defined by}

γ (t )₌

r2₋a2 r₋asint,0,

a√r2_−a2_cost

r ₋asint

.

In fact, it is enough to check that

r2

−a2

r ₋asint −r 2

+

a√r2_−a2_cost

r₋asint 2

=a2.

Representing by Rθ a rotation on R3 _{around the z}

−axis, we see that Rθ(γ (t )) is a circular torusT (r, a)ifγ is a parametrization of the circleŴgiven above. We put nowσ ₌ √r2_−a2_,

θ ₌ ru1/σ2 and t = ru2/aσ. We note that such a choice implies 0 ≤ u1 ≤ (2π σ2)/r and 0_≤u2≤(2π aσ )/r. Let us callRθ(γ (t ))ofϕ(u1, u2), i.e.

ϕ(u1, u2)=σ (r−asint )−1(σcosθ, σsinθ, acost ). (4)

Hence we have

eφ ₌ 1+ |ϕ |

2

2 =

q(t )

2(r₋asint ), (5)

whereq(t )₌a(σ2

−1)sint₊r(σ2

+1). Now a straightforward computation yields

⎧ ⎪ ⎪ ⎨

⎪ ⎪ ⎩

∂ϕ

∂u1 =

r

(r₋asint )(−sinθ,cosθ,0) , ∂ϕ

∂u2 =

r

(r₋asint )2(σcostcosθ, σcostsinθ, a−rsint ) .

From that we derive thatϕ is a conformal parametrization ofT (r, a)satisfying

∂ϕ ∂ui,

∂ϕ ∂uj

= r

2_δij

(r ₋asint )2. (6)

Moreover, a unit vector field normal toϕis given as follows:

ν (u1, u2)= − 1

(r₋asint )((a−rsint )cosθ, (a−rsint )sinθ,−σcost ) .

Therefore we conclude that

g₌ σ

2_sint

On the other hand a new computation gives us

⎧

⎪ ⎪ ⎨

⎪ ⎪ ⎩

∂ν

∂u1 = −

(a₋rsint ) σ2

∂ϕ ∂u1

,

∂ν

∂u2 =

1 a

∂ϕ ∂u2

.

(8)

From this we havek1 = (a_(r−2r₋sin_a2₎t ) andk2 = −

1

a. Taking into account (5), (7) and (8) we conclude from Lemma 1 that

H ₌ 1

4aσ2

raσ2₋1sint₊σ2₊1 2a2₋r2.

Now we have thatH is constant if and only ifσ2

=1. Moreover,σ2

=1 yieldsH _{= 2}1_aa2 −1. Sincea < r we puta ₌rsinα,r ₌secαand this completes the proof of the theorem.

We point out thatH ₌ 0 if and only ifa ₌ 1 andr ₌ √2 which corresponds to the right triangle with two equal sides.

4 THE WILLMORE MEASURE ONT (r, a)

In this section we will present a simple way to compute_{T (r,a)}H2dAby using the parametrization ofT (a, r) given by (4). We observe that ifdA denotes the element of area of T (r, a) then its Willmore measure is given by

H2₋KdA₌ r

4

4a2_σ4du1du2.

Hence, using Gauss-Bonnet theorem, we easily conclude that

T (r,a)

H2dA₌ r

4

4a2_σ4

2π aσ_r

0

2π σ

2

r

0

du1du2=

r2

a√r2_−a2π 2

. (9)

Therefore the family of toriT √

2a, a

, which corresponds to the family of right triangles with

two equal sides, yields the minimum for_{T (r,a)}H2dAamong all circular tori. Moreover, from (9) its value is (see also Willmore 1982)

T (√2a,a)

H2dA₌2π2.

Sincea < r, if we chooseαsuch that sinα₌ a_r, we conclude from (9) the following corollary.

Corollary_{1. Given a circular torusT (r, a)⊂}R3_{we have a circular torusT (secα,tanα)⊂}R3 such that _{T (r,a)}H2dA _{= T (secα,tanα)}H_α2dAα. In other words, the family of circular tori with constant mean curvature inS3_{cover all values of}

T (r,a)H

5 CONCLUDING REMARKS

We point out that Theorem 2 of K. Nomizu and B. Smyth (Nomizu and Smyth 1969) guarantees that a flat torus of constant mean curvature inS3_{is isometric to a product of circles. Thenρ}−1_{T (a, r)} is flat if and only if it has constant mean curvature. We notice if we setψ ₌ρ−1_{ϕwhereϕ was} given by (4) then we have

ψ (u1, u2)= 1 q(t )

2σ2cosθ,2σ2sinθ,2aσcost, r(σ2₋1)₊a(σ2₊1)sint,

where q(t ) ₌ a(σ2₋1)sint ₊r(σ2₊1), (see(5)). Hence by using (3), (5), (6) and putting z₌u1+iu2we conclude that

ds_ψ2 ₌e−2φds_ϕ2 ₌ 4r

2

q2_{(t )} |dz| 2_.

According to our theorem the metric ds2

ψ is flat if and only if ρ−1T (r, a) has constant mean curvature inS3_{. In this case we have}

ψ (u1, u2)= 1

√

a2₊₁(cosθ,sinθ, acost, asint ) ,

i.e. ρ−1T (r, a)is isometric to the product of circles S1(√1

a2₊₁)×S

1₍ a

√

a2₊₁). We note that this

yields cosα₌ √1

a2₊₁ and sinα=

a

√

a2₊₁, i.e. ρ(S

1_(cosα)

×S1_(sinα))

=T (secα,tanα).

ACKNOWLEDGMENTS

This work was partially supported by FINEP-Brazil.

RESUMO

Neste artigo mostraremos que a imagem inversa pela projeção estereográfica de um toro circular de

revolu-ção no espaço euclidiano de dimensão 3 tem curvatura média constante se e somente se os seus raios são o

cateto e a hipotenusa de um triângulo retângulo apropriado.

Palavras-chave: Toro plano, Curvatura média constante, Toro circular, Projeção estereográfica.

REFERENCES

Abe N._{1982. On generalized total curvature and conformal mappings. Hiroshima Math J 12: 203-207.}

Montiel S and Ros A._{1981. Compact hypersurfaces: The Alexandrov theorem for higher order mean}

curvatures, A symposium in honor of Manfredo do Carmo, Edited byB. Lawson and K. Tenenblat_, Pitman Monographs 52: 279-296.

Nomizu K and Smyth B._{1969. A formula of Simons’type and hypersurfaces with constant mean curvature.}

J Diff Geom 3: 367-377.