Introduction LAGRANGESPACESWITHINDICATRICESASCONSTANTMEANCURVATURESURFACESORMINIMALSURFACES

LAGRANGE SPACES WITH INDICATRICES AS CONSTANT MEAN CURVATURE SURFACES OR MINIMAL SURFACES

Mircea Crˆa¸sm˘areanu

Dedicated to Academician Radu Miron on the occasion of 75th birthday

Abstract

Constant mean curvature, particularly minimal, surfaces given by indicatrices of Lagrange and generalized Lagrange spaces are studied.

Introduction

The search of minimal surfaces in R³ is an old exciting problem([6]) and se- veral methods appear in the study of these surfaces: Lie groups methods([3]), theory of integrable systems via the Weierstrass representation([4]).

Also, very interesting generalizations are fruitful: minimal surfaces in Rie- mannian manifolds([5]), constant mean curvature(CMC) surfaces([4], [9]).

In this paper we search CMC surfaces, particularly minimal surfaces, pro- vided by indicatrices of Lagrange and generalized Lagrange manifolds of di- mension three. Because the indicatrices of these spaces are given in implicit form, in first section the equations of CMC and minimal surfaces in implicit form are derived. In next two sections several equations of CMC and minimal indicatrices are obtained in the Lagrange and generalized Lagrange framework and the last section is devoted to examples.

Key Words: CMC and minimal surface, indicatrix of a Lagrange space.

Mathematical Reviews subject classification: 53A10, 53C60.

Received: October, 2002

63

1 CMC surfaces equation for surfaces in implicit form

Let in R³ a surface S given in explicit form S : u = u(x, y). The mean curvature function ofS is:

H = u_xx¡ 1 +u²_y¢

+u_yy¡ 1 +u²_x¢

−2u_xu_yu_xy 2¡

1 +u²_x+u²_y¢³

2

. (1.1)

The equationH=constant is calledCMC surfaces equation and in particular the equationH = 0 is calledminimal surfaces equation.

IfSis given in implicit formS :F(x, y, z) = 0 from relationF(x, y, u(x, y)) =

0 it results: (

u_x=−^F_F^x

z

uy =−^F_F^y

z

(1.2)









uxx= ^{Fx(FxzFz−FzzFx)−F}_F3^{z(FxxFz−FxzFx)} z

uxy=^{Fx(FyzFz−FzzFy)−F}_F3 ^{z(FxyFz−FxzFy)} z

uyy= ^{Fy(FyzFz−FzzFy)−F}_F3^{z(FyyFz−FyzFy)} z

. (1.3)

After a straightforward computation we get the CMC surfaces equation:

2 (FxyFxFy+FyzFyFz+FzxFzFx)−

−£ Fxx

¡F_y²+F_z²¢ +Fyy

¡F_z²+F_x²¢ +Fzz

¡F_x²+F_y²¢¤

= 2H¡

F_x²+F_y²+F_z²¢³

2

(1.4) and the minimal surfaces equation:

2(FxyFxFy+FyzFyFz+FzxFzFx) =Fxx

¡F_y²+F_z²¢ + Fyy

¡F_z²+F_x²¢ +Fzz

¡F_x²+F_y²¢

. (1.5)

2 CMC indicatrices in Lagrange geometry

Let us denoteTR³the tangent bundle ofR³for which we use the coordinates (x, y) =¡

xⁱ, yⁱ¢

1≤i≤3 with x=¡ xⁱ¢

the coordinates in R³ and y =¡ yⁱ¢

the coordinates in the fiber TxR³. A function f ∈ C^∞¡

TR³¢

which does not depends of x i.e. f = f(y) is called Minkowskian function. A tensor field of (r, s)-type onTR³ with law of change, at a change of coordinates onTR³, exactly as a tensor field of (r, s)-type onR³is calledd-tensor fieldof (r, s)-type.

After [8] a smooth Lagrangian L:TR³→Ris called regularif the matrix g = (g_ij)_1≤i,j≤3, g_ij = ¹₂ ∂^.i

∂.j L, is of rank 3 i.e. detg 6= 0 where ∂^.i= _∂y^∂i.

The pair¡ R³, L¢

is called thenLagrange spaceand the d-tensor fieldg= (gij) is called the Lagrange metric.

For everyx∈R³we havethe indicatrixofL,Ix={y∈TxR³;L(x, y) = 1}

which appears as a surface defined byF(y) =L(x, y)−1,xbeing fixed. Using the last relations of previous section it results thatIxis CMC surface if:

2 (F12F1F2+F23F2F3+F31F3F1)−

−£ F11

¡F₂²+F₃²¢ +F22

¡F₃²+F₁²¢ +F33

¡F₁²+F₂²¢¤

=

= 2H_x¡

F₁²+F₂²+F₃²¢³

2 (2.1)

andIxis minimal surface if:

2 (F12F1F2+F23F2F3+F31F3F1) =

=F11

¡F₂²+F₃²¢ +F22

¡F₃²+F₁²¢ +F33

¡F₁²+F₂²¢

(2.2) where Fi=∂^.iLandFij=∂^.i

∂.j L. FromFij= 2gij it follows:

Proposition 2.1(i)CMC indicatrices are given by:

2 (g12F1F2+g23F2F3+g31F3F1)−

−£ g11

¡F₂²+F₃²¢ +g22

¡F₃²+F₁²¢ +g33

¡F₁²+F₂²¢¤

=

=Hx

¡F₁²+F₂²+F₃²¢³₂

(2.3) (ii)minimal indicatrices are given by:

2 (g12F1F2+g23F2F3+g31F3F1) =

=g11

¡F₂²+F₃³¢ +g22

¡F₃²+F₁²¢ +g33

¡F₁²+F₂²¢

. (2.4)

Let us remark that for a Minkowski Lagrangian if there exists a CMC (minimal) indicatrix then all indicatrices are CMC (minimal) surfaces.

A particular important case is that of a r-homogeneous Lagrangian i.e.

L(x, λy) =λ^rL(x, y) for everyλ∈R.

Proposition 2.2IfL isr-homogeneous with r6= 1then:

(i)CMC indicatrices are given by:

2 (g12g1ag2b+g23g2ag3b+g31g3ag1b)y^ay^b− {g11

h

(g2ay^a)²+ (g3ay^a)²i + +g22

h

(g3ay^a)²+ (g1ay^a)²i +g33

h

(g1ay^a)²+ (g2ay^a)²i }=

= 2Hx

r−1 h

(g1ay^a)²+ (g2ay^a)²+ (g3ay^a)² i³

2 (2.5)

(ii)minimal indicatrices are given by:

2 (g12g1ag2b+g23g2ag3b+g31g3ag1b)y^ay^b=

=g11

h

(g2ay^a)²+ (g3ay^a)² i

+g22

h

(g3ay^a)²+ (g1ay^a)² i

+ +g33

h

(g1ay^a)²+ (g2ay^a)² i

. (2.6)

ProofFrom Euler relation∂^.iLyⁱ=rLapplying∂^.j we have 2gijyⁱ+Fj=rFj which means that:

Fj = 2

r−1giay^a (2.7)

and substituting this relation in (2.3) and (2.4) we get (2.5) and (2.6). 2 The most important case isr= 2 for:

(i)Riemann spaceswheng= (gij(x)) is a Riemannian metric andLis the kinetic energy ofg i.e. L=gijyⁱy^j

(ii) Finsler spaces when g = (gij(x, y)) is a Finsler metric([8]) and L = gijyⁱy^j.

Proposition 2.3For Finsler, particularly Riemann, spaces:

(i)the CMC indicatrices are given by:

2 (g12g1ag2b+g23g2ag3b+g31g3ag1b)y^ay^b−−{g11

h

(g2ay^a)²+ (g3ay^a)² i

+ +g22

h

(g3ay^a)²+ (g1ay^a)² i

+g33

h

(g1ay^a)²+ (g2ay^a)² i

}=

= 2Hx

h

(g1ay^a)²+ (g2ay^a)²+ (g3ay^a)²i³

2 (2.8)

(ii)the minimal indicatrices are given by(2.6).

Returning to the general case of proposition 2.1 because the matrix g = (gij) is symmetric let us suppose that this matrix is diagonal: g12 =g32 = g31= 0. Let us calldiagonal Lagrange spacethis type of Lagrange spaces.

Proposition 2.4A)In a diagonal Lagrange space:

(i)the CMC indicatrices are given by:

g11

¡F₂²+F₃²¢ +g22

¡F₃²+F₁²¢ +g33

¡F₁²+F₂²¢

=−Hx

¡F₁²+F₂²+F₃²¢³

2

(2.9) (ii)the minimal indicatrices are given by:

g₁₁¡

F₂²+F₃²¢ +g₂₂¡

F₃²+F₁²¢ +g₃₃¡

F₁²+F₂²¢

= 0. (2.10)

If the diagonal Lagrange metric is positive definite i.e. gii >0,1 ≤i≤3, it results that there are no minimal indicatrices.

B) In a diagonalr-homogeneous Lagrange space:

(i)the CMC indicatrices are given by:

g11

h¡g22y²¢2

+¡

g33y³¢2i +g22

h¡g33y³¢2

+¡

g1ay¹¢2i + +g33

h¡g11y¹¢₂ +¡

g22y²¢₂i

= 2Hx

1−r

h¡g11y¹¢₂ +¡

g22y²¢₂ +¡

g33y³¢₂i³

2

(2.11) (ii)the minimal indicatrices are given by:

g11

h¡g22y²¢2

+¡

g33y³¢2i +g22

h¡g33y³¢2

+¡

g11y¹¢2i + +g33

h¡g11y¹¢₂ +¡

g22y²¢₂i

= 0 (2.12)

C)In a diagonal Finsler, particularly Riemann, space:

(i)the CMC indicatrices are given by:

g11

h¡g22y²¢₂ +¡

g33y³¢₂i +g22

h¡g33y³¢₂ +¡

g11y¹¢₂i + +g33

h¡g11y¹¢2

+¡

g22y²¢2i

=−2Hx

h¡g11y¹¢2

+¡ g22y²¢2

+¡

g33y³¢2i³

2

(2.13) (ii)the minimal indicatrices are given by(2.12).

Example 2.5(The Euclidean case) Letgij=δij be the usual Euclidean metric ofR³which is a diagonal Riemann metric. The relation (2.13) becomes:

2h¡

y¹¢₂ +¡

y²¢₂ +¡

y³¢₂i

=−2Hx

h¡y¹¢₂ +¡

y²¢₂ +¡

y³¢₂i³

2.

In this case L=¡ y¹¢₂

+¡ y²¢₂

+¡ y³¢₂

and thus the equation ofIxis¡ y¹¢₂

¡ + y²¢₂

+¡ y³¢₂

= 1 and then for everyx∈R³:

(i) the only CMC indicatrix is the unit sphereS² withHx=−1, (ii) there are not minimal indicatrices.

3 CMC indicatrices in generalized Lagrange spaces

A d-tensor field of (0,2)-type onTR³, denotedg = (gij(x, y)), is called gen- eralized Lagrange metric (GL-metric, on short) if the following properties hold([8]):

(i) symmetry,gij =gji

(ii) nondegeneracy: det (gij)6= 0

(iii) the signature of quadratic form g(ξ) = gijξⁱξ^j, ξ = ¡ ξⁱ¢

∈ R³, is constant.

The function E(g) = gijyⁱy^j is called the absolute energy of the given GL-metric.

Definition 3.1([8]) The GL-metric is calledweak regularifE(g) is a reg- ular Lagrangian.

It follows that for a weak regular GL-metric the d-tensor field of (0,2)-type:

g_ij^∗ =1 2

∂.i

∂.j E(g) (3.1)

is a Lagrange metric and then we can associate the indicatrix:

Ix={(x, y)∈TR³;E(g) (x, y) = 1}.

Applying proposition 2.1 we get:

Proposition 3.2For a weak regular GL-metric:

(i)the CMC indicatrices are given by:

2 h

g₁₂^∗ ∂^.1E(g)∂^.2E(g) +g^∗₂₃∂^.2E(g)∂^.3E(g) +g₃₁^∗ ∂^.3E(g)∂^.1E(g) i

−

−{g^∗₁₁

·³_.

∂2E(g)

´₂ +

³_.

∂3E(g)

´₂¸ +g^∗₂₂

·³_.

∂3E(g)

´₂ +

³_.

∂1E(g)

´₂¸ + +g₃₃^∗

·³_.

∂1E(g)´₂ +³_.

∂2E(g)´₂¸ }=

=Hx

·³_.

∂1E(g)

´₂ +

³_.

∂2E(g)

´₂ +

³_.

∂3E(g)

´₂¸³

2

(3.2) (ii)the minimal indicatrices are given by:

2h

g₁₂^∗ ∂^.1E(g)∂^.2E(g) +g₂₃^∗ ∂^.2E(g)∂^.3E(g) +g^∗₃₁∂^.3E(g)∂^.1E(g)i

=

=g^∗₁₁

·³_.

∂2E(g)

´₂ +

³_.

∂3E(g)

´₂¸ +g^∗₂₂

·³_.

∂3E(g)

´₂ +

³_.

∂1E(g)

´₂¸ + +g^∗₃₃

·³_.

∂1E(g)

´₂ +

³_.

∂2E(g)

´₂¸

. (3.3)

A straightforward computation gives:





g^∗_ij=gij+³_.

∂i

∂.jgab

´

y^ay^b+³_.

∂igja+∂^.jgia

´ y^a

∂.i E(g) =³_.

∂igab

´

y^ay^b+ 2giay^a . (3.4)

The above formulae become more simple in the following case:

Definition 3.3([8]) A weak regular GL-metric is calledregularif:

∂.i E(g) = 2gijy^j. (3.5) It results([8]):

g_ij^∗ =gij+

³_.

∂j gik

´

y^k (3.6)

and then:

Proposition 3.4For a regular GL-metric:

(i)the CMC indicatrices are given by:

2{h

g12+³_.

∂2g1k

´ y^ki

g1ag2b+h

g23+³_.

∂3g2k

´ y^ki

g2ag3b

+ h

g31+

³_.

∂1g3k

´ y^k

i

g3ag1b}y^ay^b−{

h g11+

³_.

∂1g1k

´ y^k

i h

(g2ay^a)²+ (g3ay^a)² i

+ +

h g22+

³_.

∂2g2k

´ y^k

i h

(g3ay^a)²+ (g1ay^a)² i

+ +

h g33+

³_.

∂3g3k

´ y^k

i h

(g1ay^a)²+ (g2ay^a)² i

}= 2Hx

h

(g1ay^a)²+ (g2ay^a)²+ (g3ay^a)² i³

2 (3.7)

(ii)the minimal indicatrices are given by:

2{

h g12+

³_.

∂2g1k

´ y^k

i

g1ag2b+ h

g23+

³_.

∂3g2k

´ y^k

i

g2ag3b+ h

g₃₁+³_.

∂1g_3k´ y^ki

g_3ag_1b}y^ay^b =h

g₁₁+³_.

∂1g_1k´ y^ki h

(g_2ay^a)²+ (g_3ay^a)²i + +h

g22+³_.

∂2g2k

´ y^ki h

(g3ay^a)²+ (g1ay^a)²i + +h

g33+³_.

∂3g3k

´ y^ki h

(g1ay^a)²+ (g2ay^a)²i

. (3.8)

Another approach in the regular case is provided by homogeneity. By multiplication of (3.5) withyⁱ we have:

∂.iE(g)yⁱ= 2gijyⁱy^j = 2E(g) (3.9) which means that E(g) is 2-homogeneous i.e. E(g) is a Finslerian function.

Then we apply proposition 2.3:

Proposition 3.5For a regular GL-metric:

(i)the CMC indicatrices are given by:

2 (g^∗₁₂g^∗_1ag_2b^∗ +g₂₃^∗ g_2a^∗ g_3b^∗ +g^∗₃₁g^∗_3ag^∗_1b)y^ay^b−

−{g^∗₁₁ h

(g^∗_2ay^a)²+ (g_3a^∗ y^a)² i

+g₂₂^∗ h

(g_3a^∗ y^a)²+ (g_1a^∗ y^a)² i

+ +g^∗₃₃h

(g^∗_1ay^a)²+ (g^∗_2ay^a)²i }=

= 2Hx

h

(g^∗_1ay^a)²+ (g^∗_2ay^a)²+ (g^∗_3ay^a)²i³

2 (3.10)

(ii)the minimal indicatrices are given by:

2 (g^∗₁₂g^∗_2ag_3b^∗ +g₂₃^∗ g_2a^∗ g_3b^∗ +g^∗₃₁g^∗_3ag^∗_1b)y^ay^b=

=g₁₁^∗ h

(g_2a^∗ y^a)²+ (g_3a^∗ y^a)²i +g^∗₂₂h

(g^∗_3ay^a)²+ (g^∗_1ay^a)²i + +g₃₃^∗

h

(g_1a^∗ y^a)²+ (g_2a^∗ y^a)² i

(3.11) where forg_ij^∗ we use the relation(3.6).

4 Beil metrics as examples

Let eg = (egij(x, y)) be a Finsler metric and B =Bⁱ(x, y)∂^.i a d-vector field for which we denoteBi=egijB^jandB0=Biyⁱ. Let alsoa, b∈C^∞¡

TR³¢ . In [1] and [2] the following GL-metric is studied:

gij=aegij+bBiBj. (4.1) These GL-metrics, calledBeil metrics, are not Lagrange metrics. From:

E(g) =aE(eg) +b(B0)² (4.2) we get:

∂.iE(g) =³_.

∂ia´

E(eg) +a³_.

∂iE(eg)´ +³_.

∂ib´

(B0)²+ 2bB0

³_.

∂iB0

´ (4.3) 2g_ij^∗ = 2aegij+∂^.i

∂.j aE(eg) +∂^.ia∂^.jE(eg) +∂^.ja∂^.iE(eg) +∂^.i

∂.j b(B0)²+ 2B0

³_.

∂i b∂^.jB0+∂^.jb∂^.iB0+b∂^.i

∂.j B0

´

+ 2b∂^.iB0

∂.jB0. (4.4) I) On T0R³=TR³\{null section}let:

a=1

2, b= 1

2kyk²_F (4.5)

where k,kF is the norm induced by the Finsler metriceg i.e. kyk²_F =E(eg) = e

gijyⁱy^j. LetB =yⁱ∂^.ibethe Liouville vector field, it resultsBi=egijy^{j denoted}= e

yi. The associated Beil metric is:

g= 1 2kyk²_F



 (ey1)²+eg11kyk²_F ey1ey2+eg12kyk²_F ye1ye3+eg13kyk²_F e

y₁ey₂+eg₁₂kyk²_F (ey₂)²+ge₂₂kyk²_F ye₂ye₃+eg₂₃kyk²_F e

y1ey3+eg13kyk²_F ey2ey3+eg23kyk²_F (ey3)²+eg33kyk²_F



.

(4.6) Thus:

E(g) =kyk²_F =E(eg) (4.7) which is 2-homogeneous and then a Finsler function. It results that the Beil metric is regular GL-metric withg_ij^∗ =egij and then the CMC(minimal) indi- catrices of Beil metric are exactly the CMC(minimal) indicatrices of Finsler metriceg.

II) (Miron-Tavakol metrics) Fora= exp (2σ), b= 0 withσ∈C^∞¡ TR³¢ and eg = eg(x) a Riemannian metric we have the so-called Miron-Tavakol metrics([7]):

gij(x, y) =e^2σ(x,y)egij(x) (4.8) for which:

∂.iE(g) = 2³

g_iay^a+³_.

∂iσ´

g_aby^ay^b´

(4.9)

g_ij^∗ =gij+

³_.

∂i

∂.j σ+ 2∂^.i σ∂^.j σ

´

gaby^ay^b+ 2

³ gja

∂.iσ+gia

∂.jσ

´

. (4.10) Particular cases:

II1. ([7, p. 219])σ= ¹₂E(eg) =¹₂egijyⁱy^j

∂.iE(g) = 2e^E(eg)(1 +E(eg))egiay^a (4.11)

g_ij^∗ =gij+¡ e

gij+ 2egiagejby^ay^b¢

guvy^uy^v+ 2 (gjaegiuy^u+giaegia)y^a. (4.12) II2. σ=γ_i(x)yⁱ withγ_i∈C^∞¡

R³¢

,1≤i≤3

∂.iE(g) = 2 (giay^a+γ_iE(g)) (4.13)

g_ij^∗ =gij+ 2γ_iγ_jE(g) + 2¡

γ_igja+γ_jgia

¢y^a. (4.14)

References

[1] Anastasiei, M., Shimada, H.,Beil metrics associated to a Finsler space, Balkan J. Geom.

Appl., 3(1998), no. 2, 1-16.

[2] Anastasiei, M., Shimada, H.,Deformations of Finsler metrics, in Antonelli, P. L.(Ed.)- Finslerian geometries. A meeting of minds, Kluwer Academic Publishers, FTPH no.

109, 2000, 53-65.

[3] Bˆıl˘a, N., Lie groups applications to minimal surfaces PDE, Differential Geometry- Dynamical Systems, 1(1999), no. 1, 1-9.

[4] H´elein, F., Constant mean curvature surfaces, harmonic maps and integrable systems ( Notes taken by Roger Moser), Birkh¨auser, Boston, 2001.

[5] Ji, M., Wang, G. Y., Minimal surfaces in Riemannian manifolds, Memoirs AMS, 104(1993), no. 495.

[6] Lucas, J., Barbosa, M., Gervasio Colares, A.,Minimal surfaces inR³, Springer, Berlin, 1986.

[7] Miron, R., Tavakol, R. K., B˘alan, V., Roxburgh, I.,Geometry of space-time and gen- eralized Lagrange gauge theory, Publ. Math. Debrecen, 42(1993), no. 3-4, 215-224.

[8] Miron, R., Anastasiei, M.,The geometry of Lagrange spaces: theory and applications, Kluwer Academic Publishers, FTPH no. 59, 1994.

[9] Wente, H. C., Constant mean curvature immersions of Enneper type, Memoirs AMS, 100(1992), no. 478.

Faculty of Mathematics, University ”Al. I. Cuza”, Ia¸si, 6600,

Romania

e-mail: [email protected]