CMC and minimal surfaces in Finsler spaces with convex indicatrix (brief account) Vladimir BALAN Abstract In the framework of locally Minkowski Finsler spaces with convex indicatrix of Randers and Kropina type, emerging from the general form of the mean curvature field, is determined and studied the mean curvature field of immersed hypersurfaces. For the 2-dimensional case, is explicitely determined the mean curvature function. The obtained results are further applied to Monge charts and to surfaces of rotation of Kropina and Randers spaces.
MSC: 53C60, 53B40, 53C42, 53B15. Keywords: Finsler space, Randers metric, Kropina metric, mean curvature, minimal surface.
1
Introduction
Recent advances in the theory of Finsler surfaces have been provided after 2002 by Z.Shen - who has intensively studied 2-dimensional Finsler metrics and their flag curvature. Recently Z.Shen ([29]), and further M.Souza and K. Tenenblat ([30, 31]) have investigated minimal surfaces immersed in Finsler spaces from differential geometric point of view. Still, earlier rigorous attempts using functional analysis exist in the works of G.Bellettini and M. Paolini (after 1995, e.g., [13, 14, 15]). In 1998, based on the notion of Hausdorff measure, Z.Shen ([29]) has introduced the notion of mean curvature on submanifolds of Finsler spaces as follows. ˜ , F˜ ) is a Finsler structure, and ϕ : (M, F ) → (M ˜ , F˜ ) is an isometric If (M ˜ immersion (hence F is induced by F ), then the mean curvature of M is given by ([29, (57), p.563]) Hϕ (X) =
´ 1 ³ G;xi − G;zai zj ϕj;ua ub − G;xj zai ϕj;ua X i , b G
where lower indices stand for corresponding partial derivatives and: • (ua , v b )a,b∈1,n are local coordinates in T M (dim M = n); ˜ (dim M ˜ = m); • (xi , y j )i,j∈1,m are local coordinates in T M 1
• zai are the entries of the Jacobian matrix [J(ϕ)] = (∂ϕi /∂ua )a=1,n,i=1,m ;
˜ , t ∈ (−ε, ε), ϕ0 = ϕ, is the variation of the surface; • ϕt : M → M t | (x) induced along ϕ attached to the variation; • X is the vector field Xx = ∂ϕ ∂t t=0 • G is the Finsler induced volume form Ge˜(z) =
vol[B n ] , vol{(v a ) ∈ R n | F˜ (v a zai e˜i ) ≤ 1}
(1.1)
where z = (zai )a=1,n,i=1,m ∈ GLm×n ( R ), e˜ = {˜ ei }i=1,m is an arbitrary basis in R m and B n ⊂ R n is the standard Euclidean ball. It was proved that the variation of the volume in M reaches a minimum for Hϕ = 0 ([29]). Recent advances in constructing minimal surfaces (n = 2) based on (1.1) were provided in ([30, 31]), by characterizing the minimal surfaces of revolution ˜ = R3 , F˜ ) with the Finsler (α, β)−fundamental function in Randers spaces (M q
F˜ (x, y) = α(x, y) + β(x, y),
aij (x)i y j , β(x, y) = bi (x)y i ,
α(x, y) =
for the particular case when aij = δij (the Euclidean metric) and β = b · dx3 , with b ∈ [0, 1). ˜ = m = n + 1. Let F : T M ˜ → R be a We further consider the case when dim M positive 1-homogeneous locally Minkowski Finsler fundamental function ([26]). In the following we shall study an extension of the following cases: ˜ , F ) with the fundamental function a) Randers space (M q
F (x, y) =
δij y i y j + bs y s ,
bs ∈ R , s = 1, n,
which is reducible to the case studied in [30, 31], F (x, y) =
qP
n+1 i 2 i=1 (y )
+ by n+1 , b ∈ [0, 1).
(1.2)
˜ , F ) with the fundamental function b) Kropina space (M F (x, y) =
δij y i y j , bs y s
bs ∈ R , s = 1, n,
n+1 X
b2i > 0,
i=1
which is reducible to F (x, y) = (by n+1 )−1 ·
n+1 X
(y i )2 , b ∈ (0, 1).
(1.3)
i=1
2
Generalized Randers-Kropina hypersurfaces.
˜ = R n+1 be a simple hypersurface. We denote Let H = Im ϕ, ϕ : D ⊂ R n → M i ∂ϕ 1 n zαi = ∂u α , u = (u , . . . , u ) ∈ D. We shall further determine the volume of the body ˜ Q ⊂ Tϕ(u) H bounded by the induced on Tϕ(u) H indicatrix from M Σ∗ = Tϕ(u) H ∩ {y ∈ Tϕ(u) R n+1 |F (y) = 1}.
If v = v α ∂u∂α ∈ Tu D, then ϕ∗,u (v) = zαi v α point u ∈ D, Q is given by
¯
∂ ¯ ¯ ∂y i ϕ(u)
∈ Tϕ(u) H and hence at some fixed
Q = {v ∈ Tu D | F (ϕ(u), ϕ∗,u (v)) ≤ 1}. We have the following: Theorem 1. If the body Q is given by Q:
n X
(zαi v α )2 + µ(zαn+1 v α )2 + 2νzαn+1 v α + ρ ≤ 0,
(2.1)
i=1
where µ, ν, ρ ∈ R , then ³ 2 ´n/2 ν τ √ V ol(Bn ) · − ρ(1 + τ ) , µ−1 δ·(1+τ )(n+1)/2
for µ 6= 1
V ol(Bn ) · (−ρ + ν 2 zan+1 zbn+1 hab )n/2 √ ,
for µ = 1,
V ol(Q) =
δ
(2.2)
where τ and δ are given by (2.3). τ = (µ − 1)zan+1 zbn+1 hab ,
δ = det(hab )a,b=1,n ,
(2.3)
and Bn ⊂ R n is the standard n-dimensional ball and hab is the dual of hab (has hsb = δba ). In particular, we obtain the following result: Corollary 1. a) In the Randers case F (x, y) =
qP
n+1 i 2 i=1 (y )
+ by n+1 , b ∈ [0, 1),
(2.4)
we obtain the known result ([31, (5), p. 627]), V ol(QR ) = √
V ol(Bn ) . δ(1 − b2 zan+1 zbn+1 hab )(n+1)/2
b) In the Kropina case F (x, y) = (by n+1 )−1 ·
n+1 X
(y i )2 , b ∈ [0, 1),
(2.5)
i=1
we have V ol(QK ) =
V ol(Bn )
³
´n/2 b2 n+1 n+1 ab z h z b 4 a
√
δ
.
Remarks. In the Kropina case, the function G in (1.1) has the expression √ δ V ol(Bn ) = n+1 n+1 ab 2 n/2 = 2n · CB −n/2 , G= V ol(QK ) (za zb h · b /4)
where we have used the notations from [31], B = b2 zan+1 zbn+1 hab , C = mean curvature vector field has the components ¯i = 1 H G
Ã
√
δ. Then the
!
∂ 2G ∂ 2ϕ , i = 1, n + 1, · ∂zεi zηj ∂uε ∂uη
and the volume form of the hypersurface H is √ δ 1 n dVF = ³ ´n/2 du ∧ . . . ∧ du . 2 b n+1 n+1 ab z zb h 4 a Theorem 2. The mean curvature vector field of the hypersurface M in the Kropina ˜ = R n+1 with the fundamental function (1.3) has the following expression space M in terms of B and C ·
n
Hi = 2 B
−(n+4)/2
µ
− nB 2
∂C ∂B ∂zεi ∂zηj
+
∂2C B2 ∂zεi ∂zηj
+
∂C ∂B ∂zηj ∂zεi
2B C ∂z∂i ∂z j η ε
+
n(n+2) ∂B ∂B C ∂z j− i 4 ε ∂zη
¶¸
∂2ϕ ,i ∂uε ∂uη
= 1, n + 1.
Corollary 2. The mean curvature vector field of the surface M in the Kropina ˜ = R n+1 with the fundamental function (1.3) has the following expression space M ·
Hi =
4C E3
2
∂C ∂C ∂E ∂E E 6E 2 ∂z + 2C 2 ∂z − C 2 E ∂z∂i ∂z j− i i ∂z j ∂z j
µ
−3CE
ε
η
∂E ∂C ∂zεi ∂zηj
where E = b2
+
¶
∂E ∂C ∂zηj ∂zεi
3 2 X X
ε
η
+ 3E
ε
2
2C C ∂z∂i ∂z j η ε
¸
(−1)α+β zαk˜ zβk˜zα3 zβ3 ,
η
∂2ϕ ,i ∂uε ∂uη
(2.6) = 1, n + 1,
α ˜ = 3 − α.
k=1 α,β=1
Corollary 3. The mean curvature of the surface M in the Kropina space (1.3) is H∗ = Hi X i , where Hi are given by (2.6), ˜ | F (y) = 1}, X ∈ Ker(G∗ Z 1 ) ∩ Ker(G∗ Z 2 ) ∩ {y ∈ Tϕ(u) M Z 1 = (z11 , z12 , z13 ), Z 2 = (z21 , z22 , z23 ), and G∗ v is defined by the equality (G∗ v)(v 0 ) = 2 i 0j hv, v 0 iF = 21 ∂y∂F i ∂y j v v . Corollary 4. Let M = Σ = Im ϕ be a surface of revolution described by ϕ(t, θ) = (f (t) cos θ, f (t) sin θ, t), (t, θ) ∈ D = R × [0, 2π). Then M is minimal iff the function f satisfies the ODE 1 + f 02 = 3f f 00 (1 + 2f 02 ).
3
DPW for the Euclidean case (F = α). Open problems for l.M.F. spaces
The 3-dimensional Euclidean case can be regarded as a particular q locally MinkowskiFinsler (l.M.F.) case with (α, β)-metric, with F (y) = α = δij y i y j . In this case one can apply the DPW method for obtaining minimal, CMC or constant Gauss curvature surfaces ([23], [24], [22]). Namely, for CMC immersions in isothermic (conformal) parametrization, the process of obtaining associated moduli (potentials) emerges by identifying the matrices in SO(3) with the ones in Aut(su(2)) and providing the lift F of the Gauss map to SU (2), which further provides the MaurerCartan form α = F −1 dF regarded as a matrix in su(2). The twisted loopification of this matrix using the spectral parameter λ ∈ S 1 provides a new matrix αλ in the loop algebra Λσ sl(2, C). The integration for g of the equation αλ = Fλ−1 dFλ followed directly by a Birkhoff splitting Fλ = Fλ− · Fλ+ of Fλ produces a meromorphic potential ξ = Fλ−1 dFλ− valid for a whole family of surfaces, which has a certain − form for each type of investigated surface. One might obtain as well a less easy to handle holomorphic potential, via performing first a Grauert-type decomposition of the loopified frame Fλ (see the diagram). Back, one can start with a given meromorphic potential ξ ∈ Λ− σ sl(2, C), as −1 follows. Integration for g− of ξ = g− g.− followed by an Iwasawa splitting g− = Fλ ·v+ , provides a loop of frames Fλ ∈ ΛSU (2) which correspond to a family of CMC surfaces which lead to the desired immersions via the Sym-Bobenko formula (in particular for minimal surfaces, by the Weierstrass formula). Like in the Euclidean case, a major goal is to classify CMC (and in particular, minimal) immersions of the locally-Minkowski Finsler framework, by using a DPW-type approach which might extend the Euclidean case, and to emphasize the physical background of the notion of mean curvature in Finsler spaces. To this aim, an open question is to find a map ϕ : Σ → G/K, which might take over the role of the Gauss map from the Euclidean case, having values into a ”reasonable” symmetric (or less, homogeneous reductive) Finsler space. As well, an open problem remains to provide a geometric meaning for an involution Σ which would yield the twisting throughout the subjacent loopified framework. However, regarding ”simple” Finsler homogeneous spaces there already exist results, which characterize the bi-invariant Finsler metrics - an essential tool in linking the PDE properties of a mapping (i.e., the harmonicity for the Gauss map ϕ : D → G/K, in the Euclidean case) with the variational characterization of a CMC surface. The study of the isometry group of a Finsler space (G, F ) plays here a distinct role ([17], [18], [19]) and the classification of homogeneous reductive Finsler spaces (G/K, Fˆ ) might provide progress in understanding the loop geometric objects which characterize the CMC surfaces immersed in l.M.F. spaces. Acknowledgement. The present work was partially supported by Grant CNCSIS 1478/2006. A detailed study of CMC surfaces in Finsler spaces with BerwaldMoor metric and with general Randers and Kropina convex metrics is in preparation.
³
´
CMC surfaces moduli 0 g dz Q/g 0
Meromorphic potential −1 ξλ = g λ dgλ
-
ξλ = λ−1
−
g meromorphic, Q holomorphic
6
¡
¡
-
−
New meromorphic potential −1 ξλ = g ˜λ d˜ gλ −
−
6
6
¡
³
¡
¡
´
Minimal surfaces moduli 0 g ξλ = λ−1 dz Q/g 0
¡
g meromorphic
?
¡
−
¡
¡
-
Birkhoff decomposition Hλ = gλ · gλ −
+
Dressing with u+ u+ · gλ = g ˜λ · g ˜λ −
−
+
6
¡
¡ ¡ ª
Grauert-type decomposition −1 Fλ = Hλ · wλ
Iwasawa decomposition gλ = Fλ · vλ −
+
-
HH
Holomorphic potential ηλ = H −1 dHλ λ
+
6
HH
HH
HH j H
?
Maurer-Cartan frame equation αλ = F −1 dFλ
¾
Iwasawa decomposition Hλ = Fλ · wλ +
λ
6
Maurer-Cartan loopified form αλ = λ−1 α01 + α0 + λα00 1 λ ∈ S 1 = spectral parameter
6
Maurer-Cartan form α = F −1 dF : D → su(2)
-
6
Lifted moving frame F : D → SU (2)
6
Minimal surfaces Holomorphic Gauss map
Integrability conds. for αλ ⇔ Integrability for α and harmonicity for ϕ
6
Maurer-Cartan form α = F −1 dF : D → g = k ⊕ p α = α0 + (α01 + α00 1)
6
-
Lifted moving frame F :D→G
6
6
CMC surfaces Harmonic Gauss map n : D → S 2 = SU (2)/U (1)
¾
-
Harmonic map ϕ : D → G/K
¾
Variational principle τ (n) = 0 ⇔ ϕ harmonic
6
Ruh-Vilms Theorem H = const. ⇔ τ (n) = 0
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Vladimir Balan, University Politehnica of Bucharest, Faculty of Applied Sciences, Department of Mathematics I, Splaiul Independentei 313, RO-060042, Bucharest, Romania. e-mail address:
[email protected] 