Abstract. We consider Finsler spaces with a Randers metric F = α + β, on the three-dimen-sional real vector space, where α is the Euclidean metric and β is a 1-form with norm b, 0 ≤b < 1. By using the notion of mean curvature for immersions in Finsler spaces, introducedby Z. Shen, we obtain the partial differential equation that characterizes the minimal surfaceswhich are graphs of functions. For each b, 0 ≤ b < 1/
√3, we prove that it is an elliptic equa-
tion of mean curvature type. Then the Bernstein type theorem and other properties, such as thenonexistence of isolated singularities, of the solutions of this equation follow from the theorydevelopped by L. Simon. For b ≥ 1/
√3, the differential equation is not elliptic. Moreover, for
every b, 1/√
3 < b < 1 we provide solutions, which describe minimal cones, with an isolatedsingularity at the origin.
1. Introduction
A classical result of S. Bernstein states that the plane is the only regular minimalsurface of R
3, which is the graph of a C2-function defined on the whole plane.This is a consequence of the partial differential equation which characterizes suchsurfaces. Many properties of the minimal surfaces can be atributed to the form ofthis equation. A major contribution to the theory of such equations was given byL. Simon [S1,S2], who considered the class of equations of mean curvature type.The pioneering work in this direction was done by R. Finn [F], who considered theequations of minimal type. Later other important results were obtained by Jenkins[J], Jenkins-Serrin [JS] and Spruck [Sp].
M. Souza∗Instituto de Matematica e Estatıstica, Universidade Federal de Goias, 74001-970, Goiania, GO,Brazil (e-mail: [email protected])
K. Tenenblat∗∗∗Departamento de Matematica, Universidade de Brasılia, 70910-900, Brasılia, DF, Brazil(e-mail: [email protected])∗ Partially supported by CAPES/PROCAD.∗∗ Partially supported by NSF grant DMS-0072242.∗∗∗ Partially supported by CNPq and CAPES/PROCAD.
292 M. Souza et al.
The theory of minimal surfaces in Finsler spaces is quite recent. Actually, thefirst non trivial examples of such surfaces were studied in [ST]. The fundamentalcontribution on this subject was given by Z. Shen in [Sh1]. He introduced thenotion of mean curvature for immersions into Finsler manifolds and he estab-lished some of its properties. As in the Riemannian case, if the mean curvature isidentically zero, then the immersion is said to be minimal.
The main purpose of this paper is to prove a Bernstein type theorem in a three-dimensional vector space, equipped with a Randers metric. This metric can beviewed as the simplest possible perturbation of the Euclidean metric in a fixeddirection. This perturbation has a norm b, where 0 ≤ b < 1 (b = 0 being theEuclidean case).
Our main result follows from the partial differential equation that character-izes the minimal surfaces, which are graphs of functions, in this Randers space.We show that for each b, 0 ≤ b < 1/
√3, this equation is an elliptic differential
equation of mean curvature type. Then the Bernstein type theorem in this Ran-ders space follows from the theory developped by L. Simon for such equations.Similary, as a consequence of this theory, one gets several results for the solutionsof the equation such as a-priori gradient estimates, a Bers-type theorem concern-ing the limiting behaviour of the gradient of solutions defined outside a compactset, a global Holder continuity estimate for solutions which continuously attainLipschitz boundary values and a theorem concerning the removability of isolatedsingularities.
When b ≥ 1/√
3, the differential equation is not elliptic and one does notknow if the Bernstein type theorem holds. However, one can show that the prop-erty of nonexistence of isolated singularities does not hold. In fact, for each b,b > 1/
√3, we provide a minimal cone with an isolated singularity at the origin.
We conclude this introduction by pointing out that the differential equationis sensitive to the fixed direction in the Randers space. In fact, a minimal sur-face which is the graph of a function f over a certain plane may not be minimalas a surface obtained as the graph of f over a different plane (see Examples inSection 3).
2. Preliminaries
We will follow the notation and terminology of [Sh1] and [ST], and we will makeuse of the following conventions: we will use Greek letters τ, η, ε for indices run-ning from 1 to n, and Latin letters i, j, k, l for indices running from 1 to n+1. Wewill also use Einstein’s convention, i.e., in general we will not write the symbolof the summand to represent the sum on repeated indices.
LetMn be aC∞ n-manifold, andπ : T M → M be the natural projection fromthe tangent bundle T M . Let (x, y) be a point of T M, x ∈ M, y ∈ TxM . We con-sider local coordinates (x1, ..., xn) on an open subset U of M . As usual, ∂/∂xi and
A Bernstein type theorem on a Randers space 293
dxi are the induced coordinate basis for TxM and T ∗x M and (xi, yi) are local coor-
dinates on π−1(U) ⊂ T M , where y = yi∂/∂xi . A function F : T M −→ [0, ∞)
is called a Finsler metric on M if F has the following properties: [i] (Regularity)F ∈ C∞ in T M \ {0}; [ii] (Positive Homogeneity) F(x, ty) = tF (x, y), ∀t >
0, (x, y) ∈ T M; [iii] (Strong Convexity) g = (gij (x, y)
) = (12 [F 2(x, y)]yiyj
)is
positive definite at each point of T M \ {0}. The pair (M, F ) is called a Finslerspace.
Examples of Finsler manifolds are Minkowski spaces and Randers spaces.Denote by V n the standard n-dimensional real vector space. A Minkowski spaceis V n equipped with a Minkowski norm F (whose indicatrix is strongly convex),i.e., F(x, y) depends only on y ∈ TxV
n. A Randers metric on M is the Finslerstructure F on T M given by
F(x, y) = α(x, y) + β(x, y),
where α(x, y) = √aij (x)yiyj , β(x, y) = bk(x)yk, and aij , aij are the compo-
nents of the Riemannian metric and of its inverse matrix respectively and bk arethe components of the 1-form β, whose norm b = √
aijbibj , satisfies 0 ≤ b < 1.If (Mn, F ) is a Finsler space, then F induces a smooth volume form defined
by
dµF = σ(x)dx1 ∧ ... ∧ dxn
where
σ(x) = vol (Bn)
vol {y ∈ TxM; F(x, y) ≤ 1} ,
Bn is a unit ball in Rn and vol is the Euclidean volume.
Let (Mm, F ) be a Finsler manifold, with local coordinates (x1, ..., xm) and letϕ : Mn −→ (Mm, F ) be an immersion. Then there is an induced Finsler metricon M , defined by
F(x, y) = (ϕ∗F )(x, y) = F (ϕ(x), ϕ∗(y)), ∀ (x, y) ∈ T M.
The notion of mean curvature was introduced by Z. Shen (see [Sh1]) asfollows. Let ϕ : Mn −→ (Mm, F ) be an immersion in a Finsler space and letϕt : Mn −→ (Mm, F ), t ∈ (−ε, ε) be a variation such that for all t , ϕt is animmersion, ϕ0 = ϕ and ϕt = ϕ outside a compact set ⊂ M . Then Ft = ϕ∗
t F
denotes the induced metric of the variation and X = ∂ϕt
∂t|t=0 is the variational
vector field. Consider the function V (t) =∫
dµFt. Then
V ′(0) =∫
M
Hϕ(X)dµF ,
294 M. Souza et al.
where Hϕ is called the mean curvature of the immersion ϕ. One can show thatHϕ(v) depends linearly on v and Hϕ vanishes on ϕ∗(T M) (cf. Lemmas in [Sh1]).The immersion ϕ is said to be minimal when Hϕ ≡ 0.
From now on, we will consider hypersurfaces in a special Randers space ϕ :Mn −→ (V n+1, Fb), where V is an+1-dimensional real vector space, Fb = α+β,where α is the Euclidean metric, and β is a 1-form with norm b, 0 ≤ b < 1.Without loss of generality we will consider β = b dxn+1. If Mn has local coordi-nates x = (xε), ε = 1, · · · , n, and ϕ(x) = (
ϕi(xε)) ∈ V , i = 1, . . . , n + 1, we
consider the application
F(x, z) = vol(Bn)
vol(Dnx)
, (1)
where x ∈ M ,
z = (ziα) =
(∂ϕi
∂xα
), (2)
Bn = unitary ball in Rn and
Dnx = {
(y1, y2, . . . , yn) ∈ Rn | F(x, yαzα|x ) < 1
}, (3)
where zα = ∂ϕ/∂xα.The induced volume element of (M, F ) is given by
dVF = F(x, z)dx, (4)
where F(x, z) is given by (1).The Euclidean volume of Dn
x is given by
vol Dnx = vol Bn
(1 − b2Aτγ zn+1
τ zn+1γ
) n+12
√detA
,
where
A = (Aτγ
) =(
n+1∑
i=1
ziτ z
iγ
)
, (5)
and (Aτγ ) = (Aτγ
)−1. It follows from (4) that the volume form dVF is given by
the following formula ([Sh2])
dVF = (1 − b2Aτγ zn+1
τ zn+1γ
) n+12
√detA dx1 · · · dxn. (6)
The mean curvature Hϕ is given by (see [Sh1])
Hϕ(v) = 1
F
{∂2F
∂ziε∂z
jη
∂2ϕj
∂xε∂xη+ ∂2F
∂xj ∂ziε
∂ϕj
∂xε− ∂F
∂xi
}
vi.
A Bernstein type theorem on a Randers space 295
Observe that whenever (V , F ) is a Minkowsky space, the expression of the meancurvature reduces to
Hϕ(v) = 1
F
{∂2F
∂ziε∂z
jη
∂2ϕj
∂xε∂xη
}
vi. (7)
3. The differential equation for a minimal surface which is the graphof a function
In this section, we recall the differential equation which characterizes the minimalhypersurfaces Mn in the Randers space (V n+1, Fb). We then restrict ourselves tosurfaces immersed in V 3 and we obtain the differential equation which charac-terizes the minimal surfaces which are the graph of a function.
Theorem 1 ([ST]). Let ϕ : Mn −→ (V n+1, Fb) be an immersion into a Randersspace, with local coordinates
(ϕj (x)
). Then ϕ is minimal, if and only if, it satisfies
the differential equation{
(n2 − 1)
4
∂B
∂ziε
∂B
∂zjη
C − n + 1
2(1 − B)
(∂2B
∂ziε∂z
jη
C + ∂B
∂zjη
∂C
∂ziε
+ ∂B
∂ziε
∂C
∂zjη
)
+(1 − B)2 ∂2C
∂ziε∂z
jη
}∂2ϕj
∂xε∂xηvi = 0, ∀v = viei ∈ V n+1, (8)
where
C =√
detA, B = b2Aεηzn+1ε zn+1
η , (9)
{ei} is the canonical basis of V n+1, ziε is given by (2), A is given by (5).
In what follows, we will restrict ourselves to studying minimal surfaces in thethree-dimensional Randers space. As a consequence of the above theorem one hasthe following result.
Theorem 2 ([ST]). Let ϕ : M2 −→ (V 3, Fb) be an immersion given in localcoordinates by
(ϕj (x)
). Then ϕ is minimal, if and only if, it satisfies the differen-
tial equation{
12E2 − (2E + C2)2
C(C2 − E)
∂C
∂ziε
∂C
∂zjη
− 3C
2
∂2E
∂zjη∂zi
ε
− 3
2
(2E − C2
C2 − E
)
×(
∂C
∂ziε
∂E
∂zjη
+ ∂C
∂zjη
∂E
∂ziε
)
+ 3C
4(C2 − E)
∂E
∂ziε
∂E
∂zjη
+ (2E + C2)
2C
∂2C2
∂zjη∂zi
ε
}
× ∂2ϕj
∂xε∂xηvi = 0, ∀ v = viei ∈ V 3, (10)
296 M. Souza et al.
where z = (ziε) and C are defined by (2) and (9) respectively, and
E = b23∑
k=1
(−1)γ+τ zkγ zk
τ z3γ z3
τ , τ = δτ2 + 2δτ1. (11)
We observe that in (11) we have introduced the notation E = C2B and τ ,which means τ = 1 if τ = 2 and τ = 2 if τ = 1.
In our next results, by considering the immersion to be a surface which is thegraph of a function f , we obtain the differential equation that characterizes suchminimal surfaces. We will first consider the special and important case, obtainedin [S], when the surface is a graph over the x1x2-plane (observe that we havechosen β = b dx3 in the Randers metric) and then we will consider the generalcase when the surface is the graph over any plane.
Theorem 3. An immersion ϕ : U ⊂ R2 −→ (V 3, Fb) given by ϕ(x1, x2) =
(x1, x2, f (x1, x2)) is minimal, if and only if, f satisfies
∑
i,j=1,2
{Tb(Tb − 3b2)
(δij − fxi
fxj
W 2
)+ 3b2(Tb + b2)
fxifxj
W 2
}fxixj
= 0, (12)
where
W 2 = 1 + f 2x1
+ f 2x2
, Tb = b2 + (1 − b2)W 2. (13)
Proof. In order to obtain equation (12), we need to compute the expressionsinvolved in (10) for the immersion ϕ. One computes the first and second orderderivatives of C, det A and E with respect to the variables zi
η, 1 ≤ i ≤ 3, η = 1, 2(see also [ST]).
From the expression of ϕ and (5) we have that
A =(
1 + f 2x1
fx1fx2
fx1fx2 1 + f 2x2
), C =
√det A = W, (14)
where W is given by (13). We now consider the vector field
v = (v1, v2, v3) = (−fx1, −fx2, 1),
which is linearly independent with ϕx1 and ϕx2 .
A Bernstein type theorem on a Randers space 297
By using the first and second order derivatives of C, det A, E, and the fact that∂2ϕj/∂xε∂xη = δj3fxεxη
, a straightforward computation implies that
∂C
∂ziε
vi = 0, ∀ε; (15)
∂E
∂ziε
vi = 2b2(δε1fx1 + δε2fx2), ∀ε; (16)
∂C
∂zjη
∂2ϕj
∂xε∂xη
= fx1fxεx1 + fx2fxεx2
W, ∀ε; (17)
∂E
∂zjη
∂2ϕj
∂xε∂xη
= 2b2 [fx1fxεx1 + fx2fxεx2
], ∀ ε; (18)
∂2E
∂ziε∂z
jη
∂2ϕj
∂xε∂xη
vi = 2b2 [(1 + f 2x1
)fx2x2 − 2fx1fx2fx1x2
+(1 + f 2x2
)fx1x1
] ; (19)
1
2
∂2detA
∂zjη∂zi
ε
∂2ϕj
∂xε∂xη
vi = [(1 + f 2
x1)fx2x2 − 2fx1fx2fx1x2 + (1 + f 2
x2)fx1x1
], (20)
where 1 ≤ i, j ≤ 3, 1 ≤ ε, η ≤ 2.As a consequence of (15) and the definition of E, equation (10) reduces to
{
−3C
2
∂2E
∂zjη∂zi
ε
− 3
2
(2E − C2
C2 − E
)∂C
∂zjη
∂E
∂ziε
+ 3C
4(C2 − E)
∂E
∂ziε
∂E
∂zjη
+(2E + C2)
2C
∂2detA
∂zjη∂zi
ε
}∂2ϕj
∂xε∂xηvi = 0. (21)
It follows from (5), (9) and (14), that B = b2(W 2 − 1)/W 2. Therefore,
−2E − C2
C2 − E= 2Tb − W 2
Tb
,3C
4(C2 − E)= 3W
4Tb
, −2E + C2
2C= 2Tb − 3W 2
2W,
(22)
where Tb is given by (13). Now it follows, from (15)-(20) and (22), that equation(21) reduces to
Tb(Tb − 3b2)[(1 + f 2
x1)fx2x2 − 2fx1fx2fx1x2 + (1 + f 2
x2)fx1x1
]
+3b2(Tb + b2)[f 2
x1fx1x1 + 2fx1fx2fx1x2 + f 2
x2fx2x2
] = 0.
This concludes the proof of the theorem, since this equation is equivalent to (12). �
Our next result will provide the differential equation which must be satisfiedfor a minimal surface which is the graph of a function over any plane of V 3.
298 M. Souza et al.
Theorem 4. An immersion ϕ : U ⊂ R2 −→ (V 3, Fb) which is the graph of a
function f (x1, x2) over a plane of V 3 is minimal, if and only if, f satisfies
2∑
i,j=1
{Tb(Tb − 3b2w2)(δij − fxi
fxj
W 2)
+3b2W 2(Tb + b2w2)(ki + fxi
W 2w)(kj + fxj
W 2w)
}fxixj
= 0. (23)
where W 2 is defined by (13), ki are real numbers such that∑3
Proof. The proof is similar (although lengthier) to the particular case proved inTheorem 3. Assume that the immersion ϕ is a graph of a function over an opensubset of a plane of V 3. Then ϕ is a function of the form
ϕ(x1, x2) = (x1, x2, f (x1, x2))(mij ), (25)
where (mij ) is a 3 × 3 orthogonal matrix, (x1, x2) ∈ U ⊂ R2 and the surface is a
graph over the plane m31x + m32y + m33z = 0.We need to compute the expressions involved in (10) for the immersion ϕ. The
first and second order derivatives of C, det A and E with respect to the variablesziη, are those computed in the proof of Theorem 3.
From (5) and the expression of ϕ given by (25), and since the matrix (mij ) isorthogonal, we have that A and C are given by (14). We now consider the vectorfield v = (v1, v2, v3)
vi = −fx1m1i − fx2m2i + m3i ,
which is linearly independent with ϕx1 and ϕx2 .Observe that
ziη = ∂ϕi
∂xη
= mηi + fxηm3i ,
∂2ϕi
∂xε∂xη
= fxεxηm3i . (26)
Moreover, for all i = 1, 2, 3 and η, γ, ε = 1, 2
3∑
i=1
ziηv
i = 0,
3∑
i=1
vim3i = 1,
3∑
i=1
ziηm3i = fxη
,
3∑
i=1
ziγ
∂2ϕi
∂xε∂xη
= fxγfxεxη
.
A Bernstein type theorem on a Randers space 299
By using the first and second order derivatives of C, det A and E, a straight-forward computation implies that
∂C
∂ziε
vi = 0, ∀ε; (27)
∂E
∂ziε
vi = 2b2(z3εAεε − z3
εAεε)w, ∀ε; (28)
∂C
∂zjη
∂2ϕj
∂xε∂xη
= fx1fxεx1 + fx2fxεx2
W, ∀ε; (29)
∂E
∂zjη
∂2ϕj
∂xε∂xη
= 2b2 [(fx1 + k1w)fxεx1 + (fx2 + k2w)fxεx2
], ∀ ε; (30)
∂2E
∂ziε∂z
jη
∂2ϕj
∂xε∂xη
vi = 2b2 {[1 + f 2x1
− k2(k2W2 + fx2w)
]fx2x2
− [(1 + k23)fx1fx2 + k1k2W
2 + k1k3fx2
+k2k3fx1 + k1k2]fx1x2
+ [1 + f 2
x2− k1(k1W
2 + fx1w)]fx1x1
} ; (31)
1
2
∂2detA
∂zjη∂zi
ε
∂2ϕj
∂xε∂xη
vi = (1 + f 2x1
)fx2x2 − 2fx1fx2fx1x2 + (1 + f 2x2
)fx1x1, (32)
where w = v3 is given by (24) and we have introduced the notation ki = mi3, fori = 1, 2, 3.
As a consequence of (27) and the definition of E, equation (10) reduces to(21). It follows from (5), (9) and (14), that
B = b2
W 2[W 2 − w2].
Therefore,
−2E − C2
C2 − E= 2Tb − W 2
Tb
,3C
4(C2 − E)= 3W
4Tb
, −2E + C2
2C= 2Tb − 3W 2
2W,
(33)
where Tb is given by (24). Now it follows from (27)–(32), (33) and the fact thatk3 = w + fx1k1 + fx2k2 that equation (21) reduces to (23). �
Observe that when k1 = k2 = 0 and k3 = 1, then equation (23) reduces to(12). Moreover, when b = 0 both equations reduce to the classical equation of aminimal surface in R
3, which is the graph of f .
300 M. Souza et al.
Examples. A surface which is the graph of a linear function f (x1, x2), is minimal∀b, 0 ≤ b < 1. Moreover, one can verify, that for any b such that 1/
√3 < b < 1,
the cone
x1, x2,
√(3b2 − 1)(x2
1 + x22)
1 − b2
, (34)
where x21 + x2
2 �= 0 is a minimal surface in (V 3, Fb), since it satisfies (12). In
particular, (x1, x2,
√x2
1 + x22), is a minimal surface, when b2 = 1/2. However,
the cone (x1,
√x2
1 + x22 , x2), is not a minimal surface in this Randers space. In
fact, by considering k1 = k3 = 0, k2 = 1 and b2 = 1/2, one shows that the lefthand side of (23) is always positive.
For a fixed plane of V 3 of the form v1x1 +v2x2 +v3x3 = 0, where∑
i v2i = 1,
the minimal graphs over subsets of this plane are the solutions of equation (23),where k1 = m13, k2 = m23, k3 = v3 and (mij ) is an orthogonal 3×3 matrix suchthat m31 = v1 and m32 = v2.
It is not difficult to prove that, for 0 ≤ b < 1/√
3, equation (12) is an ellipticequation of mean curvature type, as defined by L. Simon [S1]. In fact, one canshow that for such a b, one has Tb > 0 and Tb − 3b2 > 0. Hence, (23) can bewritten as
It is simple to verify that for all ξ ∈ R2 \ {0}, x, p ∈ R
2 and z ∈ R,
0 <|ξ |2
1 + |p|2 ≤∑
i,j=1,2
aij (x, z, p)ξiξj ≤(
1 + |p|2Sb(p)
1 + |p|2)
|ξ |2.
Therefore, (35) is an elliptic equation. Moreover, one can show that for b fixed,there exists a constant C > 0, such that Sb(p) ≤ C/W 2(p) and
∑ij �ij (p)ξiξj ≥
|ξ |2/W 2(p) where
�ij (p) = δij − pipj
W 2(p). (36)
Therefore, for all (x, z, p) ∈ R5 and ξ ∈ R
2
|ξ |2 − (p · ξ)2
1 + |p|2 ≤∑
i,j=1,2
aij (x, z, p)ξiξj ≤ (1 + C)
(|ξ |2 − (p · ξ)2
1 + |p|2)
,
i.e., (12) is an elliptic equation of mean curvature type.
A Bernstein type theorem on a Randers space 301
Our next result proves the general case, i.e. that for 0 ≤ b < 1/√
3, the differ-ential equation of a minimal surface, which is the graph of a function in (V 3, Fb),(23), is an elliptic equation of mean curvature type.
Theorem 5. Let ϕ : R2 −→ (V 3, Fb) be an immersion which is the graph of a
function f (x1, x2) over a plane. If 0 ≤ b < 1/√
3, then ϕ is minimal, if and onlyif, f satisfies the elliptic differential equation, of mean curvature type, given by
∑
i,j=1,2
aij (x, f, ∇f )fxixj= 0, (37)
where
aij = δij − fxifxj
W 2 + QbW2(ki + fxi
W 2 w) (
kj + fxj
W 2 w)
, (38)
Qb = 3b2(Tb+b2)
Tb(Tb−3b2), (39)
w and Tb are given by (24) and ki , i = 1, 2, 3, are real numbers such that∑i k
2i = 1.
Proof. From Theorem 4, ϕ is minimal if and only if f satisfies (23). Since 0 ≤b < 1/
With the notation introduced in (36), we observe that for all ξ ∈ R2
2∑
i,j=1
�ij (p)ξiξj = |ξ |2W 2
(1 + |p|2 sin2 θ), (40)
where θ is the angle function between p and ξ . Moreover,
2∑
i,j=1
aij (x, z, p)ξiξj =2∑
i,j=1
�ij ξiξj + QbW2[(k1, k2) · ξ + w
W 2p · ξ
]2, (41)
where · is the Euclidean inner product. Therefore, since Qb > 0, for all ξ ∈R
2 \ {0}, we have
∑
i,j=1,2
aij (x, z, p)ξiξj ≥2∑
i,j=1
�ij ξiξj ≥ |ξ |2W 2
> 0, (42)
302 M. Souza et al.
where the second inequality follows from (40). Therefore, (37) is an ellipticequation.
In order to prove that it is a differential equation of mean curvature type, weneed to show that there exists a constant C such that, for all (x, z, p) ∈ R
5 andξ ∈ R
2,
2∑
i,j=1
�ij (x, z, p)ξiξj ≤2∑
i,j=1
aij (x, z, p)ξiξj ≤ (1 + C)
2∑
i,j=1
�ij (x, z, p)ξiξj .
(43)
The first inequality was obtained in (42). In order to prove the second inequality,it follows from (41), that we only need to prove that there exists a constant C suchthat
QbW2[(k1, k2) · ξ + w
W 2p · ξ
]2≤ C
2∑
i,j=1
�ij (x, z, p)ξiξj , (44)
where w = −k1p1 − k2p2 + k3.From (24) and (39) we have that
Qb = 3b2[(1 − b2)W 2 + 2b2w2]
[(1 − b2)W 2 + b2w2][(1 − b2)W 2 − 2b2w2](45)
and it follows from (40) that
W 2[(k1, k2) · ξ + w
W 2p · ξ
]2
=[W 2|(k1, k2)| cos γ + w|p| cos θ
]2
1 + |p|2 sin2 θ
2∑
i,j=1
�ij (x, z, p)ξiξj ,
where γ is the angle between (k1, k2) and ξ . Hence, we need to show that thereexists a constant C such that
Qb
[W 2|(k1, k2)| cos γ + w|p| cos θ ]2
1 + |p|2 sin2 θ≤ C. (46)
Observe that W 2 ≥ 1. When W 2 = 1, i.e., p = 0, the left hand side of (46) isless than or equal to the real number Qb(0)(k2
1 + k22) ≥ 0. Whenever W 2 > 1 and
sin θ = 0, then p �= 0 and the vectors p and ξ are parallel. Hence,[W 2 |(k1, k2)| cos γ + w|p| cos θ
]2 = [|(k1, k2)| cos γ + k3|p| cos θ]2
.
Therefore, the left hand side of (46) is a rational function of |p| whose numeratoris of degree less than or equal to 4, and denominator is of degree 4 and hence itis a bounded function when |p| (or equivalently W ) tends to infinity. Whenever
A Bernstein type theorem on a Randers space 303
W 2 > 1 and sin θ �= 0, then p �= 0 and the vectors p and ξ are not parallel.Hence, the left hand side of (46) is a rational function of |p| whose numerator isof degree less than or equal to 6, and denominator is of degree 6 and hence it is abounded function when |p| (or equivalently W ) tends to infinity. This completesthe proof of the inequality (46). �We observe that when b = 1/
√3 equation (12) is not elliptic. In fact, in this case,
the equation reduces to∑
i,j=1,2 cijfxixj= 0, where
cij = 2|∇f |23
(1 + 2
3|∇f |2)(δij − fxi
fxj
W 2) + (4 + 2|∇f |2)fxi
fxj
3W 2.
Therefore,∑
i,j cij (x, z, p)ξiξj is a multiple of |p|2 and hence vanishes for p = 0.As an immediate consequence of Theorem 5 and a Bernstein type theorem
proved by Simon (see Theorem 4 in [S1], Theorem 4.1 in [S2]), we conclude that
Theorem 6. A minimal surface in a special Randers Space (V 3, Fb), 0 ≤ b <
1/√
3, which is the graph of a function defined on R2, is a plane.
Our next results, provide properties of minimal surfaces in the special Randersspace, when 0 ≤ b < 1/
√3.
Corollary 7. Assume b is in the interval [0, 1/√
3). If there exists a solutionof the Dirichlet problem for a minimal surface which is the graph of a function f
in the special Randers space (V 3, Fb), then it is unique.
Consider two minimal surfaces in a special Randers space (V 3, Fb), with0 ≤ b < 1/
√3, which are graphs of functions. Assume the surfaces are tangent
at a point p0 ∈ V 3, then both surfaces can be locally considered to be graphs offunctions f (x1, x2) and h(x1, x2) over the same plane of V 3. Let u = f − h be afunction defined in the intersection of the domains of f and h. Then
Lemma 8. The function u satisfies the differential equation
L(u) :=∑
i,j
aij (x, f, ∇f )uxixj+∑
i
ciuxi= 0, (47)
where aij is given by (38),
ci = −2∑
j,�=1
[∫ 1
0
∂βj�
∂pi
(∇f + t (∇h − ∇f )) dt
]hxj x�
(48)
and for p = (p1, p2) ∈ R2,
βj�(p) = −pjp�
W 2+ Qb(p)W 2
(kj + pjw(p)
W 2
)(k� + p�w(p)
W 2
), (49)
where Qb(p) is given by (39), W 2 = 1 + |p|2 and w(p) = −k1p1 − k2p2 + k3.Moreover, L(u) is an elliptic operator.
304 M. Souza et al.
Proof. Since the graphs of f and h are minimal surfaces, it follows from Theorem5 that
∑
j,�
aj�(x, f, ∇f )fxj x�= 0 and
∑
j,�
aj�(x, h, ∇h)hxj x�= 0,
where aj� is given by (38). Taking the difference of these two equations, addingand subtracting the expression
∑j,� aj�(x, f, ∇f )hxj x�
, we get
0 =∑
j,�
aj�(x, f, ∇f )uxj x�+∑
j,�
[βj�(∇f ) − βj�(∇h)
]hxj x�
,
where βj� is given by (49). Since
βj�(∇h) − βj�(∇f ) =2∑
i=1
∫ 1
0
∂βj�
∂pi
(∇f + t (∇h − ∇f ))(hxi− fxi
) dt,
we conclude that
∑
j,�
[βj�(∇f ) − βj�(∇h)
]hxj x�
=2∑
i=1
ciuxi,
where ci is given by (48). �
Since the functions ci given by (48) are locally bounded, the following theoremfollows from Lemma 8 and the maximum principle.
Theorem 9. Let M1 and M2 be minimal surfaces in (V 3, Fb), 0 ≤ b < 1/√
3. IfM1 is above M2 near p0 and internally tangent at p0, then M1 and M2 coincidein a neighborhood of p0.
Besides the Bernstein type theorem given in Theorem 6, as a consequence ofTheorem 5 and the theory developped by L. Simon [S1], one gets several resultsfor the solutions of the equation of mean curvature type (37); in particular we willlist some of these results such as a-priori gradient estimates, a Bers-type theoremconcerning the limiting behaviour of the gradient of solutions defined outside acompact set, a global Holder continuity estimate for solutions which continuouslyattain Lipschitz boundary values and a theorem concerning the removability ofisolated singularities.
In what follows, it is assumed that ⊂ R2, and f is a C2() solution of (37),
where we have fixed ki such that∑
k2i = 1. Let x0 denote a fixed point of , and
let p0 be the corresponding point on the surface M which is the graph of f . LetDρ(x0) = {x ∈ R
2; |x − x0| < ρ} and Sρ(p0) = {p ∈ M; |p − p0| < ρ}.
A Bernstein type theorem on a Randers space 305
Proposition 10. If Dρ(x0) ⊂ , then
supSρ/2(p0)
√1 + |∇f |2 ≤ γ inf
Sρ/2(p0)
√1 + |∇f |2,
where γ > 0 depends only on C of (43). Moreover, if f ≥ 0 on Dρ(x0), then
|∇f (x0)| ≥ γ1exp{γ2f (x0)/ρ},where γ1, γ2 depend only on C.
Proposition 11. Suppose f is defined outside of a compact subset of R2. Then
there is a vector a ∈ R2 such that ∇f (x) → a uniformly for |x| → ∞.
Proposition 12. A minimal surface in the Randers space (V 3, Fb), for 0 ≤ b <
1/√
3, cannot have an isolated singularity.
We conclude by observing that the above result fails if the condition on b doesnot hold. In fact, any minimal cone given by (34), for 1/
√3 < b < 1, has an
isolated singularity at the origin.
References
[BCS] Bao, D., Chern, S.S., Shen, Z.: An Introduction to Riemann-Finsler Geometry. GraduateTexts in Mathematics, 200, Springer-Verlag, New York, 2000
[F] Finn, R.: Equations of minimal surface type. Ann. Math. 60, 397–416 (1954)[J] Jenkins, H.: On 2-dimensional variational problems in parametric form.Arch. Rat. Mech.
Anal. 8, 181–206 (1961)[JS] Jenkins, H., Serrin, J.: Variational problems of minimal surface type I. Arch. Rat. Mech.
Anal. 12, 185–212 (1963)[Sh1] Shen, Z.: On Finsler geometry of submanifolds. Math. Ann. 311, 549–576 (1998)[Sh2] Shen, Z.: Lectures on Finsler Geometry. World Scientific Publishers, Singapore, xiv,
2001[S1] Simon, L.: Equations of Mean Curvature Type in 2 Independent Variables. Pac. J. Math.
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