Math. Ann. 329, 291–305 (2004)
Mathematische Annalen
DOI: 10.1007/s00208-003-0500-3
A Bernstein type theorem on a Randers space Marcelo Souza · Joel Spruck · Keti Tenenblat Received: 16 July 2003 / Published online: 13 March 2004 – © Springer-Verlag 2004 Abstract. We consider Finsler spaces with a Randers metric F = α + β, on the three-dimensional 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, introduced by Z. Shen, we obtain the partial differential equation √ that characterizes the minimal surfaces which are graphs of functions. For each b, 0 ≤ b < 1/ 3, we prove that it is an elliptic equation of mean curvature type. Then the Bernstein type theorem and other properties, such as the nonexistence of isolated singularities,√of the solutions of this equation follow from the theory developped√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 isolated singularity at the origin.
1. Introduction A classical result of S. Bernstein states that the plane is the only regular minimal surface of R3 , which is the graph of a C 2 -function defined on the whole plane. This is a consequence of the partial differential equation which characterizes such surfaces. Many properties of the minimal surfaces can be atributed to the form of this equation. A major contribution to the theory of such equations was given by L. 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 the equations of minimal type. Later other important results were obtained by Jenkins [J], Jenkins-Serrin [JS] and Spruck [Sp]. M. Souza∗ Instituto de Matem´atica e Estat´ıstica, Universidade Federal de Goi´as, 74001-970, Goiˆania, GO, Brazil (e-mail:
[email protected]) J. Spruck∗∗ Mathematics Department, Johns Hopkins University, Baltimore, MD 21218-2689, USA (e-mail:
[email protected]) K. Tenenblat∗∗∗ Departamento de Matem´atica, Universidade de Bras´ılia, 70910-900, Bras´ılia, 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. ∗∗
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The theory of minimal surfaces in Finsler spaces is quite recent. Actually, the first non trivial examples of such surfaces were studied in [ST]. The fundamental contribution on this subject was given by Z. Shen in [Sh1]. He introduced the notion of mean curvature for immersions into Finsler manifolds and he established some of its properties. As in the Riemannian case, if the mean curvature is identically zero, then the immersion is said to be minimal. The main purpose of this paper is to prove a Bernstein type theorem in a threedimensional vector space, equipped with a Randers metric. This metric can be viewed as the simplest possible perturbation of the Euclidean metric in a fixed direction. This perturbation has a norm b, where 0 ≤ b < 1 (b = 0 being the Euclidean case). Our main result follows from the partial differential equation that characterizes 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 Randers 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 solutions of the equation such as a-priori gradient estimates, a Bers-type theorem concerning the limiting behaviour of the gradient of solutions defined outside a compact set, a global H¨older continuity estimate for solutions which continuously attain Lipschitz boundary values and a theorem concerning the removability of isolated singularities. √ When b ≥ 1/ 3, the differential equation is not elliptic and one does not know if the Bernstein type theorem holds. However, one can show that the property 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 equation is sensitive to the fixed direction in the Randers space. In fact, a minimal surface which is the graph of a function f over a certain plane may not be minimal as a surface obtained as the graph of f over a different plane (see Examples in Section 3). 2. Preliminaries We will follow the notation and terminology of [Sh1] and [ST], and we will make use of the following conventions: we will use Greek letters τ, η, ε for indices running from 1 to n, and Latin letters i, j, k, l for indices running from 1 to n + 1. We will also use Einstein’s convention, i.e., in general we will not write the symbol of the summand to represent the sum on repeated indices. Let M n be a C ∞ n-manifold, and π : T M → M be the natural projection from the tangent bundle T M. Let (x, y) be a point of T M, x ∈ M, y ∈ Tx M. We consider local coordinates (x 1 , ..., x n ) on an open subset U of M. As usual, ∂/∂x i and
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dx i are the induced coordinate basis for Tx M and Tx∗ M and (x i , y i ) are local coordinates on π −1 (U ) ⊂ T M, where y = y i ∂/∂x i . A function F : T M −→ [0, ∞) is called a Finsler metric on M if F has the following properties: [i] (Regularity) F (x, F ∈ C ∞ in T M \ {0}; [ii] (Positive Homogeneity) ty) = tF (x, y), ∀t > 0, (x, y) ∈ T M; [iii] (Strong Convexity) g = gij (x, y) = 21 [F 2 (x, y)]yi yj is positive definite at each point of T M \ {0}. The pair (M, F ) is called a Finsler space. Examples of Finsler manifolds are Minkowski spaces and Randers spaces. Denote by V n the standard n-dimensional real vector space. A Minkowski space is V n equipped with a Minkowski norm F (whose indicatrix is strongly convex), i.e., F (x, y) depends only on y ∈ Tx V n . A Randers metric on M is the Finsler structure F on T M given by F (x, y) = α(x, y) + β(x, y),
where α(x, y) = aij (x)y i y j , β(x, y) = bk (x)y k , and aij , a ij are the components of the Riemannian metric and of its inverse matrix respectively and bk are the components of the 1-form β, whose norm b = a ij bi bj , satisfies 0 ≤ b < 1. If (M n , F ) is a Finsler space, then F induces a smooth volume form defined by dµF = σ (x)dx 1 ∧ ... ∧ dx n where σ (x) =
vol (B n ) , vol {y ∈ Tx M; F (x, y) ≤ 1}
B n is a unit ball in Rn and vol is the Euclidean volume. m , F˜ ) be a Finsler manifold, with local coordinates (x˜ 1 , ..., x˜ m ) and let Let (M n m , F ) be an immersion. Then there is an induced Finsler metric ϕ : M −→ (M on M, defined by )(x, y) = F (ϕ(x), ϕ∗ (y)), F (x, y) = (ϕ ∗ F
∀ (x, y) ∈ T M.
The notion of mean curvature was introduced by Z. Shen (see [Sh1]) as m , F ) be an immersion in a Finsler space and let follows. Let ϕ : M n −→ (M n m ϕt : M −→ (M , F ), t ∈ (−ε, ε) be a variation such that for all t, ϕt is an immersion, ϕ0 = ϕ and ϕt = ϕ outside a compact set ⊂ M. Then Ft = ϕt∗ F ∂ϕ t denotes the induced metric of the variation and X˜ = |t=0 is the variational ∂t dµFt . Then vector field. Consider the function V (t) =
V (0) =
M
˜ Hϕ (X)dµ F,
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where Hϕ is called the mean curvature of the immersion ϕ. One can show that Hϕ (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 : n+1 , Fb ), where V is a n+1-dimensional real vector space, F b = α+β, M n −→ (V 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 d x˜ n+1 . If M n has local coordi, i = 1, . . . , n + 1, we nates x = (x ε ), ε = 1, · · · , n, and ϕ(x) = ϕ i (x ε ) ∈ V consider the application vol(B n ) , vol(Dxn )
F (x, z) = where x ∈ M,
z=
(zαi )
=
∂ϕ i ∂x α
(1)
(2)
,
B n = unitary ball in Rn and Dxn = (y 1 , y 2 , . . . , y n ) ∈ Rn | F (x, y α zα|x ) < 1 ,
(3)
where zα = ∂ϕ/∂x α . 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 Dxn is given by vol Dxn =
vol B n 1 − b2 Aτ γ zτn+1 zγn+1
where
A = Aτ γ =
n+1
, √ n+1 2 detA
zτi zγi
,
(5)
i=1
−1 . It follows from (4) that the volume form dVF is given by and (Aτ γ ) = Aτ γ the following formula ([Sh2]) n+1 √ detA dx 1 · · · dx n . (6) dVF = 1 − b2 Aτ γ zτn+1 zγn+1 2 The mean curvature Hϕ is given by (see [Sh1])
1 ∂ 2 F ∂ϕ j ∂F ∂ 2F ∂ 2ϕj Hϕ (v) = + − i vi . F ∂zεi ∂zηj ∂x ε ∂x η ∂ x˜ j ∂zεi ∂x ε ∂ x˜
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Observe that whenever (V , F ) is a Minkowsky space, the expression of the mean curvature reduces to
1 ∂ 2F ∂ 2ϕj Hϕ (v) = (7) vi . F ∂zεi ∂zηj ∂x ε ∂x η 3. The differential equation for a minimal surface which is the graph of a function In this section, we recall the differential equation which characterizes the minimal hypersurfaces M n in the Randers space (V n+1 , Fb ). We then restrict ourselves to surfaces immersed in V 3 and we obtain the differential equation which characterizes 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 Randers space, with local coordinates ϕ j (x) . Then ϕ is minimal, if and only if, it satisfies the differential equation
∂ 2B n+1 ∂B ∂C ∂B ∂C (n2 − 1) ∂B ∂B (1 − B) C− C+ j i + i j j 4 ∂zεi ∂zηj 2 ∂zε ∂zη ∂zεi ∂zη ∂zη ∂zε ∂ 2ϕj i ∂ 2C v = 0, ∀v = v i ei ∈ V n+1 , (8) +(1 − B)2 j ∂zεi ∂zη ∂x ε ∂x η where C=
√ detA,
B = b2 Aεη zεn+1 zηn+1 ,
(9)
{ei } is the canonical basis of V n+1 , zεi is given by (2), A is given by (5). In what follows, we will restrict ourselves to studying minimal surfaces in the three-dimensional Randers space. As a consequence of the above theorem one has the following result. 2 3 Theorem 2 ([ST]). j Let ϕ : M −→ (V , Fb ) be an immersion given in local coordinates by ϕ (x) . Then ϕ is minimal, if and only if, it satisfies the differential equation
3C ∂ 2 E 3 2E − C 2 12E 2 − (2E + C 2 )2 ∂C ∂C − − C(C 2 − E) ∂zεi ∂zηj 2 ∂zηj ∂zεi 2 C2 − E
∂C ∂E (2E + C 2 ) ∂ 2 C 2 ∂E ∂E 3C ∂C ∂E + j i + + × j ∂zεi ∂zηj 4(C 2 − E) ∂zεi ∂zηj 2C ∂zη ∂zε ∂zη ∂zεi ∂ 2ϕj ∀ v = v i ei ∈ V 3 , (10) × ε η v i = 0, ∂x ∂x
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where z = (zεi ) and C are defined by (2) and (9) respectively, and E = b2
3 (−1)γ +τ zγk˜ zτk˜ zγ3 zτ3 ,
τ˜ = δτ 2 + 2δτ 1 .
(11)
k=1
We observe that in (11) we have introduced the notation E = C 2 B 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 the graph of a function f , we obtain the differential equation that characterizes such minimal surfaces. We will first consider the special and important case, obtained in [S], when the surface is a graph over the x1 x2 -plane (observe that we have chosen β = b dx3 in the Randers metric) and then we will consider the general case 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 fxi fxj fxi fxj 2 2 2 Tb (Tb − 3b ) δij − + 3b (Tb + b ) fxi xj = 0, (12) W2 W2 i,j =1,2 where W 2 = 1 + fx21 + fx22 ,
Tb = b2 + (1 − b2 )W 2 .
(13)
Proof. In order to obtain equation (12), we need to compute the expressions involved in (10) for the immersion ϕ. One computes the first and second order derivatives of C, det A and E with respect to the variables zηi , 1 ≤ i ≤ 3, η = 1, 2 (see also [ST]). From the expression of ϕ and (5) we have that A=
1 + fx21 fx1 fx2 fx1 fx2 1 + fx22
,
C=
√ det A = W,
where W is given by (13). We now consider the vector field v = (v 1 , v 2 , v 3 ) = (−fx1 , −fx2 , 1), which is linearly independent with ϕx1 and ϕx2 .
(14)
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By using the first and second order derivatives of C, det A, E, and the fact that ∂ 2 ϕ j /∂xε ∂xη = δj 3 fxε xη , a straightforward computation implies that ∂C i v ∂zεi ∂E i v ∂zεi ∂C ∂ 2 ϕ j j ∂zη ∂xε ∂xη ∂E ∂ 2 ϕ j j ∂zη ∂xε ∂xη
= 0, ∀ε;
(15)
= 2b2 (δε1 fx1 + δε2 fx2 ), ∀ε;
(16)
=
fx1 fxε x1 + fx2 fxε x2 , ∀ε; W
(17)
= 2b2 fx1 fxε x1 + fx2 fxε x2 , ∀ ε;
(18)
∂ 2ϕj i v = 2b2 (1 + fx21 )fx2 x2 − 2fx1 fx2 fx1 x2 j ∂x ∂x i ∂zε ∂zη ε η +(1 + fx22 )fx1 x1 ; (19) 2 2 j 1 ∂ detA ∂ ϕ v i = (1 + fx21 )fx2 x2 − 2fx1 fx2 fx1 x2 + (1 + fx22 )fx1 x1 , (20) 2 ∂zηj ∂zεi ∂xε ∂xη ∂ 2E
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 E ∂E ∂E 3 2E − C 2 ∂C ∂E 3C − − + j j 2 i 2 i 2 ∂zη ∂zε 2 C − E ∂zη ∂zε 4(C − E) ∂zεi ∂zηj (2E + C 2 ) ∂ 2 detA ∂ 2ϕj i + v = 0. (21) j 2C ∂zη ∂zεi ∂x ε ∂x η It follows from (5), (9) and (14), that B = b2 (W 2 − 1)/W 2 . Therefore, −
2Tb − W 2 2E − C 2 , = C2 − E Tb
3C 3W , = 2 4(C − E) 4Tb
−
2E + C 2 2Tb − 3W 2 = , 2C 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 + fx21 )fx2 x2 − 2fx1 fx2 fx1 x2 + (1 + fx22 )fx1 x1 +3b2 (Tb + b2 ) fx21 fx1 x1 + 2fx1 fx2 fx1 x2 + fx22 fx2 x2 = 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 satisfied for a minimal surface which is the graph of a function over any plane of V 3 .
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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 fxi fxj ) T˜b (T˜b − 3b2 w2 )(δij − W2 i,j =1
fxj fxi 2 2 ˜ +3b W (Tb + b w )(ki + 2 w)(kj + 2 w) fxi xj = 0. W W 2
2
where W 2 is defined by (13), ki are real numbers such that w = −k1 fx1 − k2 fx2 + k3 ,
3
2 i=1 ki
(23)
= 1, and
T˜b = b2 w2 + (1 − b2 )W 2 .
(24)
Proof. The proof is similar (although lengthier) to the particular case proved in Theorem 3. Assume that the immersion ϕ is a graph of a function over an open subset 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 m31 x + m32 y + m33 z = 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 variables zηi , are those computed in the proof of Theorem 3. From (5) and the expression of ϕ given by (25), and since the matrix (mij ) is orthogonal, we have that A and C are given by (14). We now consider the vector field v = (v 1 , v 2 , v 3 ) v i = −fx1 m1i − fx2 m2i + m3i , which is linearly independent with ϕx1 and ϕx2 . Observe that zηi =
∂ 2ϕi = fxε xη m3i . ∂xε ∂xη
∂ϕ i = mηi + fxη m3i , ∂xη
(26)
Moreover, for all i = 1, 2, 3 and η, γ , ε = 1, 2 3 zηi v i = 0,
3 v i m3i = 1,
3
i=1
i=1
i=1
zηi m3i
= fxη ,
3 ∂ 2ϕi zγi = fxγ fxε xη . ∂xε ∂xη i=1
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By using the first and second order derivatives of C, det A and E, a straightforward computation implies that ∂C i v ∂zεi ∂E i v ∂zεi ∂C ∂ 2 ϕ j j ∂zη ∂xε ∂xη ∂E ∂ 2 ϕ j j ∂zη ∂xε ∂xη
= 0, ∀ε;
(27)
= 2b2 (zε3 Aε˜ ε˜ − zε3˜ Aεε˜ )w, ∀ε;
(28)
fx1 fxε x1 + fx2 fxε x2 , ∀ε; W
(29)
=
= 2b2 (fx1 + k1 w)fxε x1 + (fx2 + k2 w)fxε x2 , ∀ ε; (30)
∂ 2ϕj i v = 2b2 1 + fx21 − k2 (k2 W 2 + fx2 w) fx2 x2 j ∂x ∂x i ∂zε ∂zη ε η − (1 + k32 )fx1 fx2 + k1 k2 W 2 + k1 k3 fx2 +k2 k3 fx1 + k1 k2 fx1 x2 + 1 + fx22 − k1 (k1 W 2 + fx1 w) fx1 x1 ; ∂ 2E
1 ∂ 2 detA ∂ 2 ϕ j i v = (1 + fx21 )fx2 x2 − 2fx1 fx2 fx1 x2 + (1 + fx22 )fx1 x1 , 2 ∂zηj ∂zεi ∂xε ∂xη
(31) (32)
where w = v 3 is given by (24) and we have introduced the notation ki = mi3 , for i = 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 − w2 ]. W2
Therefore, −
2E − C 2 2T˜b − W 2 = , C2 − E T˜b
3C 3W , = 2 4(C − E) 4T˜b
−
2E + C 2 2T˜b − 3W 2 = , 2C 2W (33)
where T˜b is given by (24). Now it follows from (27)–(32), (33) and the fact that k3 = w + fx1 k1 + fx2 k2 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 a minimal surface in R3 , which is the graph of f .
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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 2 2 2 x1 , x2 , (3b − 1)(x1 + x2 ) , (34) 1 − b2 where x12 + x22 = 0 is a minimal surface in (V 3 , Fb ), since it satisfies (12). In particular, (x1 , x2 , x12 + x22 ), is a minimal surface, when b2 = 1/2. However, the cone (x1 , x12 + 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 left hand side of (23) is always positive. For a fixed plane of V 3 of the form v1 x1 + v2 x2 + v3 x3 = 0, where i vi2 = 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 such that m31 = v1 and m32 = v2 . √ It is not difficult to prove that, for 0 ≤ b < 1/ 3, equation (12) is an elliptic equation of mean curvature type, as defined by L. Simon [S1]. In fact, one can show that for such a b, one has Tb > 0 and Tb − 3b2 > 0. Hence, (23) can be written as fxi fxj aij (x, f, ∇f )fxi xj = 0, where aij = δij + (Sb − 1) (35) 2 W i,j =1,2 and Sb =
3b2 (Tb + b2 ) . Tb (Tb − 3b2 )
It is simple to verify that for all ξ ∈ R2 \ {0}, x, p ∈ R2 and z ∈ R, |p|2 Sb (p) |ξ |2 ≤ aij (x, z, p)ξi ξj ≤ 1 + |ξ |2 . 0< 1 + |p|2 i,j =1,2 1 + |p|2 Therefore, (35) is an elliptic equation. Moreover, one can showthat for b fixed, there exists a constant C > 0, such that Sb (p) ≤ C /W 2 (p) and ij ij (p)ξi ξj ≥ |ξ |2 /W 2 (p) where pi pj ij (p) = δij − 2 . (36) W (p) Therefore, for all (x, z, p) ∈ R5 and ξ ∈ R2 (p · ξ )2 (p · ξ )2 2 |ξ |2 − ≤ a (x, z, p)ξ ξ ≤ (1 + C ) |ξ | − , ij i j 1 + |p|2 i,j =1,2 1 + |p|2 i.e., (12) is an elliptic equation of mean curvature type.
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√ Our next result proves the general case, i.e. that for 0 ≤ b < 1/ 3, the differential 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. which is the graph of a Theorem 5. Let ϕ : R2 −→ (V 3 , Fb ) be an immersion √ function f (x1 , x2 ) over a plane. If 0 ≤ b < 1/ 3, then ϕ is minimal, if and only if, f satisfies the elliptic differential equation, of mean curvature type, given by aij (x, f, ∇f )fxi xj = 0, (37) i,j =1,2
where aij = δij −
fxi fxj W2
+ Qb W 2 ki + Qb =
fxi w W2
kj +
fxj W2
w ,
3b2 (T˜b +b2 ) , T˜b (T˜b −3b2 )
(38) (39)
w T˜b are given by (24) and ki , i = 1, 2, 3, are real numbers such that and 2 i ki = 1. Proof. √ From Theorem 4, ϕ is minimal if and only if f satisfies (23). Since 0 ≤ b < 1/ 3, it follows that T˜b > 0 and T˜b − 3b2 w2 > T˜b − w2 = (1 − b2 )(W 2 − w2 ). One can see that W 2 − w2 = (k2 fx1 − k1 fx2 )2 + (k1 + k3 fx1 )2 + (k2 + k3 fx2 )2 . Therefore, T˜b − 3b2 w2 > 0. Dividing equation (23) by −T˜b (T˜b − 3b2 w2 )W 2 , we get (37). With the notation introduced in (36), we observe that for all ξ ∈ R2 2
ij (p)ξi ξj =
i,j =1
|ξ |2 (1 + |p|2 sin2 θ ), W2
(40)
where θ is the angle function between p and ξ . Moreover, 2
aij (x, z, p)ξi ξj =
i,j =1
2 w ij ξi ξj + Qb W 2 (k1 , k2 ) · ξ + 2 p · ξ , (41) W i,j =1 2
where · is the Euclidean inner product. Therefore, since Qb > 0, for all ξ ∈ R2 \ {0}, we have i,j =1,2
aij (x, z, p)ξi ξj ≥
2 i,j =1
ij ξi ξj ≥
|ξ |2 > 0, W2
(42)
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where the second inequality follows from (40). Therefore, (37) is an elliptic equation. In order to prove that it is a differential equation of mean curvature type, we need to show that there exists a constant C such that, for all (x, z, p) ∈ R5 and ξ ∈ R2 , 2
ij (x, z, p)ξi ξj ≤
i,j =1
2
aij (x, z, p)ξi ξj ≤ (1 + C )
i,j =1
2
ij (x, z, p)ξi ξj .
i,j =1
(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 such that 2 2 w Qb W 2 (k1 , k2 ) · ξ + 2 p · ξ ≤ C ij (x, z, p)ξi ξj , W i,j =1
(44)
where w = −k1 p1 − k2 p2 + k3 . From (24) and (39) we have that Qb =
3b2 [(1 − b2 )W 2 + 2b2 w2 ] [(1 − b2 )W 2 + b2 w2 ][(1 − b2 )W 2 − 2b2 w2 ]
(45)
and it follows from (40) that 2 w W 2 (k1 , k2 ) · ξ + 2 p · ξ W 2 2 2 W |(k1 , k2 )| cos γ + w|p| cos θ = ij (x, z, p)ξi ξj , 1 + |p|2 sin2 θ i,j =1 where γ is the angle between (k1 , k2 ) and ξ . Hence, we need to show that there exists a constant C such that Qb
[W 2 |(k1 , k2 )| cos γ + w|p| cos θ ]2 ≤ C. 1 + |p|2 sin2 θ
(46)
Observe that W 2 ≥ 1. When W 2 = 1, i.e., p = 0, the left hand side of (46) is less than or equal to the real number Qb (0)(k12 + k22 ) ≥ 0. Whenever W 2 > 1 and sin θ = 0, then p = 0 and the vectors p and ξ are parallel. Hence, 2 2 2 W |(k1 , k2 )| cos γ + w|p| cos θ = |(k1 , k2 )| cos γ + k3 |p| cos θ . Therefore, the left hand side of (46) is a rational function of |p| whose numerator is of degree less than or equal to 4, and denominator is of degree 4 and hence it is a bounded function when |p| (or equivalently W ) tends to infinity. Whenever
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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 is of degree less than or equal to 6, and denominator is of degree 6 and hence it is a bounded function when |p| (or equivalently W ) tends to infinity. This completes the 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 cij fxi xj = 0, where fxi fxj fxi fxj 2 2|∇f |2 (1 + |∇f |2 )(δij − ) + (4 + 2|∇f |2 ) . 2 3 3 W 3W 2 Therefore, i,j cij (x, z, p)ξi ξj 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 cij =
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 Randers space, when 0 ≤ b < 1/ 3. √ Corollary 7. Assume b is in the interval [0, 1/ 3). If there exists a solution of 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 ), with 0 ≤ 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 of functions f (x1 , x2 ) and h(x1 , x2 ) over the same plane of V 3 . Let u = f − h be a function defined in the intersection of the domains of f and h. Then Lemma 8. The function u satisfies the differential equation aij (x, f, ∇f )uxi xj + ci uxi = 0, L(u) := i,j
where aij is given by (38), 2 ci = − j,=1
(47)
i
1 0
∂βj (∇f + t (∇h − ∇f )) dt hxj x ∂pi
(48)
and for p = (p1 , p2 ) ∈ R2 ,
pj p pj w(p) p w(p) 2 k + , βj (p) = − 2 + Qb (p)W kj + W W2 W2
(49)
where Qb (p) is given by (39), W 2 = 1 + |p|2 and w(p) = −k1 p1 − k2 p2 + k3 . Moreover, L(u) is an elliptic operator.
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Proof. Since the graphs of f and h are minimal surfaces, it follows from Theorem 5 that aj (x, f, ∇f )fxj x = 0 and aj (x, h, ∇h)hxj x = 0, j,
j,
where aj is given by (38). Taking the difference of these two equations, adding and subtracting the expression j, aj (x, f, ∇f )hxj x , we get 0=
aj (x, f, ∇f )uxj x +
j,
βj (∇f ) − βj (∇h) hxj x , j,
where βj is given by (49). Since βj (∇h) − βj (∇f ) =
2 i=1
1 0
∂βj (∇f + t (∇h − ∇f ))(hxi − fxi ) dt, ∂pi
we conclude that 2 ci uxi , βj (∇f ) − βj (∇h) hxj x = j,
where ci is given by (48).
i=1
Since the functions ci given by (48) are locally bounded, the following theorem follows from Lemma 8 and the maximum principle. √ Theorem 9. Let M1 and M2 be minimal surfaces in (V 3 , Fb ), 0 ≤ b < 1/ 3. If M1 is above M2 near p0 and internally tangent at p0 , then M1 and M2 coincide in a neighborhood of p0 . Besides the Bernstein type theorem given in Theorem 6, as a consequence of Theorem 5 and the theory developped by L. Simon [S1], one gets several results for the solutions of the equation of mean curvature type (37); in particular we will list some of these results such as a-priori gradient estimates, a Bers-type theorem concerning the limiting behaviour of the gradient of solutions defined outside a compact set, a global H¨older continuity estimate for solutions which continuously attain Lipschitz boundary values and a theorem concerning the removability of isolated singularities. In what follows, it is assumed that ⊂ R2 , and f is a C 2 ( ) solution of (37), where we have fixed ki such that ki2 = 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 . Let Dρ (x0 ) = {x ∈ R2 ; |x − x0 | < ρ} and Sρ (p0 ) = {p ∈ M; |p − p0 | < ρ}.
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Proposition 10. If Dρ (x0 ) ⊂ , then 1 + |∇f |2 , sup 1 + |∇f |2 ≤ γ inf Sρ/2(p0 )
Sρ/2(p0 )
where γ > 0 depends only on C of (43). Moreover, if f ≥ 0 on Dρ (x0 ), then |∇f (x0 )| ≥ γ1 exp{γ2 f (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 does not 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. Graduate Texts 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. 69, 245–268 (1977) [S2] Simon, L.: A H¨older Estimate for Quasiconformal Maps Between Surfaces in Euclidean Space. Acta Math. 139, 19–51 (1977) [S] Souza, M.: Superf´ıcies m´ınimas em espa¸cos de Finsler com uma m´etrica de Randers. Thesis, Universidade de Bras´ı lia, 2001 [Sp] Spruck, J.: Gauss curvature estimates for surfaces of constant mean curvature. Comm. Pure Appl. Math. 27, 547–557 (1974) [ST] Souza, M., Tenenblat, K.: Minimal Surfaces of Rotation in a Finsler Space with a Randers Metric. Math. Ann. 325, 625–642 (2003)
