On minimal surfaces immersed in three dimensional Kropina Minkowski space
aa r X i v : . [ m a t h . DG ] F e b On minimal surfaces immersed in threedimensional Kropina Minkowski space
Ranadip Gangopadhyay ∗ , Ashok Kumar † and Bankteshwar Tiwari ‡ DST-CIMS, Banaras Hindu University, Varanasi-221005, IndiaFebruary 12, 2021
Abstract
In this paper we consider a three dimensional Kropina space and obtainthe partial differential equation that characterizes a minimal surfaces withthe induced metric. Using this characterization equation we study variousimmersions of minimal surfaces. In particular, we obtain the partial differ-ential equation that characterizes the minimal translation surfaces and showthat the plane is the only such surface.
The study of minimal surfaces in Riemannian manifolds has been extensively de-veloped [7, 12, 15]. Various well developed techniques have played key roles in thedifferential geometry and partial differential equations. For instance, the estimatesfor nonlinear equations based on the maximum principle arising in Bernstein’s clas-sical work, and even in the Lebesgue’s definition of the integral, that he developedin his thesis, is based on the Plateau problem for minimal surfaces [16]. However,minimal surfaces in Finsler spaces have not been studied and developed at the samepace. The fundamental contribution to the minimal surfaces of Finsler geometrywas made by Shen [17]. Shen introduced the notion of mean curvature for im-mersions into Finsler manifolds and established some of its properties. As in theRiemannian case, if the mean curvature is identically zero, then the immersion issaid to be minimal.The Randers metric is the simplest class of non-Riemannian Finsler metric,defined as F = α + β , where α is a Riemannian metric and β is a one-form. M.Souza and K. Tenenblat studied the surfaces of revolution in Minkowski space R with Randers metric [21] and found the condition for such surfaces to be minimal. ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] W satisfying h ( W, W ) < , in a Riemannian space ( M, h ) . In fact thesolution of a Zermello Navigation problem in a Riemannian manifold ( M, h ) witha time independent vector field W satisfying h ( W, W ) < , gives a unique Randersmetric on the manifold M and conversely. However, for vector field W , satisfying h ( W, W ) = 1 , recently R. Yoshikawa and S.V. Sabau prove that the time minimiz-ing path would be the geodesics of a Kropina metric which is given by F = α β ,where α is a Riemannian metric and β is a one-form [23]. Kropina metric was intro-duced by V.K. Kropina [13] and has numerous applications in physics and biology[1]. V. Balan studied the constant mean curvature surfaces in Minkowski spacewith Kropina metric and obtain some characteristic differential equations for thesesurfaces [2].In this paper we study the minimal surfaces immersed in three-dimensionalKropina spaces. We first obtain the characterization partial differential equationfor a surface to be minimal in the Kropina space. Then we consider three differ-ent immersions, the surfaces of revolution, the graph of a smooth functions, thetranslation surfaces in the Kropina space. In Section , we study the minimal sur-faces of revolution and find the generating curves which gives the minimal surfacesexplicitly. That is, we obtain the following theorem: Theorem 1.1
Let ( R , F = α /β ) be a Kropina space, where α is the Euclideanmetric, and β = bdx is a -form with norm b , satisfying b > . Let ϕ : M → R be an immersion given by ϕ ( x , x ) = ( f ( x ) cos x , f ( x ) sin x , x ) , f is a smoothfunction and f ( x ) > . Then ϕ is minimal, if and only if, either it is an open conewith generating curve f ( x ) = 1 √ x + c , or a surface with the generating curve x = ( − f / − a ) p a − f / + d , here a, c, d being real constants. In Section , we study the minimal surfaces of graph of a smooth function andobtain a Bernstein-type theorem as follows: Theorem 1.2
If a minimal surface in the Kropina space ( R , F ) is the graph of asmooth function f : U ⊂ R → R then it is a plane provided f x + f x ≥ . In Section , we study the minimal translation surfaces and prove that plane is theonly such surface. Theorem 1.3
If a minimal translation surface in the Kropina space ( R , F ) isgiven by the immersion ϕ : U ⊂ R → ( R , F ) , ϕ ( x , x ) = ( x , x , f ( x ) + g ( x )) then it is a plane. Preliminaries
Let M be an n -dimensional smooth manifold. T x M denotes the tangent space of M at x . The tangent bundle of M is the disjoint union of tangent spaces T M := ⊔ x ∈ M T x M . We denote the elements of T M by ( x, y ) where y ∈ T x M and T M := T M \ { } . Definition 2.1 [6] A Finsler metric on M is a function F : T M → [0 , ∞ ) satis-fying the following condition:(i) F is smooth on T M ,(ii) F is a positively 1-homogeneous on the fibers of tangent bundle T M ,(iii) The Hessian of F with element g ij = ∂ F ∂y i ∂y j is positive definite on T M .The pair ( M, F ) is called a Finsler space and g ij is called the fundamental tensor. In general, the explicit calculations of geometric objects in Finsler geometry isvery tedious and complicated. This is perhaps one of the reason why the studyof special Finsler spaces has attracted its attention and specailly the geometry ofFinsler metrics, namely, ( α, β ) -metric, introduced by M. Matsumoto [14] has takenmuch attention in recent years. The ( α, β ) -metric is defined as, F = αφ ( βα ) where α is a Riemannian metric, β is a one form and φ is a smooth function. This classof Finsler metrics contains many interesting subclass of Finsler metrics such asRanders metrics, Matsumoto mountain metric, Kropina metric etc.For an n -dimensional Finsler manifold ( M n , F ) , the Busemann-Hausdorff vol-ume form is defined as dV BH = σ BH ( x ) dx , where σ BH ( x ) = vol ( B n (1)) vol { ( y i ) ∈ T x M : F ( x, y ) < } , (1) B n (1) is the Euclidean unit ball in R n and vol is the Euclidean volume. Proposition 2.1 [5] Let F = αφ ( s ) , s = β/α , be an ( α, β ) -metric on an n -dimensional manifold M . Then the Busemann-Hausdorff volume form dV F of the ( α, β ) -metric F is given by dV F = π R sin n − ( t ) dt π R n − ( t ) φ ( b cos( t )) n dt dV α where, dV α = p det ( a ij ) dx denotes the volume form of Riemannian metric α . A Kropina metric on M is a Finsler structure F on T M is given by F = α β , where α = p a ij y i y j is a Riemannian metric and β = b i y i is a one-form with b > .Let ( ˜ M m , ˜ F ) be a Finsler manifold, with local coordinates (˜ x , . . . , ˜ x m ) and ϕ : M n → ( ˜ M m , ˜ F ) be an immersion. Then ˜ F induces a Finsler metric on M ,defined by F ( x, y ) = (cid:16) ϕ ∗ ˜ F (cid:17) ( x, y ) = ˜ F ( ϕ ( x ) , ϕ ∗ ( y )) , ∀ ( x, y ) ∈ T M. (2)3he following convention is in order: the greek letters ǫ, η, γ, τ, . . . are the indicesranging from to n and the latin letters i, j, k, l, . . . are the indices ranging from to n + 1 .A Minkowski space is the vector space R n equipped with a Minkowski norm F whose indicatrix is strongly convex. Equivalently, we can say that F ( x, y ) de-pends only on y ∈ T x ( R n ) . In this paper we consider the hypersurface M n in theMinkowski Kropina space R n +1 given by the immersion ϕ : M n → ( R n +1 , F ) , where F = α β , α is the Euclidean metric, and β is a one-form with Euclidean norm b > .Without loss of generality we consider β = bdx n +1 . If M n has local coordinates x = ( x ǫ ) , ǫ = 1 , ..., n , and ϕ ( x ) = ( ϕ i ( x ǫ )) ∈ R n +1 , i = 1 , . . . , n + 1 , we define F ( x, z ) = vol ( B n ) vol ( D nx ) , (3)where, D nx = (cid:8) ( y , y , ..., y n ) ∈ R n : F ( x, y ) < (cid:9) , y = y ǫ z iǫ and z = (cid:0) z iǫ (cid:1) = (cid:18) ∂ϕ i ∂x ǫ (cid:19) (4)The mean curvature H ϕ , for the immersion ϕ , along the vector v introduced by Z.Shen [17] and is given by H ϕ ( v ) = 1 F (cid:26) ∂ F ∂z iǫ ∂z jη ∂ ϕ j ∂x ǫ ∂x η + ∂ F ∂z iǫ ∂ ˜ x j ∂ϕ j ∂x ǫ − ∂ F ∂ ˜ x i (cid:27) v i . Here v = ( v i ) is a vector field over R n +1 . H ϕ ( v ) depends linearly on v and the meancurvature vanishes on ϕ ∗ ( T M ) . Since, ( R n +1 , F ) is a Minkowski space, F = F ( y ) .Hence, the expression of the mean curvature reduces to H ϕ ( v ) = 1 F (cid:26) ∂ F ∂z iǫ ∂z jη ∂ ϕ∂x ǫ ∂x η (cid:27) v i . (5)The immersion ϕ is said to be minimal when H ϕ = 0 .In this paper we consider an immersed surface in three dimensional Minkowskispace. Using the definition of pullback metric given in (2), we show that if ˜ F is aKropina metric, then the induced pullback metric on the surface is again a Kropinametric. Proposition 2.2
Let ϕ : M → ( R , ˜ F = α β ) , where α is the Euclidean metricand β = bdx , ( b > be an immersion in a Kropina space with local coordinates ( ϕ i ( x a )) . Then the pull back metric defined in (2) is a Kropina metric. Proof:
Let ϕ ( x , x ) = ( ϕ ( x , x ) , ϕ ( x , x ) , ϕ ( x , x )) be an immersion. Then,for any tangent vector v ∈ T M , ( φ ∗ ( ˜ F ))( v ) = ˜ F ( φ ∗ v ) = δ ij ∂φi∂xǫ ∂φj∂xδ v ǫ v δ b ∂φj∂xη v η = A ǫδ v ǫ v δ bz η v η ,where A = ( A τγ ) = X i =1 z iτ z iγ ! (6)4ence, F = φ ∗ ( ˜ F ) is again a Kropina metric where, α = A ǫδ v ǫ v δ and β = bz η v η . (cid:3) Theorem 2.1
Let ϕ : M → ( R , F ) be an immersion in a Kropina space withlocal coordinates ( ϕ i ( x a )) . Then ϕ is minimal if and only if ∂ ϕ j ∂x ǫ ∂x η v i {− C E ∂ E∂z iǫ ∂z jη + 4 C ∂E∂z iǫ ∂E∂z jη − CE (cid:18) ∂C∂z iǫ ∂E∂z jη + ∂E∂z iǫ ∂C∂z jη (cid:19) − CE ∂C∂z iǫ ∂C∂z jη + 3 E ∂ C ∂z iǫ ∂z jη } = 0 , (7) where E = b X k =1 ( − γ + τ z k ˜ γ z k ˜ τ z γ z τ , ˜ τ = δ τ + 2 δ τ , C = √ detA (8) Proof:
For Kropina surface we have φ ( s ) = s and n = 2 . Therefore, we have dV BH = π R dt π R ( b ′ cos t ) dt p det ( A ) dx = 4 b ′ p det ( A ) dx (9)Here, b ′ = b A ǫδ z ǫ z δ is the norm of β with respect to the pullback Kropina metric F . Hence, the Euclidean volume of D nx is given by vol ( D nx ) := 4 vol ( B n ) √ detAb A ǫη z ǫ z η , (10) A = ( A τγ ) = X i =1 z iτ z iγ ! (11)Therefore, using (3) and (8) in (10) we have F ( x, z ) = 4 C E (12)It should be noted that ∂ C ∂z iǫ ∂z jη = ∂∂z jη (cid:18) C ∂C∂z iǫ (cid:19) = 2 ∂C∂z iǫ ∂C∂z jη + 2 C ∂ C∂z iǫ ∂z jη (13)Now differentiating (12) twice first with respect to z iǫ and then with respect to z jη and using (13) we get ∂ F ∂z iǫ ∂z jη = 8 C E ∂E∂z iǫ ∂E∂z jη − C E (cid:18) ∂C∂z jη ∂E∂z iǫ + ∂E∂z jη ∂C∂z iǫ (cid:19) − C E ∂ E∂z iǫ ∂z jη + 12 CE ∂C∂z jη ∂C∂z iǫ + 6 CE ∂ C ∂z iǫ ∂z jη (14)Now using (14) in (5) we obtain the proof of the theorem. (cid:3) Minimal surfaces of revolution
In this section we consider the surface of revolution given by the immersion ϕ : M → R such that ϕ ( x , x ) := ( ϕ i ( x ǫ )) = ( f ( x ) cos x , f ( x ) sin x , x ) , with f ( x ) > . The curve ( x , f ( x )) is the generating curve for the surface of revolution.Here we note that, the mean curvature vanishes on tangent vectors of the immersion ϕ . Therefore, we only need to consider a vector field v such that the set { ϕ x , ϕ x , v } is linearly independent. Hence, we consider v = ( − cos x , − sin x , f ′ ( x )) . There-fore, the coordinates of the vector v can be written as v i = − δ i cos x − δ i sin x + δ i f ′ ( x ) . (15)We also have z iǫ := ∂ϕ i ∂x ǫ = δ ǫ [ δ i f ′ ( x ) cos x + δ i f ′ ( x ) sin x + δ i ]+ δ ǫ [ − δ i f ( x ) sin x + δ i f ( x ) cos x ] (16)Therefore, in view of (11) we obtain the followings: A = (cid:18) f ′ ( x ))
00 ( f ( x ) (cid:19) (17) C = f ( x )(1 + f ′ ( x )) , E = b f ( x ) . (18)From the above values of A , C and E we further obtain ∂C∂z iǫ v i = 0 (19) ∂E∂z iǫ v i = 2 b δ ǫ f ( x ) f ′ ( x ) (20) ∂C∂z jη ∂ ϕ j ∂x ǫ ∂x η = f ′ ( x ) p f ′ ( x ) δ ǫ (cid:2) f ( x ) f ′′ ( x ) + 1 + f ′ ( x ) (cid:3) (21) ∂E∂z jη ∂ ϕ j ∂x ǫ ∂x η = 2 b δ ǫ f ( x ) f ′ ( x ) (22) ∂ E∂z iǫ ∂z jη ∂ ϕ j ∂x ǫ ∂x η v i = 2 b f ( x ) (cid:2) f ′ ( x ) (cid:3) (23) ∂ C ∂z iǫ ∂z jη ∂ ϕ j ∂x ǫ ∂x η v i = f ( x ) (cid:2) − f ( x ) f ′′ ( x ) + 1 + f ′ ( x ) (cid:3) (24)Then putting all these values together in (7) and after some simplifications we have (cid:0) − f ′ (cid:1) (cid:0) f f ′′ + f ′ (cid:1) = 0 (25)Therefore, either (1 − f ′ ) = 0 , or, f f ′′ + f ′ = 0 (1 − f ′ ) = 0 then f ′ ( x ) = √ . Therefore, the minimal surface is an opencone with the generating line f ( x ) = √ x + c . Here c is a positive real number.Now suppose f f ′′ + f ′ = 0 . Let us consider p = f ′ . Then f ′′ = p dpdf .Substituting these values in the above ODE it reduces to f p dpdf + p = 0 (26) i.e. p p dp + 1 f df = 0 (27)Integrating we have,
32 ln(1 + p ) + ln f = 32 ln a, where a is an integrating constant . Simplifying we have, p = p af − / − (28) i.e. dfdx = p af − / − (29)Integrating, we have x = ( − f / − a ) p a − f / + d, where d is an integrating constant . (30)This is the precise generating curve for which the surface becomes a Finsler minimalsurface and therefore we obtain the proof of the Theorem 1.1. In this section we study the graph of a smooth function f : U ⊂ R → R in theKropina space ( R , F ) , where ˜ α is the Euclidean metric and ˜ β = bdx is a one-formwith b > . Here we consider the immersion ϕ : U ⊂ R → ( R , F ) given by ϕ ( x , x ) = ( x , x , f ( x , x )) . Before proving Theorem 1.2. we need the followingproposition: Proposition 4.1
An immersion ϕ : U ⊂ R → ( R , F ) given by ϕ ( x , x ) =( x , x , f ( x , x )) , where f is a real valued smooth function on U ⊂ R is minimal,if and only if, f satisfies X ǫ,η =1 , (cid:20) ( W − W − (cid:18) δ ǫη − f x ǫ f x η W (cid:19) + 2( W + 3) f x ǫ f x η W (cid:21) f x ǫ x η = 0 , (31) where, W = 1 + f x + f x . roof: As we already know that the mean curvature vanishes along the tangentvectors of the immersed surface, we need to consider a vector field v such that theset { v, ϕ x , ϕ x } is linearly independent. Therefore, we consider v = ϕ x × ϕ x .Then v = ( v , v , v ) = ( − f x , − f x , . From (11) we obtain the following: A = (cid:18) f x f x f x f x f x f x (cid:19) , C = √ detA = W, E = b (cid:0) W − (cid:1) , (32)By some simple calculations we can have ∂C∂z iǫ v i = 0 , ∂E∂z iǫ v i = 2 b ( δ ǫ f x + δ ǫ f x ) , (33) ∂C∂z jη ∂ ϕ j ∂x ǫ ∂x η v i = f x f x ǫ x + f x f x ǫ x W , (34) ∂E∂z jη ∂ ϕ j ∂x ǫ ∂x η = 2 b ( f x f x ǫ x + f x f x ǫ x ) , (35) ∂ E∂z iǫ ∂z jη ∂ ϕ j ∂x ǫ ∂x η v i = 2 b (cid:2) (1 + f x ) f x x − f x f x f x x + (1 + f x ) f x x (cid:3) , (36) ∂ C ∂z iǫ ∂z jη ∂ ϕ j ∂x ǫ ∂x η v i = (cid:2) (1 + f x ) f x x − f x f x f x x + (1 + f x ) f x x (cid:3) . (37)Then putting all these values in (7) we get (31). Hence, we complete the proof. (cid:3) Remark 4.1 If W is constant then we have (cid:0) ∂f∂x (cid:1) + (cid:0) ∂f∂x (cid:1) = k , where k issome real number. Then solving this partial differential equation we get f ( x , x ) = ax + √ k − a x + c , where a and c are arbitrary constants. Therefore, the surfacebecomes a plane and clearly it satisfies (31) . Hence, it is minimal. Definition 4.1 [19] A differential equation is said to be a elliptic equation of meancurvature type on a domain Ω ⊂ R if X ǫ,η =1 , a ǫη ( x, f, ∇ f ) f x ǫ x η = 0 (38) where a ǫη , ǫ, η = 1 , are given real-valued functions on Ω × R × R , x ∈ Ω , f : Ω → R with | ξ | − ( p · ξ ) | p | X ǫ,η =1 , a ǫη ( x, u, p ) ξ ǫ ξ η ≤ (1 + C ) (cid:20) | ξ | − ( p · ξ ) | p | (cid:21) (39) for all u ∈ R , p ∈ R and ξ ∈ R \ { } . Therefore, we can write equation (4.1) as X ǫ,η =1 , a ǫη ( x, f, ∇ f ) f x ǫ x η = 0 (40)8here, a ǫη := ( δ ǫη − f xǫ f xη W + 2 W +3( W − W − (cid:0) f xǫ f xη W (cid:1) if W > W + 3) (cid:0) f xǫ f xη W (cid:1) if W = 3 (41) Case-1:
Now let us assume W = 3 . Then clearly, (31) satisfies (39). Case-2:
Now assume W > . Let us consider ξ ∈ R \ { } , x, t ∈ R and u ∈ R and we define h ǫη ( u ) = δ ǫη − t ǫ t η W ( u ) . (42)Hence, we have, X ǫ,η =1 h ǫη ( t ) ξ ǫ ξ η = | ξ | W (1 + | t | sin θ ) , (43)where, θ is the angle function between t and ξ . We also have from X ǫ,η =1 a ǫη ( x, u, t ) ξ ǫ ξ η = X ǫη =1 h ǫη ( t ) ξ ǫ ξ η + 2 W + 3( W − W − (cid:20) t · ξW (cid:21) , (44)where · represents the Euclidean inner product.Hence, it is evident from (43) and (44) that for all ξ ∈ R \ { } , X ǫ,η =1 a ǫη ( x, u, t ) ξ ǫ ξ η > . (45)Hence, (40) is an elliptic equation.Now we prove that it is a differential equation of mean curvature type for which weneed to show that there exists a constant C such that, for all X ǫ,η =1 h ǫη ( x, u, t ) ξ ǫ ξ η ≤ X ǫ,η =1 a ǫη ( x, u, t ) ξ ǫ ξ η ≤ (1 + C ) X ǫ,η =1 h ǫη ( x, u, t ) ξ ǫ ξ η . (46)The first inequality is immediate from (45). To prove the second inequality we needto show that W + 3( W − W − (cid:20) t · ξW (cid:21) ≤ C X ǫ,η =1 h ǫη ( x, u, t ) ξ ǫ ξ η , (47)That is, we need to show that W + 3) | t | ( W − W − | t | sin θ ) ≤ C (48)The left hand side of (48) is a rational function of | t | . Here the numerator is ofdegree less than or equal to , and denominator is of degree . Therefore, it is abounded function of | t | as | t | goes to infinity.9ow if W > m , for some real number m > , then again the left hand side of (48)is bounded by the similar reason given above.Therefore, from the above two cases and by the continuity of W it can be saidthat (31) satisfies (39) whenever W ≥ .Now the theorem proved by L. Simon (Theorem 4.1 of [20]) and from the abovediscussion, we conclude Theorem 1.2. In this section we study the minimal translation surface M in Kropina space ( R , F ) , where R is a real Minkowski space and F = ˜ α ˜ β is a Kropina metric, where ˜ α is the Euclidean metric and ˜ β = bdx is a one-form with b > . Here we considerthe immersion ϕ : U ⊂ R → ( R , F ) given by ϕ ( x , x ) = ( x , x , f ( x ) + g ( x )) .Let us consider the following immersion: ϕ ( x , x ) = ( ϕ , ϕ , ϕ ) = (cid:0) x , x , f ( x ) + g ( x ) (cid:1) (49)Then we can write ϕ j = δ j x + δ j x + ( f + g ) δ j , ≤ ǫ ≤ . (50)Therefore, from (11) and (49) we get A = (cid:18) f x f x g x f x g x g x (cid:19) , C = √ detA = q f x + g x and E = b ( f x + g x ) (51)Here we choose v = ϕ x × ϕ x . Then v = ( v , v , v ) = ( − f x , − g x , . Hence, v i = − δ i f x − δ i g x + δ i , ≤ i ≤ .By some simple calculations we can have ∂C∂z iǫ v i = 0 , ∂E∂z iǫ v i = 2 b ( δ ǫ f x + δ ǫ g x ) , (52) ∂C∂z jη ∂ ϕ j ∂x ǫ ∂x η = δ ǫ f x f x x + δ ǫ g x g x x C , (53) ∂E∂z jη ∂ ϕ j ∂x ǫ ∂x η = 2 b ( δ ǫ f x f x x + δ ǫ g x g x x ) , (54) ∂ E∂z iǫ ∂z jη ∂ ϕ j ∂x ǫ ∂x η v i = 2 b (cid:2) (1 + g x ) f x x + (1 + f x ) g x x (cid:3) , (55) ∂ C ∂z iǫ ∂z jη ∂ ϕ j ∂x ǫ ∂x η v i = 2 (cid:2) (1 + g x ) f x x + (1 + f x ) g x x (cid:3) , (56)10hen substituting all these above values in (7) we obtain λf x x + µg x x = 0 (57)where, λ = 8 f x + 2 f x ( f x + g x ) + (cid:0) g x (cid:1) ( f x + g x )( f x + g x − (58)and µ = 8 g x + 2 g x ( f x + g x ) + (cid:0) f x (cid:1) ( f x + g x )( f x + g x − (59)Let r = ( f x ) and s = ( g x ) . Then f x x = r f , g x x = s g . (60)Then (58) and (59) becomes λ = 8 r + 2 r ( r + s ) + (1 + s )( r + s )( r + s − (61)and µ = 8 s + 2 s ( r + s ) + (1 + r )( r + s )( r + s − (62)And (57) becomes r f λ + s g µ = 0 (63)Therefore, we have two cases: Case 1: If r f = 0 or, s g = 0 , then r and s are constant functions. And hence f and g are linear functions. Therefore, M is a plane in ( V , F ) locally. Case 2:
Let r f = 0 and s g = 0 . Then we have, λ = 0 and µ = 0 . Suppose κ = r f µ = − s g λ . (64)Which implies that ( r f ) g = µ g κ + µκ g = 0 and ( s g ) f = λ f κ + λκ f = 0 Hence, we have, (log κ ) g = κ g κ = − µ g µ and (log κ ) f = κ f κ = − λ f λ . (65)Since, (log κ ) fg = (log κ ) gf , we have, (cid:18) λ f λ (cid:19) g = (cid:18) µ g µ (cid:19) f . (66)We can easily observe that, r g = ( r f ) g = 0 and s f = ( s g ) f = 0 . Therefore, we have, (cid:18) λ f λ (cid:19) g = (cid:18) λ r r f λ (cid:19) g = (cid:18) λ r λ (cid:19) g r f = (cid:18) λ r λ (cid:19) s r f s g (67)11nd (cid:18) µ g µ (cid:19) f = (cid:18) µ s s g µ (cid:19) f = (cid:18) µ s µ (cid:19) f s g = (cid:18) µ s µ (cid:19) r r f s g (68)Using (67) and (68) in (66) we get, (cid:18) λ r λ (cid:19) s = (cid:18) µ s µ (cid:19) r . (69)That is, (cid:18) log λµ (cid:19) rs = 0 (70)Let p = r + s and q = r − s . Then we have λ = K ( p ) − L ( p ) q, µ = K ( p ) + L ( p ) q (71)where, K ( p ) = 4 p + p + p ( p −
2) + 12 p ( p − (72) L ( p ) = 12 p ( p − − p − (73)Now from (70) it follows that (cid:18) log λµ (cid:19) rs = (cid:18) log λµ (cid:19) pp − (cid:18) log λµ (cid:19) qq = 0 (74)Now substitute the values of λ and µ in (74) we get q (cid:0) K pp L − KL L pp − K p L p L + 2 KLL p (cid:1) + q (cid:0) − K pp K L + K L pp − K p K L p + 2 K p KL − KL (cid:1) = 0 . (75)Since, q is an arbitrary function we get, K pp L − KL L pp − K p L p L + 2 KLL p = 0 (76) − K pp K L + K L pp − K p K L p + 2 K p KL − KL = 0 (77)Multiplying (76) by NM , and (77) by MN and then adding them together, we obtain "(cid:18) KL (cid:19) p = 1 . (78)Again from (72) and (73) we have KL = p + 4 p + 12 pp − p − (79)12ow differentiating (79) with respect to p we get (cid:18) KL (cid:19) p = 1 − p + 34 p + 12( p − p − (80)Squaring both sides of (80) gives "(cid:18) KL (cid:19) p = 1 − p + 34 p + 12( p − p − + 16 (cid:18) p + 34 p + 12( p − p − (cid:19) (81)Hence, the right hand side of (81) is a rational function of p . Therefore, it can notbe equal to identically. This argument contradicts (78). Hence it leads to the factthat case 1 is true and hence we obtain Theorem 1.3. References [1] Antonelli, P.L., Ingarden, R.S., Matsumoto, M.: The Theory of Sprays andFinsler Spaces with Applications in Physics and Biology. Kluwer, Dordrecht(1993)[2] Balan, V.: Constant mean curvature submanifolds in ( α, β ) -Finsler spaces. Dif-ferential Geom. Appl. 45–55 (2007)[3] Bao, D., Robbles, C., Shen Z.: Zermelo navigation on Riemannian manifolds.J. Differential Geom. (3), 377–435 (2004).[4] Caratheodory, C.: Calculus of Variations and Partial Differential Equations ofthe First Order, (Translated by Robert B. Dean). AMS Chelsea Publishing,Berlin (2006). (Originally published 1935, Berlin)[5] Cheng, X., Shen, Z., A class of Finsler metrics with isotropic S-curvature. Israel.J. Math. , 317–340 (2009)[6] Chern, S.S., Shen, Z.: Riemannian-Finsler geometry. World Scientific Publisher,Singapore (2005).[7] Colding, T., Minicozzi, W.P.: A Course in Minimal Surfaces. American Math-ematical Society (2011).[8] Cui, N.: On minimal surfaces in a class of Finsler 3-spheres. Geom. Dedicata. , 87–100 (2014)[9] Cui, N.: Nontrivial minimal surfaces in a class of Finsler 3-spheres. Int. J. Math. (4), 1450034(17 pages) (2014).[10] Cui, N., Shen, Y.B.: Nontrivial minimal surfaces in a hyperbolic Randersspace. Math. Nachr. (4), 570–582 (2017).1311] Gangopadhyay, R., Tiwari, B.: Minimal surfaces in three-dimensional Mat-sumoto space. arXiv:2007.09439 [math.DG]. (2020).[12] Itoh, T., Minimal surfaces in a Riemannian manifold of constant curvature.Kodai Math. Sem. Rep. (2), 202-214 (1973)[13] Kropina, V.K.: On projective two-dimensional finsler spaces with special met-ric. Trudy Sem. Vector. Tensor. Anal. , 277 (1961).[14] Matsumoto, M., Theory of Finsler spaces with ( α, β ) -metric. Rep. Math. Phys. (1), 43-83 (1992).[15] Osserman, R.: A Survey of minimal surfaces. Dover Publication (2002).[16] Rado, T.: On Plateau’s proble. Ann. of Math. , 457-469 (1930).[17] Shen, Z.: On Finsler geometry of submanifolds. Math. Ann., , 549-576(1998).[18] da Silva, R.M., Tenenblat, K.: Helicoidal Minimal Surfaces in a Finsler Spaceof Randers Type, Canad. Math. Bull. (4), 765–779 (2014)[19] Simon, L.: Equations of mean curvature type in 2 independent variables. Pac.J. Math. , 245–268 (1977)[20] Simon, L.: A Holder Estimate for Quasiconformal Maps Between Surfaces inEuclidean Space. Acta Math. , 19–51 (1977).[21] Souza, M., Tenenblat, K.: Minimal surfaces of rotation in Finsler space witha Randers metric. Math. Ann. , 625–642 (2003).[22] Souza, M., Spruck, J., Tenenblat, K.: A Berstein type theorem on a Randersspace. Math. Ann. , 291–305 (2004).[23] Yoshikawa, R., Sabau, S.V.: Kropina metrics and Zermelo navigation on Rie-mannian manifolds. Geom. Dedicata.171