Advances in Pure Mathematics, 2012, 2, 220225 http://dx.doi.org/10.4236/apm.2012.24032 Published Online July 2012 (http://www.SciRP.org/journal/apm) Minimal Surfaces and Gauss Curvature of Conoid in Finsler Spaces with (α, β)Metrics* Dinghe Xie, Qun He Department of Mathematics, Tongji University, Shanghai, China Email: x_dinghe8707@126.com Received December 26, 2011; received March 18, 2012; accepted March 26, 2012 ABSTRACT In this paper, minimal submanifolds in Finsler spaces with (α, β)metrics are studied. Especially, helicoids are also minimal in (α, β)Minkowski spaces. Then the minimal surfaces of conoid in Finsler spaces with (α, β)metrics are given. Last, the Gauss curvature of the conoid in the 3dimension RandersMinkowski space is studied. Keywords: Isometrical Immersion; Minimal Submanifold; (α, β)Metric; Conoid Surface; Gauss Curvature 1. Introduction In recent decades, geometry of submanifolds in Finsler geometry has been rapidly developed. By using the BusemannHausdorff volume form, Z. Shen [1] intro duced the notions of mean curvature and normal curva ture for Finsler submanifolds. Being based on it, Bern stein type theorem of minimal rotated surfaces in Rand ersMinkowski space was considered in [2]. Later, Q. He and Y. B. Shen used another important volume form, i.e., HolmesThompson volume form, to introduce notions of another mean curvature and the second fundamental form [3]. Thus, Q. He and Y. B. Shen constructed the corre sponding Bernstein type theorem in a general Minkowski space [4]. The theory of minimal surfaces in Euclidean space has developed into a rich branch of differential geometry. A lot of minimal surfaces have been found in Euclidean space. Minkowski space is an analogue of Euclidean space in Finsler geometry. A natural problem is to study minimal surfaces with BusemannHausdorff or Holmes Thompson volume forms. M. Souza and K. Tenenblat first studied the minimal surfaces of rotation in Randers Minkowski spaces, and used an ODE to characterize the BHminimal rotated surfaces in [5]. Later, the nontrivial HTminimal rotated hypersurfaces in quadratic (α, β) Minkowski space are studied [6]. N. Cui and Y. B. Shen used another method to give minimal rotational hyper surface in quadratic Minkowski (α, β)space [7]. How ever, these examples only consider the special (α, β) metrics either Randers or quadratic. Therefore, what is the case with the general (α, β)metric? The main purpose of this paper is to study the conoid in (α, β)space. It includes minimal submanifolds in Finsler spaces with general (α, β)metric (F ) and the Causs curvature in RandersMinkowski 3space. We present the equations that characterize the minimal hypersurfaces in general (α, β)Minkowski space. We prove that the conoid in Minkowski 3space with metric F :0,FTM is minimal if and only if it is a helicoid or a plane under some conditions. Finally, similar to [7], we give the Gauss curvature of conoid in RandersMin kowski 3space and point out that the Gauss curvature is not always nonpositive on minimal surfaces. 2. Preliminaries Let M be an ndimensional smooth manifold. A Finsler metric on M is a function satisfying the following properties: 1) F is smooth on \0TM ; 2) ,, xy Fxy 0 for all ; 3) The induced quadratic form g is positively definite, where 2 :,dd, 1 :. 2ij ij ij ij yy gxyx x gF (1) Here and from now on, i , ij y F mean , i y 2 ij yy 1,,;1 ,,.ijnm n , and we shall use the following convention of index ranges unless otherwise stated: *Project supported by NNSFC (no. 10971239, no. 10771160) and the atural Science Foundation of Shanghai (no. 09ZR1433000). C opyright © 2012 SciRes. APM
D. H. XIE, Q. HE 221 The projection gives rise to the pull back bundle and its dual , which sits over . We shall work on and rigidly use only objects that are invariant under positive re scaling in y, so that one may view them as objects on the projective sphere bundle SM using homogeneous coor dinates. π:TM M ** πTM \0TM ** πTM * πTM \0TM In there is a global section d i i y x , called the Hilbert form, whose dual is , i i ll ii lyF dV ˆ , called the distinguished field.The volume element SM of SM with respect to the Riemannian metric , the pullback of the Sasaki metric on , can be expressed as \0TM ddd,Vx SM (2) where 1 dd, n :det, d ij g xx d d i n y y F (3) 11 1 d:1 d nii i yy . (4) The volume form of a Finsler nmanifold (M, F) is defined by 1 1d, x SM n c d: d,: M Vxxx 1n c (5) where denotes the volume of the unit Euclidean (n − 1)sphere , 1n S xx y yTM=SM , . Let (M, F) and F be Finsler manifolds, and : MM , be an immersion. If ,d xyF fxf y for all ,\0TM ,, i j xy , then f is called an isometric immersion. It is clear that , ij xygxy ff , (6) for the isometric immersion :, MF MF , where fx , i i fy , ii f . Let * πTM be the orthogonal complement of in * πTM *1 π TM with respect to , and set 2 , ,, k k fGG h h ij ij hfyy hgh x (7) where 2 ij ij f x k G G, and are the geodesic coefficients of F and * πhT respectively. We can see that (see (1.14) in [3]), which is called the normal curvature. Recall that for an isometric immersion M :, , MF MF , we have (see formulae (2.14) and (3.14) of Chapter V in [8]) , kk ij ij GfyyG . klk l (8) gg where From (2.7), it follows that , ij ij hpfyyG :. i i pf (9) where Set 2 1 1dd, SM x n h cF (10) which is called the mean curvature form of f. An isomet ric immersion :, , MF MF 0 is called a minimal immersion if any compact domain of M is the critical point of its volume functional with respect to any varia tion vector field. Then f is minimal if and only if . 3. Minimal Hypersurfaces of (α, β)Spaces Here and from now on, we consider general (α, β)metric. Let s C is a positive s , , where function on 00 ,bb , ,, ij i ij i axyy bxy 0 0. ij ij abbbb b 1 If s , then F is a Randers metric. If is an Euclidean metric and is parallel with respect to , F is a locally Minkowski metric and (M, F) is called an (α, β)Minkowski metric. By [9], F is a Finsler metric if and only if satisfies 22 0 () 0, .ss bsssbb (11) Let det ,det,. ijij n Aag g , n (12) It have been proved ([9]) that sHsA 222 . n Hsssb s (13) where In the following part, we will discuss minimal hyper surfaces in Minkowski space with (α, β)metric. Let :, ,MFMF be an isometric immersion, Fs , where Copyright © 2012 SciRes. APM
D. H. XIE, Q. HE 222 , ,.yy bya tric immer Since f is an isomesion, we get * Ff ,F where ,, ,. j i iii ayyaa f f bybbf ij ijij i hat (M, F) is a hype of , Note trsurface F , let nne be the unit normeld of al vector fi M with respect to and nne it nmal vector fiel ect to be the unor d of M with resp ely. That is , respectiv 0, 0, ii nfg nf ,1,1.nnangnngn n ,n There exist a function , y on SM , sch that , u nan 1 ,,ann where gnn . Then ga.n n 14) From above, we know that f is minial if and only if ( m 2d0. h F (15) ca G x SM n From (3.3) and (3.4), a nd in a similar way as in [5], we n get ij ij ij hghg fyy gnn 2. ij fy yG an an 1 2. n1 22 n AHA gg HA AA Then (3.5) is equivalent to d0. ij ij fy y G 122 2 x n n SM ssbs na (16) If F is an (α, β)Minkowski metric, 0. In Minkowski(α, β) space, f is minimal if and only if then G 122 n ssbs 2 xn d0. ij ij SM yy fn em 1 Let (M, F) be a hypersurface of (17) Theor F , and F be an (α, β)Minkowski metric. Then :, , MF MF is a minimal immersion if and only if 22 () ij ij Sx fn yy b 1n () d 0, (18) S is a sphere such that 1 where . Now, we consider the conoid in 3dimensional (α, β) Minkowski space paralleling to x3axis. Set , F where 222 123 3 ,, yy by and b is a constant. Let sin , cos , uvu, vhv where hv is a unknown function. Then 23 cossin0 , sin cos i vv fuvu vh 1 12122 cossin 0 s cossin sincos. vv vh yvuyvyvuy vyh 23 12 sin co yy yyyuvu Assume that 12 2 2 1 cos ,sinyy uh , 0, 2π, then 222 123 22 2 12 2 1. yy y yuhy Note thatal vector of the surface is the norm 22 2 22 2 sin cos ,, , hvh vu n hu huhu and 11 13 12 21 13 13 22 13 000, sincos0 , =cos sin. f ff vv fuvuvh Set 22 d, ijij Sx Wyy b (19) Copyright © 2012 SciRes. APM
D. H. XIE, Q. HE 223 Then (3.8) is equivalent to 22 22 20.nW (20) Since 312 12 1 fnW f S is symmetric with resp 1 ect to and ision onlyding on a funct depen 2 , 121 2 22 Sx Wyy b d 22 0, .W u ver, 22 0W is impossible. Recall th 0, Therefore, (3.10) becomes to uh Howe at 2 2 22 2 2d, 0 xn S g Wys s , and 2 is not identically vanishing, we can obtain W. Then 0h , ,hcvd wThe Let 3,VF be an (α, β)Minkowski 22 0 here c, d are arbitrary constants. orem 2 space, 3 , by , and sin,cos , uv if and only if f i uvhv be a conoid. Then f imal s a helicoid or a plane. ma helicoid al nt in (α, β)ski swhere 3 by . This is an interesting result for minisurf ition 3 by is we co tion: s mini Rerk 3.1 From theorem 2, we can affirm that a is minim o only in Euclidean space but also Minkowpace, mal aces. But whether the result hold if the cond not satisfied? Nownsider the following condi 123 12 3 cossin byb yby bvbv 12 12 21 cossin ,ybvbvy where 123 ,,bbb are not all zeros. To simplify the com putation, we only discuss quadratic (α, β)metric: 2 Fk . Set 112 cos sinBb vbv , 3 3 0 0, Cvu uCv (21) where 22 1 cos sinBbvb v . Then (3.8) becomes an equation respect to u: 54 54 2 21 Cv u Cvu Cvu Cv 4 52 15 , 8 CBh 2 22 133 2 , 2B bbhB 4 15 C hh 23 12 32224222 23321 2222 2 11 15 4539 π 2484 π π21, 2 B Bb hbBkB hkB B kBkbB kbh 2 322 3 121212 3342CkBkBBkBBkbBBh 1 2 5 3222 22313223 15 9 9, 22 CBbhhkBbBhkBbhh 4 4 13 3 23222 2 211 13 242222 111 2 22 2 31 15 8 342 2 3ππ 8 921 , 4 Cbhh BkBkBkbBkbh kBkBk bB kbBkbh h 4 22 0311 31 .CkbBBh Since (3.11) is valid for any u, we can obtain 00,,5,. i Ci v If 10b or 20b , then 10B or 20B , such that . Therefore, when 12 ,bb are not all zeros, 0hv hv const. That is to say a minimal conoid hypersur face is a plane with respect to the given metric above. Theorem 3 Let 3,VF be an (α, β)Minkowski space, where 2 Fk b satisfying 123 byb yby (,bb are not all zeros). Then a mne. 4. Gauss Curvature of Conoid in Randers all kn m dy the Gauss curvatun Min kowskiRanders 3space around x 3axis in the direction # 12 312 inimal conoid hypersurface in 3,VF is a pla 3Space As we own, the Gauss curvature of a minimal sur face is nonpositive everywhere in Euclidean space. Then, a fact honatural problem arises: whether this lds for inimal surfaces in MinkowskiRanders 3space? In this section, we sture of conoid i , that is #3 by . Consider the conoid ,cos,sin, uvuvuvh v, w0 and vShere u1 . Let 1def u , 2def v . Then 12 ye e es a natur gival coordinates ,,,uv on its tangent bundle. In this section we shall use the convention that 1, 2ij and 1, 3 . Besides, the notations 12 :, :uv and 12 :, :yy are also used. Note that the induced 1form * on the surface is closed. Then the Ricci curvature tensor of * fF is given by ([10], Page 118) Copyright © 2012 SciRes. APM
D. H. XIE, Q. HE 224 23 4 Ric Ricr F where 2 00 000 12 ,Fr (22) Ric denotes the Ricci curvature tensor of the ced Riemindu* annian metric , 00 r efficients of the covarie  ij ij byy and b denote the cnt derivatives ij of o with respect to . Thuss curvature of the surface is given by en the Ga 2 2 132, 4 F KrFr 00 000 4 Ricy K,xy (23) where ,, fuv denotes the Gau to ss curvature with respect . te Denoii z and 2 . ij ij f z Then 23 cossin0 , sin cos i vv zu vhv 000, cos0 ,vv ature is computed in Euclid ean space as follows: v u 11 13 z 12 21 1313 sinzz cos sizuv 22 13 e Gauss curv n .uvh Noting that th 2 2, LN M EG F wher 22 ,,,LznMznNz n K e 11 12 22 .Gzz 11 12 ,,Ezz Fzz n obtain By direct computation, we ca 2 2 22 . h (24) cients of * K i uh Meanwhile, the coeff are given by 2 10 , uh ij 2 0 a 1 0 kl kl aa where iji j azz 2 2 10 1, uh . It is easy to verify that kkl ijl ij azz . By a direct computation, we have 100 , 0 ij u 2 2 2 22 22 0 . ij u uh uhh uh uh Since 3 ii bbz, 22 0 . s s ij hu uh b b , 22 22 j ij i bb uh h uh uh h From   , k s ijkisjksj ik ij bb bb , we have 2 11 22 22 2, bu h b uh 2 211212 22 , buh bb uh 1 22 1 2 221 22 22 , buh bb hu uh 3 221 2 22 2, buh b uh 2 2 222 2222 21 . bh u bh uhuh Besides, 22 12 2 00 22 22 2, ij ij buhu h rbyyyyy uh uh 22 12 000 2 22 2 2212 2 22 23 22 2222 4 22 21 ijk ijk bu h rbyyyyy uh buh huyy uh bh uhy uh uh (4.2) and (4.3), we obtain the following theorem. Theorem 4 Let Then, from 3,VF be an RandersMinkow space with 3 by , the Gauss curvature of the conoid ski ,cos,sin,uvuvuvhv at , fuv in direc tion of 12 yee is given by 0212222 4 334253 64 1 ,128 4 12443, Kxyb bF F bbbF where Copyright © 2012 SciRes. APM
D. H. XIE, Q. HE APM 225 Copyright © 2012 SciRes. 222 01 22 22 22 3 34 2 2 2 22 2 22 22 22 24 526 2222 2 22 ,,, ,2, 1,. huhuh uh uhuh uhhuh hu uh uh uh uh h uhuh uh to (α, 3 2 Note that a helicoid is minimal if and only if it is a conoid with respectβ)metrics (where by ). Let hvcv d (c is a constant), then the Gauss cur vate y ur of this surface is given b 222 4 12,y bF F (25 where 02122 ,3Kx b ) 222 2 12 22 2 22 2222 ,, . ccucu uc ucuc However, for , 0 a given point fuv, in which directions of TS, ,0?y ,0Kxy, ,0Kxy, 1) Kx If 0 , then ,<0Kxy for any 0c; 2) If 0 , Since 22ij i ij i ayy byu ccb , Equation (4.4) becomes 2 3 3 cbu Kxy c uc uc cb 0, let 0 2 2 22 22 12 ,. If c , then 2 3 22 . cbu u c we ca 2 2 22 12 ,Kxy c uc cb an also mke 2 u 2 3 22 20 cb c cbu c , then ,0Kxyrwise, let 0; Othe , then 2 2 23 22 22 12 ,. cbu Kxy c uc cbuc we can make 2 2 3 2 20 cbu cb c 2 c u , then ,0Kxy. In sum, the Gauss curvature is not nonposi tive anywhere. REFERENCES Finsler Geometry of Submanifolds,” Mathematische Annalen, Vol. 311, No. 3, 1998, pp. 549 576. do 50200 [1] Z. Shen, “On i:10.1007/s0020800 a, J. Spruck and K[2] M. Souz. Tenenblat, “A Bernstein Type theorem on a Randers Space,” Mathematische Annalen, Vol. 329, No. 2, 2004, pp. 291305. doi:10.1007/s0020800305003 [3] Q. He and Y. B. Shen, “On the Mean Curvature of Finsler Submanifolds,” Chinese Journal of Contemporary Mathe l. 27C, 2006, pp. 431442. [4] Q. He and Y. B. Shen, “On Bernstein Type Theorems in Finsler Spaces with the Volume form Induced from the Projecti /S0002993905080172 matics, Vo ve Sphere Bundle,” Proceedings of the American Mathematical Society, Vol. 134, No. 3, 2006, pp. 871880. doi:10.1090 [5] M. Souza and Kl Surfaces of Rota tion in a FinslMetric,” Mathema . Tenenblat, “Minima er Space with a Randers tische Annalen, Vol. 325, No. 4, 2003, pp. 625642. doi:10.1007/s0020800203927 [6] Q. He and W. Yang, “The Volume Forms and Minimal Surfaces of Rotation in Finsler Spaces with (α, β)Met rics,” International Journal of Mathematics, Vol. 21, No. 11, 2010, pp. 14011411. doi:10.1142/S0129167X10006483 [7] N. Cui and Y. B. Shen, “Minimal Rotated Hypersurface in Minkowski (α, β)Space,” Geometriae Dedicata, Vol. 151, No. 1, 2010, pp. 2739. doi:10.1007/s1071101095174 [8] H. Rund, “The Differential Geometry of Finsler Spaces,” SpringerVerlag, Berlin, 1959. [9] Z. Shen, “Landsberg Curvature, Scurvature and Riemann Curvature,” In: Z. Shen, Ed., A Sampler of Finsler Ge ometry, MSRI Series, Cambridge University Press, Cam bridge, 2004. [10] Z. Shen, “Differtial Geometry of Spray and Finsler Spaces,” Kluwer Academic Publishers, Berlin, 2001.
