常曲率黎曼流行中曲面的Weierstrass表示
摘要
本文给出H~3中给定Gauss曲率曲面的Weierstrass表示公式；讨论了双曲Gauss映照、法Gauss映照和平均曲率(或Gauss曲率)的联系；并导出一个有趣的二阶非线性方程，它的解所对应的图的法Gauss映照关于由第二基本形式所诱导的共形结构是共形映照；给出H~n中由法Gauss映照给出的Weierstrass表示公式；给出S~n中极小曲面的Weierstrass表示公式。
引文
[1] R.Aiyama and K.Akutagawa, Kenmotsu type representation formula for surfaces with prescribed mean curvature in the 3-sphere, Thoku Math.J.,52(2000), 95-105
[2] R.Aiyama and K.Akutagawa, Kenmotsu-Bryant type Representation formulas for GMC surfaces in H~3(-c~2) and S_1~3~(c~2), Ann. Global Anal. Geom. 17(1999), 49-75
[3] R..Aiyama and K.Akutagawa, Kenmotsu type representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space,J.Math.Soc.Japan,52(2000) no.4 877-898
[4] E.F.Beckenbach, Minimal surfaces in euclidean n-space, Amer.J.Math, 55(1933),458-468
[5] R.L.Bryant,Surfaces of mean curvature one in hyperbolic space, Astérisque, 154-155(1987), 321-347
[6] Do Carmo,M and Warner, F.W. Rigidity and convexity of hypersurfaces in spheres, J.Diff.Geom, 4(1970), 133-144
[7] C.L.Epstein, The hyperbolic Gauss map and quasiconformal reflections,J.Reine.Angew.Math., 372(1986) 96-135
[8] Frédéric Hélein and Pascal Romon, Weierstrass representation of Lagrangian surfaces in rourdimensional space using spinors and quaternions, Comment. Math. Heiv, 75(2000),668-680
[9] H.Fujimoto, On the number of exceptional values of the Gauss map of minimal surfaces, J.Math.Soc.Japan,40(1988) no.2 235-247
[10] J.A.Gávez and A.Martínez, The Gauss map and second Fundamental form of surfaces in R~3, Geometriae. Dedicata, 81(2000), 181-192
[11] J.A.Gálvez and A.Martínez and F.Milán, Flat surfaces in the hyperbolic 3-space, Math. Ann., 316(2000) no.3, 419-435
[12] R.Harvey and H.B.Lawson,Jr, Calibrated Geometries, Acta Math. 148(1982) 47-157
[13] M.Haskins,B.A., Constructing of Special Lagrangian cone, Ph.D.Thesis, The University of Texas at Austin, May 2000
[14] D.A.Hoffman and R.Osserman,The Gauss Map of surfaces in R~n, J.Diff.Geom., 18(1983),733-754
[15] D.A.Hoffman and R.Osserman, The Gauss map of surfaces in R~3 and R~4, Proc.London. Math. Soc., 50(1985), no.3, 2%56
[16] D.A.Hoffman and R.Osserman,The area of the generalized Gaussian image and the stability of minimal surfzces in S~n and R~n, Math.Ann.,260(1982),437-452
[17] K.Kenmotsu, Weiezstrass formula for surfaces of prescribed mean curvature, Math. Ann.,245(1979) 89-99
[18] T.Klotz, Some uses of the second conformal structure on strictly convex surfaces, Proc. AMS 14(1963),793-799
[19] M.Kokubu,Weierstrass representation for minimal surfaces in hyperbolic space, Thoku Math.J.49(1997) 367-377
[20] B.G.Konopelchenko, Induced surfaces and their integrable dynamics, Studies in applied mathemetics, 96(1996), 9-51
[21] M.Obata, The Gauss map of immersions of Riiemannian Manifolds in spaces of constant cur-
vature,J.Diff. Geom.,2(1968) 217-223
[22] R.Osserman, Global properties of minimal surfaces E~3 and E~n, Ann of Math, 80(1964),340-364
[23] R.W.Smyth and T.Weinstein, Conformally Homeomorphic Lorentz surfaces aeed not be conformally diffeomorphic,Proc. AMS, 123(1995), no.11, 3499-3506
[24] M.Spivak, A comprehensive introduction to differential geometry, Ⅳ, 1979
[25] Sun Cun-Jin, On the Gauss map of surfaces in R~n, Math.Ann.,283(1989),375-380
[26] T.Takahashi, Minimal immersion of Riemannian manifolds, J.Math.Soc.Japan, 18(1966),380-385
[27] M.Umeh~ra and K.Yamada, A parametrization of the Weierstrass formula and perturbation of some complete minimal surfaces in R~3 into the hyperbolic 3-space, J.Reine.Angew.Math, 432(1992), 93-116
[28] M.Umehara,K.Yamada, Surfaces of CMC-c in H~3(-c~2) with prescribed hyperbolic Gauss map, Math.Ann., 304(1996) 203-224
[29] M.Umehara and K.Yamada, Complete surfaces of CMC-1 in the hyperbolic 3-space, Ann of Math, 137(1993), 611-638
[30] Yu Zuhuan, Value distribution of hyperbolic Gauss maps, Proc.AMS,125(1997) no.10,2997-3001
[31] 洪剑峭，各种常平均曲率曲面的表示公式及其应用，博士论文，复旦大学数学研究所，1993，9
[32] 于祖焕，黎镇琦，H~3(-1)中的常中曲率曲面。科学通报，43(1998)，no．2，138-141
[33] 史淑国，3维双曲空间中给定平均曲率曲面的Weierstrass表示，数学年刊，22A(2001)，no．6，691-700
[2] R.Aiyama and K.Akutagawa, Kenmotsu-Bryant type Representation formulas for GMC surfaces in H~3(-c~2) and S_1~3~(c~2), Ann. Global Anal. Geom. 17(1999), 49-75
[3] R..Aiyama and K.Akutagawa, Kenmotsu type representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space,J.Math.Soc.Japan,52(2000) no.4 877-898
[4] E.F.Beckenbach, Minimal surfaces in euclidean n-space, Amer.J.Math, 55(1933),458-468
[5] R.L.Bryant,Surfaces of mean curvature one in hyperbolic space, Astérisque, 154-155(1987), 321-347
[6] Do Carmo,M and Warner, F.W. Rigidity and convexity of hypersurfaces in spheres, J.Diff.Geom, 4(1970), 133-144
[7] C.L.Epstein, The hyperbolic Gauss map and quasiconformal reflections,J.Reine.Angew.Math., 372(1986) 96-135
[8] Frédéric Hélein and Pascal Romon, Weierstrass representation of Lagrangian surfaces in rourdimensional space using spinors and quaternions, Comment. Math. Heiv, 75(2000),668-680
[9] H.Fujimoto, On the number of exceptional values of the Gauss map of minimal surfaces, J.Math.Soc.Japan,40(1988) no.2 235-247
[10] J.A.Gávez and A.Martínez, The Gauss map and second Fundamental form of surfaces in R~3, Geometriae. Dedicata, 81(2000), 181-192
[11] J.A.Gálvez and A.Martínez and F.Milán, Flat surfaces in the hyperbolic 3-space, Math. Ann., 316(2000) no.3, 419-435
[12] R.Harvey and H.B.Lawson,Jr, Calibrated Geometries, Acta Math. 148(1982) 47-157
[13] M.Haskins,B.A., Constructing of Special Lagrangian cone, Ph.D.Thesis, The University of Texas at Austin, May 2000
[14] D.A.Hoffman and R.Osserman,The Gauss Map of surfaces in R~n, J.Diff.Geom., 18(1983),733-754
[15] D.A.Hoffman and R.Osserman, The Gauss map of surfaces in R~3 and R~4, Proc.London. Math. Soc., 50(1985), no.3, 2%56
[16] D.A.Hoffman and R.Osserman,The area of the generalized Gaussian image and the stability of minimal surfzces in S~n and R~n, Math.Ann.,260(1982),437-452
[17] K.Kenmotsu, Weiezstrass formula for surfaces of prescribed mean curvature, Math. Ann.,245(1979) 89-99
[18] T.Klotz, Some uses of the second conformal structure on strictly convex surfaces, Proc. AMS 14(1963),793-799
[19] M.Kokubu,Weierstrass representation for minimal surfaces in hyperbolic space, Thoku Math.J.49(1997) 367-377
[20] B.G.Konopelchenko, Induced surfaces and their integrable dynamics, Studies in applied mathemetics, 96(1996), 9-51
[21] M.Obata, The Gauss map of immersions of Riiemannian Manifolds in spaces of constant cur-
vature,J.Diff. Geom.,2(1968) 217-223
[22] R.Osserman, Global properties of minimal surfaces E~3 and E~n, Ann of Math, 80(1964),340-364
[23] R.W.Smyth and T.Weinstein, Conformally Homeomorphic Lorentz surfaces aeed not be conformally diffeomorphic,Proc. AMS, 123(1995), no.11, 3499-3506
[24] M.Spivak, A comprehensive introduction to differential geometry, Ⅳ, 1979
[25] Sun Cun-Jin, On the Gauss map of surfaces in R~n, Math.Ann.,283(1989),375-380
[26] T.Takahashi, Minimal immersion of Riemannian manifolds, J.Math.Soc.Japan, 18(1966),380-385
[27] M.Umeh~ra and K.Yamada, A parametrization of the Weierstrass formula and perturbation of some complete minimal surfaces in R~3 into the hyperbolic 3-space, J.Reine.Angew.Math, 432(1992), 93-116
[28] M.Umehara,K.Yamada, Surfaces of CMC-c in H~3(-c~2) with prescribed hyperbolic Gauss map, Math.Ann., 304(1996) 203-224
[29] M.Umehara and K.Yamada, Complete surfaces of CMC-1 in the hyperbolic 3-space, Ann of Math, 137(1993), 611-638
[30] Yu Zuhuan, Value distribution of hyperbolic Gauss maps, Proc.AMS,125(1997) no.10,2997-3001
[31] 洪剑峭，各种常平均曲率曲面的表示公式及其应用，博士论文，复旦大学数学研究所，1993，9
[32] 于祖焕，黎镇琦，H~3(-1)中的常中曲率曲面。科学通报，43(1998)，no．2，138-141
[33] 史淑国，3维双曲空间中给定平均曲率曲面的Weierstrass表示，数学年刊，22A(2001)，no．6，691-700