乘积流形中子流形的整体性质研究
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摘要
本论文探讨乘积流形中的子流形的若干整体性质,并给出了应用.它由两部分构成.在第一部分(即第三章)里,我们探讨了H2×R中给定平均曲率曲面的Weierstrass表示,得到
定理1([37])设x=(x1,x2,x3):∑→H2×R是等距浸入,G1：∑→U1(其中Ul如(3.5)中定义)是法高斯映照.则在U1上有
定理2([37])设∑是单连通的黎曼曲面H：∑→R是C1函数且G1：∑→U1(其中Ul如(3.5)中定义)是光滑映照,假设G1,H满足方程(3.27).令
则x=(x1,x2,x3)是具有分枝点的曲面，它的平均曲率为H，法高斯映照为G1.而且若G1z≠0,则x是正则曲面.
在第二部分(即第四章和第五章),我们将分别给出在κ-Ricci曲率具有强二次衰减的完备黎曼流形中的平均曲率可控的完备逆紧浸入子流形和在某些乘积流形中的平均曲率可控的完备逆紧浸入子流形上的Omori-Yau极值原理.我们也得到平均曲率有界的完备平均曲率流上的极值原理.利用广义极值原理我们得到某些乘积流形N1×N2中的在N1上的投影有界的逆紧浸入子流形的平均曲率的估计：
定理3([38])设N1,N2分别为n1,n2维完备黎曼流形且N2的径向截面曲率满足(?)其中c是正常数，ρ2是N2上到固定点的距离函数.设(?)是k(k>n2)维完备黎曼流形到N1×N2的等距浸入,其平均曲率向量为H.给定(?)设BN1(r)是N1中以p为心,以r为半径的测地球.假设沿N1中的从p出发的径向测地线的径向截面曲率kN1rad满足(?)且(?)对其中若b<0,我们用+∞代替(?).(1)若φ：(?)是逆紧的,则(2)若则M是随机不完备的,其中Cb在第五章的开始给出定义.
另外我们也给出了广义极值原理的其它应用.
In the thesis, we shall study several global properties of submanifolds in some product manifolds and give their applications. The thesis consists of two parts. In the first part(Chapter 3), we explore the Weierstrass representation for surfaces of pre-scribed mean curvature in H2 x R and obtain
Theorem 1 ([37]) Let x = (x1, x2, x3):∑→H2×R be an isometric immersion and G1:∑→U1 be the normal Gauss map, where U1 is defined in (3.5). Then we have, on U1,
Theorem 2 ([37]) Let E be a simply connected Riemann surface, H:∑→R be a C1-function, and G1:∑→U1 be a smooth mapping, where U1 is defined in (3.5). Assume that G1 satisfies the differential equation (3.27) for the above H. we set
Then x= (x1,x2, x3) is a branched surface such that the mean curvature is H and the normal Gauss map of x is G1. Moreover, if(?), then x is a regular surface.
In the second part (Chapter 4 and Chapter 5), we obtain various versions of Omori-Yau's maximum principle on complete properly immersed submanifold with controlled mean curvature of complete Riemannian manifolds whose k-Ricci curvature has strong quadratic decay, and on complete properly immersed submanifold with controlled mean curvature of certain product manifolds. We also obtain a maximum principle for mean curvature flow of complete manifolds with bounded mean curvature. Using the gener-alized maximum principle we give an estimate of the mean curvature of properly im-mersed submanifolds with bounded projection in Ni in the product manifold N1×N2:
Theorem 3 ([38]) Let N1,N2 be complete Riemannian manifolds of dimen-sion n1, n2 respectively and the radial sectional curvature of N2 satisfy (?) (?), where c is a positive constant,ρ2 is the distance function from a fixed point on N2. Letφ:Mk→N1×N2 be an isometric immersion of a complete Riemannian manifold of dimension k> n2 with mean curvature vector H. Given q∈M, p=π1(φ(g))∈N1. Let BN1(r) be the geodesic ball of N1 centered at p with radius r. Assume that the radial sectional curvature (?) along the radial geodesics issuing from p is bounded as (?) in BN1(r). Suppose thatφ(M) (?) BN1(r)×N2 for (?), where we replace (?).
(1)Ifφ:Mk→N1×N2 is proper, then
(2) If then M is stochastically incomplete, where Cb is defined in the beginning of§4.
We also give other applications of the generalized maximum principles.
定理1([37])设x=(x1,x2,x3):∑→H2×R是等距浸入,G1：∑→U1(其中Ul如(3.5)中定义)是法高斯映照.则在U1上有
定理2([37])设∑是单连通的黎曼曲面H：∑→R是C1函数且G1：∑→U1(其中Ul如(3.5)中定义)是光滑映照,假设G1,H满足方程(3.27).令
则x=(x1,x2,x3)是具有分枝点的曲面，它的平均曲率为H，法高斯映照为G1.而且若G1z≠0,则x是正则曲面.
在第二部分(即第四章和第五章),我们将分别给出在κ-Ricci曲率具有强二次衰减的完备黎曼流形中的平均曲率可控的完备逆紧浸入子流形和在某些乘积流形中的平均曲率可控的完备逆紧浸入子流形上的Omori-Yau极值原理.我们也得到平均曲率有界的完备平均曲率流上的极值原理.利用广义极值原理我们得到某些乘积流形N1×N2中的在N1上的投影有界的逆紧浸入子流形的平均曲率的估计：
定理3([38])设N1,N2分别为n1,n2维完备黎曼流形且N2的径向截面曲率满足(?)其中c是正常数，ρ2是N2上到固定点的距离函数.设(?)是k(k>n2)维完备黎曼流形到N1×N2的等距浸入,其平均曲率向量为H.给定(?)设BN1(r)是N1中以p为心,以r为半径的测地球.假设沿N1中的从p出发的径向测地线的径向截面曲率kN1rad满足(?)且(?)对其中若b<0,我们用+∞代替(?).(1)若φ：(?)是逆紧的,则(2)若则M是随机不完备的,其中Cb在第五章的开始给出定义.
另外我们也给出了广义极值原理的其它应用.
In the thesis, we shall study several global properties of submanifolds in some product manifolds and give their applications. The thesis consists of two parts. In the first part(Chapter 3), we explore the Weierstrass representation for surfaces of pre-scribed mean curvature in H2 x R and obtain
Theorem 1 ([37]) Let x = (x1, x2, x3):∑→H2×R be an isometric immersion and G1:∑→U1 be the normal Gauss map, where U1 is defined in (3.5). Then we have, on U1,
Theorem 2 ([37]) Let E be a simply connected Riemann surface, H:∑→R be a C1-function, and G1:∑→U1 be a smooth mapping, where U1 is defined in (3.5). Assume that G1 satisfies the differential equation (3.27) for the above H. we set
Then x= (x1,x2, x3) is a branched surface such that the mean curvature is H and the normal Gauss map of x is G1. Moreover, if(?), then x is a regular surface.
In the second part (Chapter 4 and Chapter 5), we obtain various versions of Omori-Yau's maximum principle on complete properly immersed submanifold with controlled mean curvature of complete Riemannian manifolds whose k-Ricci curvature has strong quadratic decay, and on complete properly immersed submanifold with controlled mean curvature of certain product manifolds. We also obtain a maximum principle for mean curvature flow of complete manifolds with bounded mean curvature. Using the gener-alized maximum principle we give an estimate of the mean curvature of properly im-mersed submanifolds with bounded projection in Ni in the product manifold N1×N2:
Theorem 3 ([38]) Let N1,N2 be complete Riemannian manifolds of dimen-sion n1, n2 respectively and the radial sectional curvature of N2 satisfy (?) (?), where c is a positive constant,ρ2 is the distance function from a fixed point on N2. Letφ:Mk→N1×N2 be an isometric immersion of a complete Riemannian manifold of dimension k> n2 with mean curvature vector H. Given q∈M, p=π1(φ(g))∈N1. Let BN1(r) be the geodesic ball of N1 centered at p with radius r. Assume that the radial sectional curvature (?) along the radial geodesics issuing from p is bounded as (?) in BN1(r). Suppose thatφ(M) (?) BN1(r)×N2 for (?), where we replace (?).
(1)Ifφ:Mk→N1×N2 is proper, then
(2) If then M is stochastically incomplete, where Cb is defined in the beginning of§4.
We also give other applications of the generalized maximum principles.
引文
[1]K. Akutagawa, S. Nishikawa. The Gauss map and spacelike surfaces with prescribed mean curvature in Minkowski 3-space. Tohoku Math. J. (2)42(1990),67-82.
[2]L. J. Alias, G. P. Bessa, M. Dajczer, The mean curvature of cylindrically bounded submanifolds, Math. Ann.345(2009),367-376.
[3]L. J. Alias, M. Dajczer, Constant mean curvature hypersurfaces in warped product spaces, Proc. Edinb. Math. Soc.50(2007),511-526.
[4]S. Bernstein, Sur un theoreme de Gemetrie et ses applications aux equations aux derivees partielles du type elliptique, Comm. de la Soc. Math. de Kharkov (2e ser) 15(1915-1917),38-45.
[5]D.A. Berdinsky, I. A. Taimanov. Surfaces in three-dimensional Lie groups. translated in Siberian Math. J.46(2005),1005-1019.
[6]E. Bombieri, E. De Giorgi, E. Giusti, Minimal cones and the Bernstein problem, In-vent. Math.7 (1969),243-268.
[7]A. A. Borisenkko, V. Miquel, Mean curvature flow of graphs in warped products, arXiv: math.DG/0906.2981v1,2009.
[8]R.L. Bryant. Surfaces of mean curvature one in hyperbolic space. Asterisque,154-155(1987),321-347.
[9]E. Calabi, An extension of E.Hopf's maximum principle with an application to Rie-mannian geometry, Duke Math. J.25(1958),45-56.
[10]E. Calabi, Problems in differential geometry. In:S. Kobayashi, J. Jr. Eells (eds.) Pro-ceedings of the United States-Japan Seminar in Differential Geometry, Kyoto, Japan, 1965, p.170. Nippon Hyoronsha, Tokyo(1966).
[11]Q. Chen, H. B. Qiu. Weierstrass representation for surfaces in the three-dimensional Heisenberg group. Chin. Ann. Math.31B(2010),119-132.
[12]Q. Chen, Y. L. Xin, A generalized maximum principle and its applications in geometry, Amer. J. Math.114(1992),355-366.
[13]S. Y. Cheng, S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math.28(1975),333-354.
[14]S.S. Chern, The geometry of G-structures, Bull. Amer. Math. Soc.72(1966),167-219. [15] T. H. Colding, W. P. Minicozzi Ⅱ, The Calabi-Yau conjectures for embedded surfaces, Ann. Math.167(2008),211-243. [16] B. Daniel. The Gauss map of minimal surfaces in the Heisenberg group. arXiv:math.DG/0606299,2006. [17] J. Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J.32(1983),703-716. [18] K. Ecker, G. Huisken, Mean curvature evolution of entire graphs, Ann. of Math. 130(1989),453-471. [19] I. Fernandez, P. Mira. Harmonic map and constant mean curvature surfaces in H2 x R. Amer. J. Math.129(2007),1145-1181. [20] H. Fujimoto, On the number of exceptional values of the Gauss map of minimal sur-faces, J. Math. Soc. Japan,40 (1988) 235-247. [21] A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc.36(1999), 135-249. [22] J. Inoguchi. Minimal surfaces in the 3-dimensional Heisenberg group. Differ. Geom. Dyn. Syst.10(2008),163-169. [23] J. Inoguchi, S. Lee. A Weierstrass type representation for minimal surfaces in Sol. Proc. Amer. Math. Soc.136(2008),2209-2216. [24] L. Jorge, F. Xavier, A complete minimal surface in R3 between two parallel planes, Ann. of Math.112 (1980),203-206. [25] K. Kenmotsu. Weierstrass formula for surfaces of prescribed mean curvature. Math. Ann.245(1979),89-99. [26] K. Kenmotsu, C. Y. Xia, Hadamard-Frankel type theorems for manifolds with partially positive curvature, Pacific J. Math.176(1996),129-139. [27] O. Kobayashi. Maximal surfaces in the 3-dimensional Minkowski space L3. Tokyo J. Math.6(1983),297-309. [28] M. Kokubu. Weierstrass representation for minimal surfaces in hyperbolic space. To-hoku Math J.49(1997),367-377. [29] B. Lawson, The equivariant Plateau problem and interior regularity, Trans. AMS 173 (1972),231-250.
[30]F. Mercuri, S. Montaldo, P. Piu. A weierstrass representation formula for minimal surfaces in H3 and H2 x R. Acta Math. Sin. (Engl. Ser.) 22(2006),1603-1612.
[31]N. Nadirashvili, Hadamard's and Calabi-Yau's conjectures on negatively curved and minimal surfaces, Invent. Math.126(1996),457-465.
[32]B. O'Neill, Semi-Riemannian Geometry, Academic Press,1983.
[33]H. Omori, Isometric immersion of Riemannian manifolds, J. Math. Soc. Japan. 19(1967),205-214.
[34]R. Osserman, Minimal surfaces in the large, Comment. Math. Helv.35(1961),65-76.
[35]S. Pigola, M. Rigoli, A. Setti, A remark on the maximm principle and stochastic com-pleteness, Proc. A. M. S.131(2003),1283-1288.
[36]S. Pigola, M. Rigoli, A. Setti, Maximum principle on Riemannian manifolds and ap-plications, Mem. Amer. Math. Soc.822(2005).
[37]H. B. Qiu, Weierstrass Representation for Surfaces of prescribed mean curvature in H2 × R (In Chinese), Chin. Ann. Math.31 A(2010),1-12.
[38]H. B. Qiu, Y. L. Xin, Proper submanifolds in product manifolds, preprint.
[39]A. Ratto, M. Rigoli, L. Veron, Scalar curvature and conformal deformation of hyper-bolic space, J. Funct. Anal.121(1994),15-77.
[40]Z. Shen, On complete manifolds of nonnegative kth-Ricci curvature, Trans. Amer. Math. Soc.338(1993),289-310.
[41]S. G. Shi. Weierstrass representation for surfaces of prescribed mean curvature in the hyperbolic 3-dimensional space (In Chinese), Chin. Ann. Math.22A(2001),691-700.
[42]J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math.88 (1968),62-105.
[43]W. P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geom-etry. Bull. Amer. Math. Soc.6(1982),357-381.
[44]H. Wu, Manifolds of partially positve curvature, Indiana Univ. Math. J.36(1987),525-548.
[45]F. Xavier, The Gauss map of a complete non-flat minimal surface cannot omit 7 points of the sphere, Ann. of Math,113 (1981),211-214. [46] Y. L. Xin. Minimal submanifolds and related topics. Nankai Tracts in Mathematics-Vol.8., World Scientific,2003. [47] Y. L. Xin, Geometry of harmonic maps, Birkhauser,1994. [48] Y. L. Xin, Mean curvature flow with bounded Gauss image, preprint. [49] S. T. Yau, Harmonic function on complete Riemannian manifolds, Comm. Pure. Appl. Math.28(1975),201-228.
[2]L. J. Alias, G. P. Bessa, M. Dajczer, The mean curvature of cylindrically bounded submanifolds, Math. Ann.345(2009),367-376.
[3]L. J. Alias, M. Dajczer, Constant mean curvature hypersurfaces in warped product spaces, Proc. Edinb. Math. Soc.50(2007),511-526.
[4]S. Bernstein, Sur un theoreme de Gemetrie et ses applications aux equations aux derivees partielles du type elliptique, Comm. de la Soc. Math. de Kharkov (2e ser) 15(1915-1917),38-45.
[5]D.A. Berdinsky, I. A. Taimanov. Surfaces in three-dimensional Lie groups. translated in Siberian Math. J.46(2005),1005-1019.
[6]E. Bombieri, E. De Giorgi, E. Giusti, Minimal cones and the Bernstein problem, In-vent. Math.7 (1969),243-268.
[7]A. A. Borisenkko, V. Miquel, Mean curvature flow of graphs in warped products, arXiv: math.DG/0906.2981v1,2009.
[8]R.L. Bryant. Surfaces of mean curvature one in hyperbolic space. Asterisque,154-155(1987),321-347.
[9]E. Calabi, An extension of E.Hopf's maximum principle with an application to Rie-mannian geometry, Duke Math. J.25(1958),45-56.
[10]E. Calabi, Problems in differential geometry. In:S. Kobayashi, J. Jr. Eells (eds.) Pro-ceedings of the United States-Japan Seminar in Differential Geometry, Kyoto, Japan, 1965, p.170. Nippon Hyoronsha, Tokyo(1966).
[11]Q. Chen, H. B. Qiu. Weierstrass representation for surfaces in the three-dimensional Heisenberg group. Chin. Ann. Math.31B(2010),119-132.
[12]Q. Chen, Y. L. Xin, A generalized maximum principle and its applications in geometry, Amer. J. Math.114(1992),355-366.
[13]S. Y. Cheng, S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math.28(1975),333-354.
[14]S.S. Chern, The geometry of G-structures, Bull. Amer. Math. Soc.72(1966),167-219. [15] T. H. Colding, W. P. Minicozzi Ⅱ, The Calabi-Yau conjectures for embedded surfaces, Ann. Math.167(2008),211-243. [16] B. Daniel. The Gauss map of minimal surfaces in the Heisenberg group. arXiv:math.DG/0606299,2006. [17] J. Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J.32(1983),703-716. [18] K. Ecker, G. Huisken, Mean curvature evolution of entire graphs, Ann. of Math. 130(1989),453-471. [19] I. Fernandez, P. Mira. Harmonic map and constant mean curvature surfaces in H2 x R. Amer. J. Math.129(2007),1145-1181. [20] H. Fujimoto, On the number of exceptional values of the Gauss map of minimal sur-faces, J. Math. Soc. Japan,40 (1988) 235-247. [21] A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc.36(1999), 135-249. [22] J. Inoguchi. Minimal surfaces in the 3-dimensional Heisenberg group. Differ. Geom. Dyn. Syst.10(2008),163-169. [23] J. Inoguchi, S. Lee. A Weierstrass type representation for minimal surfaces in Sol. Proc. Amer. Math. Soc.136(2008),2209-2216. [24] L. Jorge, F. Xavier, A complete minimal surface in R3 between two parallel planes, Ann. of Math.112 (1980),203-206. [25] K. Kenmotsu. Weierstrass formula for surfaces of prescribed mean curvature. Math. Ann.245(1979),89-99. [26] K. Kenmotsu, C. Y. Xia, Hadamard-Frankel type theorems for manifolds with partially positive curvature, Pacific J. Math.176(1996),129-139. [27] O. Kobayashi. Maximal surfaces in the 3-dimensional Minkowski space L3. Tokyo J. Math.6(1983),297-309. [28] M. Kokubu. Weierstrass representation for minimal surfaces in hyperbolic space. To-hoku Math J.49(1997),367-377. [29] B. Lawson, The equivariant Plateau problem and interior regularity, Trans. AMS 173 (1972),231-250.
[30]F. Mercuri, S. Montaldo, P. Piu. A weierstrass representation formula for minimal surfaces in H3 and H2 x R. Acta Math. Sin. (Engl. Ser.) 22(2006),1603-1612.
[31]N. Nadirashvili, Hadamard's and Calabi-Yau's conjectures on negatively curved and minimal surfaces, Invent. Math.126(1996),457-465.
[32]B. O'Neill, Semi-Riemannian Geometry, Academic Press,1983.
[33]H. Omori, Isometric immersion of Riemannian manifolds, J. Math. Soc. Japan. 19(1967),205-214.
[34]R. Osserman, Minimal surfaces in the large, Comment. Math. Helv.35(1961),65-76.
[35]S. Pigola, M. Rigoli, A. Setti, A remark on the maximm principle and stochastic com-pleteness, Proc. A. M. S.131(2003),1283-1288.
[36]S. Pigola, M. Rigoli, A. Setti, Maximum principle on Riemannian manifolds and ap-plications, Mem. Amer. Math. Soc.822(2005).
[37]H. B. Qiu, Weierstrass Representation for Surfaces of prescribed mean curvature in H2 × R (In Chinese), Chin. Ann. Math.31 A(2010),1-12.
[38]H. B. Qiu, Y. L. Xin, Proper submanifolds in product manifolds, preprint.
[39]A. Ratto, M. Rigoli, L. Veron, Scalar curvature and conformal deformation of hyper-bolic space, J. Funct. Anal.121(1994),15-77.
[40]Z. Shen, On complete manifolds of nonnegative kth-Ricci curvature, Trans. Amer. Math. Soc.338(1993),289-310.
[41]S. G. Shi. Weierstrass representation for surfaces of prescribed mean curvature in the hyperbolic 3-dimensional space (In Chinese), Chin. Ann. Math.22A(2001),691-700.
[42]J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math.88 (1968),62-105.
[43]W. P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geom-etry. Bull. Amer. Math. Soc.6(1982),357-381.
[44]H. Wu, Manifolds of partially positve curvature, Indiana Univ. Math. J.36(1987),525-548.
[45]F. Xavier, The Gauss map of a complete non-flat minimal surface cannot omit 7 points of the sphere, Ann. of Math,113 (1981),211-214. [46] Y. L. Xin. Minimal submanifolds and related topics. Nankai Tracts in Mathematics-Vol.8., World Scientific,2003. [47] Y. L. Xin, Geometry of harmonic maps, Birkhauser,1994. [48] Y. L. Xin, Mean curvature flow with bounded Gauss image, preprint. [49] S. T. Yau, Harmonic function on complete Riemannian manifolds, Comm. Pure. Appl. Math.28(1975),201-228.