【作者】 邱红兵；

【导师】 忻元龙；

【作者基本信息】 复旦大学 ， 基础数学， 2010， 博士

【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.更多还原

【Abstract】 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 obtainTheorem 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.更多还原