子流形几何中的刚性及变分问题
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摘要
在子流形几何中,刚性问题和变分问题是两类重要问题,被几何学家广泛研究。刚性问题可以通过各种拼挤(pinching)定理来反映。对变分问题,我们可以研究临界点的稳定性和Jacobi算子的特征值。在本文中,我们运用子流形几何研究中的一系列方法,研究了乘积流形中的极小子流形的刚性和稳定性,球面中线性Weingarten超曲面的稳定性和特征值问题,以及乘积流形到任意黎曼流形的稳定调和映照。主要结果有三个方面：
首先,我们研究了Sm(1)×R中的紧致极小子流形的刚性。我们得到了一个Simons型等式,进而分别证明了在Ricci曲率和截面曲率的拼挤条件下,关于Sm(1)×R中的紧致无边极小子流形的拼挤定理。通过上述拼挤条件,我们推出极小子流形实际上位于Sm(1)×{t0}≌Sm(1)中。通过Ricci曲率的拼挤条件,我们刻画了Clifford极小超曲面。通过截面曲率的拼挤条件,我们刻画了Veronese子流形。
其次,我们用泛函F=∫M(a+nH)dv在保体积变分下的临界点刻画了Sn+1(1)中满足(n-1)H2+aH=b的线性Weingarten超曲面,其中a,b是常数。我们计算了该泛函的第一变分公式和第二变分公式,证明了此类线性Weingarten超曲面是稳定的当且仅当它是全脐但非全测地的超曲面,这推广了关于球面中的常平均曲率和常数量曲率超曲面的稳定性结果。我们还得到了与该变分问题相关联的Jacobi算子的第一特征值和第二特征值的最优上界估计。
最后,我们研究了乘积流形中的紧致极小子流形的稳定性。我们证明了一个关于M1×M2中的稳定紧致极小子流形的分类定理,其中M1是欧几里得空间中的紧致超曲面,其维数m1≥3且截面曲率KM1满足1/(?)m1-1≤Km11≤1,M2是任意黎曼流形。这推广了Torralbo和Urbano的关于Sm(r)×M中的稳定紧致极小子流形的分类结果。特别地,我们证明了,当外围空间是M是欧几里得空间中的m维(m≥3)紧致超曲面且截面曲率满足1/(?)M+1≤KM≤1时,M中不存在稳定的紧致无边极小子流形。
Rigidity and variational problems are two kinds of important problems in geometry of submanifolds, which are widely studied by many geometers. Rigidity can be reflected by various pinching theorems. For a variational problem, we can study stability of the critical points and estimate the eigenvalues of the Jacobi operator. In this dissertation, we use a series of methods in geometry of submanifolds to study rigidity and stability of minimal submanifolds in Riemannian product manifolds, stability and eigenvalues of lin-ear Weingarten hypersurfaces in a sphere, and stable harmonic maps from a Riemannian product manifold to any Riemannian manifold. The main results are the following three parts:
Firstly, we study the rigidity of compact minimal submanifolds in Sm(1)×R. We obtain a Simons'type equation, and then we prove some pinching theorems by the Ricci curvature and the sectional curvature pinching conditions, respectively. By our pinch-ing conditions, we conclude that, the minimal submanifolds lie in Sm(1)×{t0}=Sm(1). By the Ricci curvature pinching condition, we characterize the Clifford minimal hyper-surfaces. By the sectional curvature pinching condition, we characterize the Veronese submanifolds.
Secondly, we show that linear Weingarten hypersurfaces in Sn+1(1) satisfying (n-1)H2+aH=b, where a and b are constants, can be characterized as critical points of the functional F=∫M (a+nH) dv for volume-preserving variations. We compute the first and second variational formulae of this functional, and prove that such a linear Weingarten hypersurface is stable if and only if it is totally umbilical and non-totally geodesic. This generalizes the stability results about hypersurfaces with constant mean curvature or with constant scalar curvature. We also obtain optimal upper bounds for the first and second eigenvalues of the Jacobi operator corresponding to the variational problem.
Finally, we study stability of compact minimal submanifolds in Riemannian product manifolds. We prove a classification theorem for stable compact minimal submanifolds in M1×M2, where M1is an m1-dimensional (m1≥3) compact hypersurface in Euclidean space with the sectional curvature Km1satisfying1/(?)m1-1≤KM1≤1, and M2is any Rie-mannian manifold. This generalizes the result of Torralbo and Urbano for stable compact minimal submanifolds in Sm(r)×M. In particular, we prove that, when the ambient s- pace M is an m-dimensional (m≥3) compact hypersurface in Euclidean space with the sectional curvature satisfying1(?)m+1≤KM≤1, there exist no stable compact minimal submanifolds in M.
首先,我们研究了Sm(1)×R中的紧致极小子流形的刚性。我们得到了一个Simons型等式,进而分别证明了在Ricci曲率和截面曲率的拼挤条件下,关于Sm(1)×R中的紧致无边极小子流形的拼挤定理。通过上述拼挤条件,我们推出极小子流形实际上位于Sm(1)×{t0}≌Sm(1)中。通过Ricci曲率的拼挤条件,我们刻画了Clifford极小超曲面。通过截面曲率的拼挤条件,我们刻画了Veronese子流形。
其次,我们用泛函F=∫M(a+nH)dv在保体积变分下的临界点刻画了Sn+1(1)中满足(n-1)H2+aH=b的线性Weingarten超曲面,其中a,b是常数。我们计算了该泛函的第一变分公式和第二变分公式,证明了此类线性Weingarten超曲面是稳定的当且仅当它是全脐但非全测地的超曲面,这推广了关于球面中的常平均曲率和常数量曲率超曲面的稳定性结果。我们还得到了与该变分问题相关联的Jacobi算子的第一特征值和第二特征值的最优上界估计。
最后,我们研究了乘积流形中的紧致极小子流形的稳定性。我们证明了一个关于M1×M2中的稳定紧致极小子流形的分类定理,其中M1是欧几里得空间中的紧致超曲面,其维数m1≥3且截面曲率KM1满足1/(?)m1-1≤Km11≤1,M2是任意黎曼流形。这推广了Torralbo和Urbano的关于Sm(r)×M中的稳定紧致极小子流形的分类结果。特别地,我们证明了,当外围空间是M是欧几里得空间中的m维(m≥3)紧致超曲面且截面曲率满足1/(?)M+1≤KM≤1时,M中不存在稳定的紧致无边极小子流形。
Rigidity and variational problems are two kinds of important problems in geometry of submanifolds, which are widely studied by many geometers. Rigidity can be reflected by various pinching theorems. For a variational problem, we can study stability of the critical points and estimate the eigenvalues of the Jacobi operator. In this dissertation, we use a series of methods in geometry of submanifolds to study rigidity and stability of minimal submanifolds in Riemannian product manifolds, stability and eigenvalues of lin-ear Weingarten hypersurfaces in a sphere, and stable harmonic maps from a Riemannian product manifold to any Riemannian manifold. The main results are the following three parts:
Firstly, we study the rigidity of compact minimal submanifolds in Sm(1)×R. We obtain a Simons'type equation, and then we prove some pinching theorems by the Ricci curvature and the sectional curvature pinching conditions, respectively. By our pinch-ing conditions, we conclude that, the minimal submanifolds lie in Sm(1)×{t0}=Sm(1). By the Ricci curvature pinching condition, we characterize the Clifford minimal hyper-surfaces. By the sectional curvature pinching condition, we characterize the Veronese submanifolds.
Secondly, we show that linear Weingarten hypersurfaces in Sn+1(1) satisfying (n-1)H2+aH=b, where a and b are constants, can be characterized as critical points of the functional F=∫M (a+nH) dv for volume-preserving variations. We compute the first and second variational formulae of this functional, and prove that such a linear Weingarten hypersurface is stable if and only if it is totally umbilical and non-totally geodesic. This generalizes the stability results about hypersurfaces with constant mean curvature or with constant scalar curvature. We also obtain optimal upper bounds for the first and second eigenvalues of the Jacobi operator corresponding to the variational problem.
Finally, we study stability of compact minimal submanifolds in Riemannian product manifolds. We prove a classification theorem for stable compact minimal submanifolds in M1×M2, where M1is an m1-dimensional (m1≥3) compact hypersurface in Euclidean space with the sectional curvature Km1satisfying1/(?)m1-1≤KM1≤1, and M2is any Rie-mannian manifold. This generalizes the result of Torralbo and Urbano for stable compact minimal submanifolds in Sm(r)×M. In particular, we prove that, when the ambient s- pace M is an m-dimensional (m≥3) compact hypersurface in Euclidean space with the sectional curvature satisfying1(?)m+1≤KM≤1, there exist no stable compact minimal submanifolds in M.
引文
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[6] Ejiri N. Compact minimal submanifolds of a sphere with positive Ricci curvature. J. Math.Soc. Japan,1979,31(2):251–256.
[7] Xu Hongwei, Tian Ling. A diferentiable sphere theorem inspired by rigidity of minimal sub-manifolds. Pacific J. Math.,2011,254(2):499–510.
[8] Shen Yibing. Curvature pinching for three-dimensional minimal submanifolds in a sphere.Proc. Amer. Math. Soc.,1992,115(3):791–795.
[9] Gauchman H. Minimal submanifolds of a sphere with bounded second fundamental form.Trans. Amer. Math. Soc.,1986,298(2):779–791.
[10] Alencar H, do Carmo M. Hypersurfaces with constant mean curvature in spheres. Proc. Amer.Math. Soc.,1994,120(4):1223–1229.
[11] Xu Hongwei. A rigidity theorem for submanifolds with parallel mean curvature in a sphere.Arch. Math.(Basel),1993,61(5):489–496.
[12] Xu Hongwei, Fang Wang, Xiang Fei. A generalization of Gauchman’s rigidity theorem. PacificJ. Math.,2006,228(1):185–199.
[13] Xu Hongwei, Huang Fei, Xiang Fei. An extrinsic rigidity theorem for submanifolds withparallel mean curvature in a sphere. Kodai Math. J.,2011,34(1):85–104.
[14] Xu Hongwei, Gu Juanru. Rigidity of submanifolds with parallel mean curvature in space froms.2011. http://arxiv.org/abs/1105.2920v1.
[15] Shen Chunli. A global pinching theorem of minimal hypersurfaces in the sphere. Proc. Amer.Math. Soc.,1989,105(1):192–198.
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