In the present paper we obtain the following result: Theorem Let M^R be a compact submanifold with parallel mean curvature vector in a locally symmetric and conformally flat Riemannian manifold N^(n+p)(p>1). If the...In the present paper we obtain the following result: Theorem Let M^R be a compact submanifold with parallel mean curvature vector in a locally symmetric and conformally flat Riemannian manifold N^(n+p)(p>1). If then M^n lies in a totally geodesic submanifold N^(n+1).展开更多
We investigate the integral conditions to extend the mean curvature flow in a Riemannian manifold. We prove that the mean curvature flow solution with finite total mean curvature at a finite time interval [0,T) can be...We investigate the integral conditions to extend the mean curvature flow in a Riemannian manifold. We prove that the mean curvature flow solution with finite total mean curvature at a finite time interval [0,T) can be extended over time T. Moreover,we show that the condition is optimal in some sense.展开更多
Abstract The author introduces the w-function defined on the considered spacelike graph M. Under the growth conditions w = o（log z） and w = o（r）, two Bernstein type theorems for M in Rm^n＋m are got, where z and r...Abstract The author introduces the w-function defined on the considered spacelike graph M. Under the growth conditions w = o（log z） and w = o（r）, two Bernstein type theorems for M in Rm^n＋m are got, where z and r are the pseudo-Euclidean distance and the distance function on M to some fixed point respectively. As the ambient space is a curved pseudo- Riemannian product of two Riemannian manifolds （∑1,g1） and （∑2,g2） of dimensions n and m, a Bernstein type result for n =2 under some curvature conditions on E1 and E2 and the growth condition w = o（r） is also got. As more general cases, under some curvature conditions on the ambient space and the growth condition w = o（r） or w = o（√r）, the author concludes that if M has parallel mean curvature, then M is maximal.展开更多
The authors obtain various versions of the Omori-Yau＇s maximum principle on complete properly immersed-submanifolds with controlled mean curvature in certain product manifolds, in complete Riemannian manifolds whose ...The authors obtain various versions of the Omori-Yau＇s maximum principle on complete properly immersed-submanifolds with controlled mean curvature in certain product manifolds, in complete Riemannian manifolds whose k-Ricci curvature has strong quadratic decay, and also obtain a maximum principle for mean curvature flow of complete manifolds with bounded mean curvature. Using the generalized maximum principle, an estimate on the mean curvature of properly immersed submanifolds with bounded projection in N1 in the product manifold N1 x N2 is given. Other applications of the generalized maximum principle are also given.展开更多
In this paper,the authors consider a family of smooth immersions Ft : Mn→Nn+1of closed hypersurfaces in Riemannian manifold Nn+1with bounded geometry,moving by the Hkmean curvature flow.The authors show that if the s...In this paper,the authors consider a family of smooth immersions Ft : Mn→Nn+1of closed hypersurfaces in Riemannian manifold Nn+1with bounded geometry,moving by the Hkmean curvature flow.The authors show that if the second fundamental form stays bounded from below,then the Hkmean curvature flow solution with finite total mean curvature on a finite time interval [0,Tmax)can be extended over Tmax.This result generalizes the extension theorems in the paper of Li(see "On an extension of the Hkmean curvature flow,Sci.China Math.,55,2012,99–118").展开更多
文摘In the present paper we obtain the following result: Theorem Let M^R be a compact submanifold with parallel mean curvature vector in a locally symmetric and conformally flat Riemannian manifold N^(n+p)(p>1). If then M^n lies in a totally geodesic submanifold N^(n+1).
基金supported by National Natural Science Foundation of China (Grant Nos. 10771187, 11071211)the Trans-Century Training Programme Foundation for Talents by the Ministry of Education of China+1 种基金the Natural Science Foundation of Zhejiang Province (Grant No. 101037)the China Postdoctoral Science Foundation (Grant No. 20090461379)
文摘We investigate the integral conditions to extend the mean curvature flow in a Riemannian manifold. We prove that the mean curvature flow solution with finite total mean curvature at a finite time interval [0,T) can be extended over time T. Moreover,we show that the condition is optimal in some sense.
文摘Abstract The author introduces the w-function defined on the considered spacelike graph M. Under the growth conditions w = o（log z） and w = o（r）, two Bernstein type theorems for M in Rm^n＋m are got, where z and r are the pseudo-Euclidean distance and the distance function on M to some fixed point respectively. As the ambient space is a curved pseudo- Riemannian product of two Riemannian manifolds （∑1,g1） and （∑2,g2） of dimensions n and m, a Bernstein type result for n =2 under some curvature conditions on E1 and E2 and the growth condition w = o（r） is also got. As more general cases, under some curvature conditions on the ambient space and the growth condition w = o（r） or w = o（√r）, the author concludes that if M has parallel mean curvature, then M is maximal.
基金supported by the National Natural Science Foundation of China (No. 10971028)
文摘The authors obtain various versions of the Omori-Yau＇s maximum principle on complete properly immersed-submanifolds with controlled mean curvature in certain product manifolds, in complete Riemannian manifolds whose k-Ricci curvature has strong quadratic decay, and also obtain a maximum principle for mean curvature flow of complete manifolds with bounded mean curvature. Using the generalized maximum principle, an estimate on the mean curvature of properly immersed submanifolds with bounded projection in N1 in the product manifold N1 x N2 is given. Other applications of the generalized maximum principle are also given.
基金supported by the National Natural Science Foundation of China(Nos.11301399,11126189,11171259,11126190)the Specialized Research Fund for the Doctoral Program of Higher Education(No.20120141120058)+1 种基金the China Postdoctoral Science Foundation(No.20110491212)the Fundamental Research Funds for the Central Universities(Nos.2042011111054,20420101101025)
文摘In this paper,the authors consider a family of smooth immersions Ft : Mn→Nn+1of closed hypersurfaces in Riemannian manifold Nn+1with bounded geometry,moving by the Hkmean curvature flow.The authors show that if the second fundamental form stays bounded from below,then the Hkmean curvature flow solution with finite total mean curvature on a finite time interval [0,Tmax)can be extended over Tmax.This result generalizes the extension theorems in the paper of Li(see "On an extension of the Hkmean curvature flow,Sci.China Math.,55,2012,99–118").
