A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian （n ＋ p）-dimensional manifold N^n＋p with negative sectional curvature is proved. For ...A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian （n ＋ p）-dimensional manifold N^n＋p with negative sectional curvature is proved. For given positive integers n（≥ 2）, p and for a constant H satisfying H 〉 1 there exists a negative number τ（n,p, H） ∈ （-1, 0） with the property that if the sectional curvature of N is pinched in [-1, τ-（n,p, H）], and if the squared length of the second fundamental form is in a certain interval, then N^n＋p is isometric to the hyperbolic space H^n＋P（-1）. As a consequence, this submanifold M is congruent to S^n（1√H^2 - 1） or the Veronese surface in S^4（1/√H^2-1）.展开更多
In this paper,we firstly verify that if Mn is an n-dimensional complete self-shrinker with polynomial volume growth in Rn+1,and if the squared norm of the second fundamental form of M satisfies 0≤S-1≤1/18,then S≡1 ...In this paper,we firstly verify that if Mn is an n-dimensional complete self-shrinker with polynomial volume growth in Rn+1,and if the squared norm of the second fundamental form of M satisfies 0≤S-1≤1/18,then S≡1 and M is a round sphere or a cylinder.More generally,let M be a complete λ-hypersurface of codimension one with polynomial volume growth in Rn+1 with λ≠0.Then we prove that there exists a positive constant γ,such that if |λ|≤γ and the squared norm of the second fundamental form of M satisfies0≤S-βλ≤1/18,then S≡βλ,λ> 0 and M is a cylinder.Here βλ=1/2(2+λ2+|λ|(λ2+4)1/2).展开更多
We study microscopic spacetime convexity properties of fully nonlinear parabolic partial differential equations. Under certain general structure condition, we establish a constant rank theorem for the spacetime convex...We study microscopic spacetime convexity properties of fully nonlinear parabolic partial differential equations. Under certain general structure condition, we establish a constant rank theorem for the spacetime convex solutions of fully nonlinear parabolic equations. At last, we consider the parabolic convexity of solutions to parabolic equations and the convexity of the spacetime second fundamental form of geometric flows.展开更多
Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constan...Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constantγ(n)depending only on n such that if|H|≤γ(n)andβ(n,H)≤S≤β(n,H)+n/18,then S≡β(n,H)and M is a Clifford torus.Here,β(n,H)=n+n^(3)/2(n-1)H^(2)+n(n-2)/2(n-1)(1/2)n^(2)H^(4)+4(n-1)H^(2).展开更多
In this paper, we study the complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded λ-hypersurfaces with |A|...In this paper, we study the complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded λ-hypersurfaces with |A|≤α and get some applications of the volume comparison theorem. Secondly, we consider the relation among λ, extrinsic radius k, intrinsic diameter d, and dimension n of the complete λ-hypersurface,and we obtain some estimates for the intrinsic diameter and the extrinsic radius. At last, we get some topological properties of the bounded λ-hypersurface with some natural and general restrictions.展开更多
基金Research supported by the NSFC (10231010)Trans-Century Training Programme Foundation for Talents by the Ministry of Education of ChinaNatural Science Foundation of Zhejiang Province (101037).
文摘A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian （n ＋ p）-dimensional manifold N^n＋p with negative sectional curvature is proved. For given positive integers n（≥ 2）, p and for a constant H satisfying H 〉 1 there exists a negative number τ（n,p, H） ∈ （-1, 0） with the property that if the sectional curvature of N is pinched in [-1, τ-（n,p, H）], and if the squared length of the second fundamental form is in a certain interval, then N^n＋p is isometric to the hyperbolic space H^n＋P（-1）. As a consequence, this submanifold M is congruent to S^n（1√H^2 - 1） or the Veronese surface in S^4（1/√H^2-1）.
基金National Natural Science Foundation of China (Grant Nos. 11531012, 11371315 and 11601478)the China Postdoctoral Science Foundation (Grant No. 2016M590530)。
文摘In this paper,we firstly verify that if Mn is an n-dimensional complete self-shrinker with polynomial volume growth in Rn+1,and if the squared norm of the second fundamental form of M satisfies 0≤S-1≤1/18,then S≡1 and M is a round sphere or a cylinder.More generally,let M be a complete λ-hypersurface of codimension one with polynomial volume growth in Rn+1 with λ≠0.Then we prove that there exists a positive constant γ,such that if |λ|≤γ and the squared norm of the second fundamental form of M satisfies0≤S-βλ≤1/18,then S≡βλ,λ> 0 and M is a cylinder.Here βλ=1/2(2+λ2+|λ|(λ2+4)1/2).
基金supported by National Natural Science Foundation of China(Grant No.10871187)
文摘We study microscopic spacetime convexity properties of fully nonlinear parabolic partial differential equations. Under certain general structure condition, we establish a constant rank theorem for the spacetime convex solutions of fully nonlinear parabolic equations. At last, we consider the parabolic convexity of solutions to parabolic equations and the convexity of the spacetime second fundamental form of geometric flows.
基金supported by National Natural Science Foundation of China(Grant No.11531012)China Postdoctoral Science Foundation(Grant No.BX20180274)Natural Science Foundation of Zhejiang Province(Grant No.LY20A010024)。
文摘Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constantγ(n)depending only on n such that if|H|≤γ(n)andβ(n,H)≤S≤β(n,H)+n/18,then S≡β(n,H)and M is a Clifford torus.Here,β(n,H)=n+n^(3)/2(n-1)H^(2)+n(n-2)/2(n-1)(1/2)n^(2)H^(4)+4(n-1)H^(2).
基金supported by National Natural Science Foundation of China (Grant No. 11271343)
文摘In this paper, we study the complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded λ-hypersurfaces with |A|≤α and get some applications of the volume comparison theorem. Secondly, we consider the relation among λ, extrinsic radius k, intrinsic diameter d, and dimension n of the complete λ-hypersurface,and we obtain some estimates for the intrinsic diameter and the extrinsic radius. At last, we get some topological properties of the bounded λ-hypersurface with some natural and general restrictions.
