Relative isoperimetric inequality and¶linear isoperimetric inequality for minimal submanifolds

We prove an optimal relative isoperimetric inequality

Erratum to: “On biminimal submanifolds in nonpositively curved manifolds” [Differ. Geom. Appl. 35 (2014) 1–8]

We correct theorems of Luo (2014) [1], concerning nonexistence of complete biminimal submanifolds in nonpositive curvature space forms, and Lemma 4.2 in Luo (2014) [1].     

Examples of 4-manifolds with almost nonpositive curvature

We construct an infinite family of non-homeomorphic 4-manifolds with almost nonpositive sectional curvature whose universal covering space is not contractible. As a consequence, these manifolds do not support metrics with nonpositive sectional curvature. To achieve this, we use a generalization of Bavard's surgery construction, combined with an open book decomposition and knot theory.     

Improved Chen–Ricci inequality for curvature-like tensors and its applications

We present Chen–Ricci inequality and improved Chen–Ricci inequality for curvature like tensors. Applying our improved Chen–Ricci inequality we study Lagrangian and Kaehlerian slant submanifolds of complex space forms, and C-totally real submanifolds of Sasakian space forms.     

非正曲率流形及其子流形上有界区域的特征值

本文研究了完备单连通具有非正曲率黎曼流形及其子流形上有界区域的特征值问题.利用广义Hessian比较定理,获得了局部特征值的下界估计式,将McKean[2]的定理在局部上推广到了非正曲率的情形.     

Submanifolds with restrictions on extrinsic <Emphasis Type="Italic">q</Emphasis>th scalar curvature

We study the structure of the minimum set of the normal curvature for a symmetric bilinear map on Euclidean or Hilbert space, the conditions when this set contains strongly umbilical, conformal nullity, etc. linear subspaces. The main goals are estimates from above of the codimension of these subspaces for a symmetric bilinear map with positive normal curvature and the inequality type restriction on the extrinsic qth scalar curvature. We estimate from above the codimension of asymptotic and relative nullity subspaces for a symmetric bilinear map with nonpositive extrinsic qth scalar curvature. Applying the algebraic results to the second fundamental form of a submanifold with low codimension, we characterize the totally umbilical and totally geodesic submanifolds, prove local nonembedding theorems for the products of Riemannian manifolds and global extremal theorem for the space of positive curvature. On the way we generalize results by Florit (1994), Borisenko (1977, 1987) and Okrut (1991) about Riemannian and Hilbert submanifolds. The research was supported by the Center for Absorption in Science, Ministry of Immigrant Absorption, State of Israel; and by Center for Computational Mathematics and Scientific Computation, University of Haifa.     

Complete submanifolds with parallel mean curvature vector in hyperbolic spaces

In this paper we proved a better estimate as well as generalized to higher codimensions of a theorem of Y.B. Shen on complete submanifolds with parallel mean curvature vector in a hyperbolic space.     

Submanifolds with flat normal bundle in spaces of constant curvature

In the present paper parallel submanifolds and focal points of a given submanifold with flat normal bundle are discussed provided that the ambient space has constant sectional curvature. We present shape operators of parallel submanifolds with respect to arbitrary normal vectors. Furthermore, we prove that the focal points of a submanifold with flat normal bundle form totally geodesic hypersurfaces in the normal submanifolds.Supported by Hungarian Nat. Found. for Sci. Research Grant No. 1615 (1991).Dedicated to Professor J. Strommer on the occasion of his 75th birthday     

The lightlike flat geometry on spacelike submanifolds of codimension two in Minkowski space

We introduce the notion of the lightcone Gauss–Kronecker curvature for a spacelike submanifold of codimension two in Minkowski space, which is a generalization of the ordinary notion of Gauss curvature of hypersurfaces in Euclidean space. In the local sense, this curvature describes the contact of such submanifolds with lightlike hyperplanes. We study geometric properties of such curvatures and show a Gauss–Bonnet type theorem. As examples we have hypersurfaces in hyperbolic space, spacelike hypersurfaces in the lightcone and spacelike hypersurfaces in de Sitter space.     

黎曼流形中紧子流形的拼挤定理

本利用几何不等式和曲率估计的方法，证明了黎曼流形N^n＋p,上的具有平行平均曲率的紧子流形M^n上的一个拼挤定理。若N上的截曲率KN满足- 1≤ KN≤δ≤0，且‖S- nH2‖n/2, ‖ S-nH^2‖n/n-s满足一些不等式，则δ= - 1。     

Convex functions on Grassmannian manifolds and Lawson-Osserman problem

We derive estimates of the Hessian of two smooth functions defined on Grassmannian manifold. Based on it, we can derive curvature estimates for minimal submanifolds in Euclidean space via Gauss map as in [Y.L. Xin, Ling Yang, Curvature estimates for minimal submanifolds of higher codimension, arXiv: 0709.3686; 24]. In this way, the result for Bernstein type theorem done by Jost and the first author could be improved.     

The Gauss map of submanifolds in the Heisenberg group

We obtain criteria for the harmonicity of the Gauss map of submanifolds in the Heisenberg group and consider the examples demonstrating the connection between the harmonicity of this map and the properties of the mean curvature field. Also, we introduce a natural class of cylindrical submanifolds and prove that a constant mean curvature hypersurface with harmonic Gauss map is cylindrical.     

Rigidity and sphere theorem for manifolds with positive Ricci curvature

LetM be a complete Riemannian manifold with Ricci curvature having a positive lower bound. In this paper, we prove some rigidity theorems forM by the existence of a nice minimal hypersurface and a sphere theorem aboutM. We also generalize a Myers theorem stating that there is no closed immersed minimal submanifolds in an open hemisphere to the case that the ambient space is a complete Riemannian manifold withk-th Ricci curvature having a positive lower bound. Supported by the JSPS postdoctoral fellowship and NSF of China     

Angle theorems for the Lagrangian mean curvature flow

We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle for the corresponding Lagrangian submanifold in the cross product space satisfies . If one considers a 4-dimensional K?hler-Einstein manifold of nonpositive scalar curvature that admits two complex structures J, K which commute and assumes that is a compact oriented Lagrangian submanifold w.r.t. J such that the K?hler form w.r.t.K restricted to L is positive and , then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. . Received: 11 April 2001 / Published online: 29 April 2002     

About topology of saddle submanifolds

The (k,ε)-saddle (in particular, k-saddle, i.e. ε=0) submanifolds are defined in terms of eigenvalues of the second fundamental form. This class extends the class of submanifolds with extrinsic curvature bounded from above, i.e. ?ε² (in particular, non-positive) and small codimension. We study s-connectedness and (co)homology properties of compact submanifolds with ‘small’ normal curvature and saddle submanifolds in Riemannian spaces of positive (sectional or qth Ricci) curvature. The main results are that a submanifold or the intersection of two submanifolds is s-connected under some assumption. By the way, theorems by T. Frankel and some recent results by B. Wilking, F. Fang, S. Mendonça and X. Rong are generalized.     

A rigidity theorem for a space-like graph¶of higher codimension

We prove a rigidity theorem for a space-like graph with parallel mean curvature of arbitrary dimension and codimension in pseudo-Euclidean space via properties of its harmonic Gauss map. We also give an estimate of the squared norm of the second fundamental form in terms of the mean curvature and the image diameter under the Gauss map for space-like submanifolds with parallel mean curvature in pseudo-Euclidean space. The estimate also implies the former theorem. Received: 10 December 1999     

A Möbius characterization of submanifolds in real space forms with parallel mean curvature and constant scalar curvature

In this paper, we give a Möbius characterization of submanifolds in real space forms with parallel mean curvature vector fields and constant scalar curvatures, generalizing a theorem of H. Li and C.P. Wang in [LW1].Supported by NSF of Henan, P. R. China     

共形空间${\mathbb Q}^n_s$中的正则Blaschke拟全脐子流形

[Nie C X,Wu C X,Regular submanifolds in the conformal space Q_p~n,ChinAnn Math,2012,33B(5):695-714]中研究了共形空间Q_s~n中的正则子流形,并引入了共形空间Q_s~n中的子流形理论.本文作者将分类共形空间Q_s~n中的Blaschke拟全脐子流形,证明伪Riemann空间形式中具有常数量曲率和平行的平均曲率向量场的正则子流形是共形空间中的Blaschke拟全脐子流形;反之,共形空间中的Blaschke拟全脐子流形共形等价于伪Riemann空间形式中具有常数量曲率和平行的平均曲率向量场的正则子流形.这一结论可看作是共形空间Q_s~n中共形迷向子流形分类定理的推广.     

The convex hull property and topology of hypersurfaces with nonnegative curvature

We prove that, in Euclidean space, any nonnegatively curved, compact, smoothly immersed hypersurface lies outside the convex hull of its boundary, provided the boundary satisfies certain required conditions. This gives a convex hull property, dual to the classical one for surfaces with nonpositive curvature. A version of this result in the nonsmooth category is obtained as well. We show that our boundary conditions determine the topology of the surface up to at most two choices. The proof is based on uniform estimates for radii of convexity of these surfaces under a clipping procedure, a noncollapsing convergence theorem, and a gluing procedure.     

Wu's theorem for Kähler–Finsler spaces

In this paper, we study strongly convex Kähler–Finsler manifolds. We prove two theorems: A strongly convex Kähler–Berwald manifold with a pole is a Stein manifold if it has nonpositive horizontal radial flag curvature; A strongly convex Kähler–Finsler manifold with a complex pole is a Stein manifold if it has nonpositive horizontal radial flag curvature.