A Note on the Stability of Geodesics on Diffeomorphism Groups with One-side Invariant Metric

In this note,we consider the stability of geodesics on volume-preserving diffeomorphism groups with one-side invariant metric.We showed that for non-Beltrami fields on a three-dimensional compact manifold,there does not exist Eulerian stable flow which is Lagrangian exponential unstable.We noticed that a stationary flow corresponding to the KdV equation can be Eulerian stable while the corresponding motion of the fluid is at most exponentially unstable.     

HYPERSURFACES IN SPACE FORMS WITH SCALAR CURVATURE CONDITIONS

Let f : M^n→S^n 1真包含于R^n 2 be an n-dimensional complete oriented Riemannian manifold minimally immersed in an (n 1)-dimensional unit sphere S^n 1. Denote by S^n 1 the upper closed hemisphere. If f(M^n)包含于S ^n 1, then under some curvature conditions the authors can get that the isometric immersion is a totally embedding. They also generalize a theorem of Li Hai Zhong on hypersurface of space form with costant scalar curvature.     

Liouville Type Theorem About p-Harmonic Function and p-Harmonic Map with Finite L-Energy

This paper deals with the p-harmonic function on a complete non-compact submanifold M isometrically immersed in an (n + k)-dimensional complete Riemannian manifold (M) of non-negative (n-1)-th Ricci curvature.The Liouville type theorem about the p-harmonic map with finite Lq-energy from complete submanifold in a partially nonnegatively curved manifold to non-positively curved manifold is also obtained.     

Translating Solutions of the Nonparametric Mean Curvature Flow with Nonzero Neumann Boundary Data in Product Manifold Mn × R*

In this paper, the authors can prove the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary data in a prescribed product manifold Mⁿ × R, where Mⁿ is an n-dimensional (n ≥ 2) complete Riemannian manifold with nonnegative Ricci curvature, and R is the Euclidean 1-space.     

RANDERS SPACES WITH SCALAR FLAG CURVATURE

Let(M, F) be an n-dimensional Randers space with scalar flag curvature. In this paper, we will introduce the definition of a weak Einstein manifold. We can prove that if(M, F) is a weak Einstein manifold, then the flag curvature is constant.     

New geometric flows on Riemannian manifolds and applications to Schrödinger-Airy flows

In this paper,a class of new geometric flows on a complete Riemannian manifold is defined. The new flow is related to the generalized(third order) Landau-Lifshitz equation. On the other hand it could be thought of as a special case of the Schr¨odinger-Airy flow when the target manifold is a K¨ahler manifold with constant holomorphic sectional curvature. We show the local existence of the new flow on a complete Riemannian manifold with some assumptions on Ricci tensor. Moreover,if the target manifolds are Einstein or some certain type of locally symmetric spaces,the global results are obtained.     

Volume-preserving Fields and Reeb Fields on 3-manifolds

Volume-preserving field X on a 3-manifold is the one that satisfies LxΩ = 0 for some volume Ω. The Reeb vector field of a contact form is of volume-preserving, but not conversely. On the basis of Geiges-Gonzalo＇s parallelization results, we obtain a volume-preserving sphere, which is a triple of everywhere linearly independent vector fields such that all their linear combinations with constant coefficients are volume-preserving fields. From many aspects, we discuss the distinction between volume-preserving fields and Reeb-like fields. We establish a duality between volume-preserving fields and h-closed 2-forms to understand such distinction. We also give two kinds of non-Reeb-like but volume-preserving vector fields to display such distinction.     

Existence and bifurcation of integral manifolds with applications

In this paper a bifurcation theorem on the existence of integral manifolds is obtained by using contracting principle. As an application, sufficient conditions for a higher dimensional system to have an integral manifold are given. Especially the existence and uniqueness of a 3-dimensional invariant torus appearing in a 4-dimensional autonomous system with singularity of codimension two are proved.     

W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds Dedicated to Professor Zhiming Ma on His Seventieth Birthday

In this survey paper, we give an overview of our recent works on the study of the W-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on the Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula for the Ricci flow, we prove the W-entropy formula for the heat equation associated with the Witten Laplacian on n-dimensional complete Riemannian manifolds with the CD(K, m)-condition, and the W-entropy formula for the heat equation associated with the time-dependent Witten Laplacian on n-dimensional compact manifolds equipped with a(K, m)-super Ricci flow, where K ∈ R and m ∈ [n, ∞]. Furthermore, we prove an analogue of the W-entropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds.Our result improves an important result due to Lott and Villani(2009) on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two W-entropy formulas, we introduce the Langevin deformation of geometric flows on the tangent bundle over the Wasserstein space and prove an extension of the W-entropy formula for the Langevin deformation. We also make a discussion on the W-entropy for the Ricci flow from the point of view of statistical mechanics and probability theory. Finally, to make this survey more helpful for the further development of the study of the W-entropy, we give a list of problems and comments on possible progresses for future study on the topic discussed in this survey.     

INVOLUTIONS FIXING ^mUi=1 CPi（1） ×HPi（n）＊

Let(M,T) be a closed manifold with an involution T.The fixed point set of T is F.In this article,bordism classes of the involutions with fixed point set F = ∪(from i=1 to m) CP i(1) × HP i(n) are determined,where CP(1) and H P(n) denote the 1-dimensional complex projective space and n-dimensional quaternionic projective space respectively,and n = 2p-2 or n = 2p-1(p > 1).     

保持n-形式系统的Lie对称群约化及应用

本文考虑保持ｎ－形式（面积、体积的高维推广概念）的ｎ维向量场,应用Ｌｉｅ群方法对其约化问题进行了系统性研究,得到了下列结果,第一,如何保持ｎ－ｇａａａ的ｎ维向量场具有一个单参数保持ｎ－形式的空间对称群,则可具体地构造出一个与向量场无关变换,便得原向量场约化掉一维,并且该约化向量场保持相应的（ｎ－１）形式,特别ｎ＝３时可直接得到「１」中的重要结果,第二,上述ｎ维向量场如果具有一个ｒ参数保持ｎ－ｇａａ     

A Note on Diffeomorphism Groups of Closed Manifolds

In this short note, we construct a retracting deformation from group of orientation-preserving diffeomorphisms onto the group of volume-preserving diffeomorphismsof a compact smooth manifold. The retracting deformation isrealized by a moment map type of flow with the aid of the theory of quasi-linear parabolic equations.     

On the integrability and perturbation of three-dimensional fluid flows with symmetry

Summary The purpose of this paper is to develop analytical methods for studyingparticle paths in a class of three-dimensional incompressible fluid flows. In this paper we study three-dimensionalvolume preserving vector fields that are invariant under the action of a one-parameter symmetry group whose infinitesimal generator is autonomous and volume-preserving. We show that there exists a coordinate system in which the vector field assumes a simple form. In particular, the evolution of two of the coordinates is governed by a time-dependent, one-degree-of-freedom Hamiltonian system with the evolution of the remaining coordinate being governed by a first-order differential equation that depends only on the other two coordinates and time. The new coordinates depend only on the symmetry group of the vector field. Therefore they arefield-independent. The coordinate transformation is constructive. If the vector field is time-independent, then it possesses an integral of motion. Moreover, we show that the system can be further reduced toaction-angle-angle coordinates. These are analogous to the familiar action-angle variables from Hamiltonian mechanics and are quite useful for perturbative studies of the class of systems we consider. In fact, we show how our coordinate transformation puts us in a position to apply recent extensions of the Kolmogorov-Arnold-Moser (KAM) theorem for three-dimensional, volume-preserving maps as well as three-dimensional versions of Melnikov's method. We discuss the integrability of the class of flows considered, and draw an analogy with Clebsch variables in fluid mechanics.     

A NEW LAPLACIAN COMPRISON THEOREM AND THE ESTIMATE OF EIGENVALUES

Ａ ＮＥＷ ＬＡＰＬＡＣＩＡＮ ＣＯＭＰＲＩＳＯＮ ＴＨＥＯＲＥＭ ＡＮＤ ＴＨＥ ＥＳＴＩＭＡＴＥ ＯＦ ＥＩＧＥＮＶＡＬＵＥＳ ￥ＤＩＮＧＱＩＮＧＡｂｓｔｒａｃｔ：ＴｈｉｓｐａｐｅｒｅｓｔａｂｌｉｓｈｅｓａｎｅｗＬａｐｌａｃｉａｎｃｏｍｐａｒｉｓ...     

关于可三角剖分的紧致流形的模2示嵌类

<正> 引言 关于复合形或更一般的空間在欧氏空間中的实現問題,Whitney和Thom分別有下面的結果: 定理.(Whitney)n維紧致微分流形M~n可微分实現于R~N中的必要条件为 W~k(M~n)=0,k≥N-n.(1) 定理.(Thom)一个有可数基而局部可縮的紧致Hausdorff空間X可以拓扑实現     

球面空间单形的两个几何不等式（英文）

In this paper, by using the theory and method of distance geometry, we study the geometric inequality of a n-dimensional simplex in the spherical space and establish two geometric inequalities involving the edge-length and volume of one simplex and the volume,height and(n-1)-dimensional volume of the side of another simplex in the n-dimensional spherical space. They are the extensions of the results [10] in the n-dimensional Euclidean geometry to the n-dimensional spherical space.     

n-Dimensional submanifolds with an (n−1)-dimensional asymptotic distribution and with 1-asymptotic lines in a space form Rm(k)

In this paper we prove a theorem about the reduction of the codimension of n-dimensional submanifolds of a space form R^m(k), with an (n-1)-dimensional asymptotic distribution and with 1-asymptotic lines (i.e. asymptotic curves for which the (1+1)-dimensional osculating spaces are subspaces of the tangent spaces). We also give a result for (n–1)-asymptotic lines in connection with Enneper's formula. Finally we construct an example.Dedicated to Prof.dr.J.Bilo on his 65th birthday     

Liouville Type Theorem About p-Harmonic Function and p-Harmonic Map with Finite L~q-Energy

This paper deals with the p-harmonic function on a complete non-compact submanifold M isometrically immersed in an(n + k)-dimensional complete Riemannian manifold M of non-negative(n-1)-th Ricci curvature. The Liouville type theorem about the p-harmonic map with finite L~q-energy from complete submanifold in a partially nonnegatively curved manifold to non-positively curved manifold is also obtained.     

Covering lattice points by subspaces

We find tight estimates for the minimum number of proper subspaces needed to cover all lattice points in an n-dimensional convex body C , symmetric about the origin 0. This enables us to prove the following statement, which settles a problem of G. Halász. The maximum number of n-wise linearly independent lattice points in the n-dimensional ball r B n of radius r around 0 is O(rn/(n-1)). This bound cannot be improved. We also show that the order of magnitude of the number of diferent (n - 1)-dimensional subspaces induced by the lattice points in r&Bgr;n is rn/(n-1).