紧致齐性空间上的调和分析（Ⅰ）：富里埃级数的Poisson求和

首先介绍了紧致齐性空间上调和分析的若干基础性结果,并给出这些结果的较简洁的证明。接着,我们定义了紧致齐性空间上函数的卷积(熟知n维球面是一个紧致齐性空间),这一定义看来对研究紧致齐性空间上的调和分析向题是相当有用的。最后,用定义的卷积,研究了紧致齐性空间上Fourier级数的Poisson求和。     

一类调和映射的边界正则性

讨论一类映入球面的满足拟单调不等式的弱调和映射的边界正则性。利用函数的延拓技巧以及Hardy空间和BMO空间的对偶性，对这类弱调和映射的边界正则性给出一个简明的证明。     

带有齐性核的分数次积分算子在Hardy型空间上的有界性

该文研究了带有齐性核的分数次积分算子T_(Ω,α)在一些Hardy空间上的映射性质,其中核Ω在球面S~(n-1)上满足一些L~S-Dini条件.作者将前人的一些结果改进到0αn情形,同时还得到了算子T_(Ω,α)在Herz型Hardy空间上的一个端点估计.     

Grassmann流形作为子流形的微分几何

本文把Grassmann流形看作等距地嵌入在单位球面内的子流形,建立它的基本公式,然后证明它的极小性质.此外,利用这种嵌入把欧氏空间中子流形的Gauss映射看作到单位球面内的映射,并建立了这种广义的Gauss映射是调和映射的条件.     

靶流形为球面子流形的调和映射的量子化现象(英文)

本文研究了调和映射和极小子流形的量子化性质.通过运用谱分解方法,获得了靶流形为球面子流形的调和映射的量子化性质,然后将其应用到球面的极小子流形的高斯映射,得到了极小子流形的第二基本形式的量子化性子.     

一类特殊空间上映射的梯度几何流柯西问题解的存在唯一性

通过几何分析方法与抛物型方程组解的逼近理论,研究特殊空间(一维球面S~1到二维球面S~2)上映射的梯度几何流柯西问题解的存在唯一性.利用能量法和空间本身特有的性质来解决能量守恒的问题,并利用适当的抛物型方程组逼近该梯度几何流,在适当的Sobolev空间中建立先验估计,找到其时间的一致正下界和抛物型方程组一列解的Sobo1ev范数的一致边界,借助于抛物型偏微分方程的理论,以此决定该柯西问题解的存在唯一性.     

二维严格凸赋范空间单位球面间等距映射的线性延拓

主要研究二维严格凸实赋范空间E和F的单位球面S_1(E)和S_1(F)之间的等距映射的线性延拓问题.利用二维严格凸赋范空间单位球面的性质得到:若等距映射V_0:S_1(E)→S_1(F)满足一定条件,则V_0可延拓为全空间E上的线性等距映射V:E→F.     

关于sn-度量空间

本文利用ｓｎ－度量空间的一些等价刻划及ｓｎ－度量空间与ｇ－度量空间、度量空间之间的关系，研究了ｓｎ－度量空间的一些映射性质，证明了ｓｎ－度量空间的闭象是ｓｎ－度量空间当且仅当它是ｓｎ－第一可数的，利用这一结果证明了有限到一闭映射和开闭映射均保持ｓｎ－度量空间，并给出反例说明完备映射不保持ｓｎ－度量空间，本文还证明了ｓｎ－变量空问满足完备逆象Ｇδ－对角线定理．     

bp(2)空间中的等距映射

度量与线性性质是赋范空间的重要性质,因此,研究线性算子与等距算子的关系成为了泛函分析领域重要的研究课题.本文首先研究一类特殊的赋准范空间,即bp(2)空间的重要性质.然后给出bp(2)空间单位球面间满等距映射的表示定理及延拓性质.     

复 Grassmann 流形在球面中的等距嵌入

本文考察复 Grassmann 流形作为单位球面内的等距嵌入子流形的几何性质,并且给出了从体积有限的 Riemann 流形到复 Grassmann 流形的一个调和映射为常值映射的条件。     

On p -harmonic morphisms

In this paper, we study the characterisation of p -harmonic morphisms between Riemannian manifolds, in the spirit of Fuglede-Ishihara. After a result establishing that p -harmonic morphisms are precisely horizontally weakly conformal p -harmonic maps, we compare ( 2 -)harmonic morphisms and p -harmonic morphisms ( p>2 ).     

f-Harmonic Morphisms Between Riemannian Manifolds

f-Harmonic maps were first introduced and studied by Lichnerowicz in 1970. In this paper, the author studies a subclass of f-harmonic maps called f-harmonic morphisms which pull back local harmonic functions to local f-harmonic functions. The author proves that a map between Riemannian manifolds is an f-harmonic morphism if and only if it is a horizontally weakly conformal f-harmonic map. This generalizes the well-known characterization for harmonic morphisms. Some properties and many examples as well as some non-existence of f-harmonic morphisms are given. The author also studies the f-harmonicity of conformal immersions.     

Bubbling phenomena of Palais-Smale-like sequences ofm-harmonic type systems

In this paper, we study the bubbling phenomena of weak solution sequences of a class of degenerate quasilinear elliptic systems ofm-harmonic type. We prove that, under appropriate conditions, the energy is preserved during the bubbling process. The results apply tom-harmonic maps from a closed Riemannian manifoldM to a Riemannian homogeneous space, and tom-harmonic maps with constant volumes, and also to certain Palais-Smale sequences.     

Existence and Liouville theorems for V -harmonic maps from complete manifolds

We establish existence and uniqueness theorems for V-harmonic maps from complete noncompact manifolds. This class of maps includes Hermitian harmonic maps, Weyl harmonic maps, affine harmonic maps, and Finsler harmonic maps from a Finsler manifold into a Riemannian manifold. We also obtain a Liouville type theorem for V-harmonic maps. In addition, we prove a V-Laplacian comparison theorem under the Bakry-Emery Ricci condition.     

Harmonic maps of foliated Riemannian manifolds

We study ${({\mathcal{F}}, {\mathcal{G}})}$ -harmonic maps between foliated Riemannian manifolds ${(M, {\mathcal{F}}, g)}$ and ${(N, {\mathcal{G}}, h)}$ i.e. smooth critical points ? : M → N of the functional ${E_T (\phi ) = \frac{1}{2} \int_M \| d_T \phi \|^2 \,d \, v_g}$ with respect to variations through foliated maps. In particular we study ${({\mathcal{F}}, {\mathcal{G}})}$ -harmonic morphisms i.e. smooth foliated maps preserving the basic Laplace equation Δ _B u =  0. We show that CR maps of compact Sasakian manifolds preserving the Reeb flows are weakly stable ${({\mathcal{F}}, {\mathcal{G}})}$ -harmonic maps. We study ${({\mathcal{F}}, {\mathcal{G}}_0 )}$ -harmonic maps into spheres and give foliated analogs to Solomon’s (cf., J Differ Geom 21:151–162, 1985) results.     

A remark on convex functions andp-harmonic maps

We prove that the constant maps are the onlyp-harmonic maps for anyp 2 from an arbitrary compact Riemannian manifold into a complete Riemannian manifold which admits a strictly convex function.     

Regularity for the evolution of p-harmonic maps

This paper presents our study of regularity for p-harmonic map heat flows. We devise a monotonicity-type formula of scaled energy and establish a criterion for a uniform regularity estimate for regular p-harmonic map heat flows. As application we show the small data global in the time existence of regular p-harmonic map heat flow.     

f-Harmonic maps of doubly warped product manifolds

In this paper, we study f-harmonicity of some special maps from or into a doubly warped product manifold. First we recall some properties of doubly twisted product manifolds. After showing that the inclusion maps from Riemannian manifolds M and N into the doubly warped product manifold M × _μ,λ N can not be proper f-harmonic maps, we use projection maps and product maps to construct nontrivial f-harmonic maps. Thus we obtain some similar results given in [21], such as the conditions for f-harmonicity of projection maps and some characterizations for non-trivial f-harmonicity of the special product maps. Furthermore, we investigate non-trivial f-harmonicity of the product of two harmonic maps.     

Heat Flows of Subelliptic Harmonic Maps into Riemannian Manifolds with Nonpositive Curvatures

In this paper, we discuss the heat flows of subelliptic harmonic maps into Riemannian manifolds with nonpositive curvatures, and prove the homotopic existence which is a generalization of the Eells–Sampson theorem.     

Liouville Theorems for F-Harmonic Maps and Their Applications

We prove several Liouville theorems for F-harmonic maps from some complete Riemannian manifolds by assuming some conditions on the Hessian of the distance function, the degrees of F(t) and the asymptotic behavior of the maps at infinity. In particular, the results can be applied to F-harmonic maps from some pinched manifolds, and can deduce a Bernstein type result for an entire minimal graph.