Non-Positive Curvature and Global Invertibility of Maps

Let F:M→N be a C ¹ map between Riemannian manifolds of the same dimension, M complete, N Cartan–Hadamard. We show that F is a C ¹ diffeomorphism if inf _x∈M |d(B _ζ °F)(x)|>0 for all ζ∈N(∞) and Busemann functions B _ζ . This generalizes the Cartan–Hadamard theorem and the Hadamard invertibility criterion, which requires inf _x∈M ∥DF(x)^?1∥^?1=inf _ζ∈N(∞)inf _x∈M |d(B _ζ °F)(x)|>0. Our proofs use a version of the shooting method for two-point boundary value problems. These ideas lead to new results about the size of the critical set of a function f∈C ²(? ⁿ ,?): a) If \(\inf_{x\in \mathbb{R}^{n}}|\operatorname{Hess} f(x)v|>0\) for all v≠0 then the function f has precisely one critical point. (b) If g∈C ²(? ⁿ ,?) is the C ¹ local uniform limit of functions as in a), and \(\operatorname{Hess} g(x)\) is nowhere singular, then g has at most one critical point. The totality of functions described in (b) properly contains the class consisting of all C ² strictly convex functions defined on ? ⁿ .     

Harmonic Maps Over Rings

For the torsion-free modules over noncommutative principal ideal domains von Staudt's theorem is proved. Moreover, more general (nonbijective) harmonic maps with the classical definition of harmonic quadruple is calculated.     

指数型调和映射

证明了在适当条件下,指数型能量和指数型调和映射是共形不变的.我们主要研究了指数型调和的黎曼淹没和等距浸入,还研究了与黎曼等距浸入相关的高斯映射是指数型调和.     

Uniqueness of Subelliptic Harmonic Maps

Let R^m be an open set, Nⁿ a Riemannian manifold, X a collection of vector fields on , and f a smooth map from into Nⁿ. We call f a subelliptic harmonic map if it is a critical point of the energy functional with respect to X. In this paper, we calculate the first and the second variations of the energy functional, and use them to prove the partial uniqueness of a subelliptic harmonic map under the condition that Nⁿ has the non-positive curvature. Then, we utilize the maximum principle for subelliptic PDEs to verify the global uniqueness of a subelliptic harmonic map under some other conditions.     

The Curvature and Index of Completely Positive Maps

We study conjugacy invariants for completely positive maps thatare inspired by the concept of curvature introduced for commutingd-tuples of contractions by Arveson. 2000 Mathematics SubjectClassification 46L53, 46L55, 46L57, 46L87, 47L55, 46L07, 46L08     

A Remark on Harmonic Maps

本文以二维球面到Kahler流形的调和映照为研究对象。这个课题是近年来引起很多数学家和物理学家关注的题目。 利用稳定性条件，我们获得了一个积分不等式。作为它的一个应用，我们可以证明肖荫堂和丘成桐的一个关于调和映照全纯性的著名定理。     

Harmonic Maps Between Alexandrov Spaces

In this paper, we shall discuss the existence, uniqueness and regularity of harmonic maps from an Alexandrov space into a geodesic space with curvature \(\leqslant 1\) in the sense of Alexandrov.     

Harmonic Maps on Hadamard Surfaces

Suppose (,g) is a simply connected, complete and noncompact surface. We call it is aHadamard surface if its Gaussian curwature is between two negative constants. By the Uniformization Theorem, we know that such surfaces are always conformal to 2-dAn hyperbolicspaces (D, ds2p). Here D is the unit disc on the plane or the upper half plane, and ds2p is thePoincare metric on it. In this note, we are interested in the Dirichlet problem of harmonic mapson such surfaces.Due to the conformal invar…     

稳定调和映照的不存在性

讨论了从Finsler流形到Riemannian流形的稳定调和映照的不存在性,得到了一个拼挤定理.     

Asymptotic Boundary Value Problem of Harmonic Maps via Harmonic Boundary

We prove that given any continuous data f on the harmonic boundary of a complete Riemannian manifold with image within a ball in the normal range, there exists a harmonic map from the manifold into the ball taking the same boundary value at each harmonic boundary point as that of f.     

三种变形调和级数的收敛性

给出了三种变形调和级数,证明了相应的收敛性,并给出了其收敛值的计算方法.     

Harmonic Maps and Critical Points of Penalized Energy

We discuss a sequence solutions u_ε for the E-L equations of the penalized energy defined by Chen-Struwe. We show that the blow-up set of u_ε is a H^{m-2} - rectifiable set and its weak limit satisfies a blow-up formula. Consequently, the weak limit will be a stationary harmonic map if and only if the blow-up set is stationary.     

Stable Stationary Harmonic Maps to Spheres

For k ≥ 3, we establish new estimate on Hausdorff dimensions of the singular set of stable-stationary harmonic maps to the sphere S^k. We show that the singular set of stable-stationary harmonic maps from B5 to 83 is the union of finitely many isolated singular points and finitely many HSlder continuous curves. We also discuss the minimization problem among continuous maps from B^n to S^2.     

关于调和映照的一个Liouville型定理

令M、N是完备Remann流形,设M上不存在任何非平凡的有界调和函数,N的截面曲率具有上界其为k＞0.设uM→N是一个调和映照且u(M)∈BR(P),其中R＝·如果BR(P)位于P的割迹之内,并且μ（M）∩эBR(P)最多只有一个点,则u必是一个常值映照.     

Harmonic Maps from Complete Noncompact Manifolds

ln this paper we prove some general existence theorems of harmonic maps from complete noncompact manifolds with tho positive lower bounds of spectrum into convex balls. We solve the Dirichlet problem in classical domains and some special complete noncompact manifolds for harmonic maps into convex balls. We also study the existence of harmonic maps from some special complete noncompact manifolds into complete manifolds with nonpositive sectional curvature which are not simply connected.     

A Bridge Principle for Harmonic Maps

We prove a bridge principle for harmonic maps between generalmanifolds.     

A Semigroup Approach to Harmonic Maps

We present a semigroup approach to harmonic maps between metric spaces. Our basic assumption on the target space (N,d) is that it admits a barycenter contraction, i.e. a contracting map which assigns to each probability measure q on N a point b(q) in N. This includes all metric spaces with globally nonpositive curvature in the sense of Alexandrov as well as all metric spaces with globally nonpositive curvature in the sense of Busemann. It also includes all Banach spaces.The analytic input comes from the domain space (M,) where we assume that we are given a Markov semigroup (p_t)_t>0. Typical examples come from elliptic or parabolic second-order operators on Rⁿ, from Lévy type operators, from Laplacians on manifolds or on metric measure spaces and from convolution operators on groups. In contrast to the work of Korevaar and Schoen (1993, 1997), Jost (1994, 1997), Eells and Fuglede (2001) our semigroups are not required to be symmetric.The linear semigroup acting, e.g., on the space of bounded measurable functions u:MR gives rise to a nonlinear semigroup (P_t^*)_t acting on certain classes of measurable maps f:MN. We will show that contraction and smoothing properties of the linear semigroup (p_t)_t can be extended to the nonlinear semigroup (P_t^*)_t, for instance, L_p–L_q smoothing, hypercontractivity, and exponentially fast convergence to equilibrium. Among others, we state existence and uniqueness of the solution to the Dirichlet problem for harmonic maps between metric spaces. Moreover, for this solution we prove Lipschitz continuity in the interior and Hölder continuity at the boundary.Our approach also yields a new interpretation of curvature assumptions which are usually required to deduce regularity results for the harmonic map flow: lower Ricci curvature bounds on the domain space are equivalent to estimates of the L₁-Wasserstein distance between the distribution of two Brownian motions in terms of the distance of their starting points; nonpositive sectional curvature on the target space is equivalent to the fact that the L₁-Wasserstein distance of two distributions always dominates the distance of their barycenters.Dedicated to the memory of Professor Dr. Heinz Bauer