Boundary Multifractal Behaviour for Harmonic Functions in the Ball

It is well known that if h is a nonnegative harmonic function in the ball of $\mathbb R^{d+1}$ or if h is harmonic in the ball with integrable boundary values, then the radial limit of h exists at almost every point of the boundary. In this paper, we are interested in the exceptional set of points of divergence and in the speed of divergence at these points. In particular, we prove that for generic harmonic functions and for any β?∈?[0,d], the Hausdorff dimension of the set of points ξ on the sphere such that h(rξ) looks like (1???r)^???β is equal to d???β.     

Harmonic maps on Trans-Sasaki manifolds

In this paper we prove that a (?,J)-holomorphic mapf:M→N (i.e.f _*o?=Jof _*) from a Trans-Sasaki manifold to a nearly Kähler manifold is a harmonic map. We also study the stability of a such map whenM is a compact Trans-Sasaki manifold andN is a Kähler manifold.     

Levi Harmonic Maps of Contact Riemannian Manifolds

We study Levi harmonic maps, i.e., C ^∞ solutions f:M→M′ to \(\tau_{\mathcal{H}} (f) \equiv \operatorname{trace}_{g} ( \varPi_{\mathcal{H}}\beta_{f} ) = 0\) , where (M,η,g) is an (almost) contact (semi) Riemannian manifold, M′ is a (semi) Riemannian manifold, β _f is the second fundamental form of f, and \(\varPi_{\mathcal{H}} \beta_{f}\) is the restriction of β _f to the Levi distribution \({\mathcal{H}} = \operatorname{Ker}(\eta)\) . Many examples are exhibited, e.g., the Hopf vector field on the unit sphere S ²ⁿ⁺¹, immersions of Brieskorn spheres, and the geodesic flow of the tangent sphere bundle over a Riemannian manifold of constant curvature 1 are Levi harmonic maps. A CR map f of contact (semi) Riemannian manifolds (with spacelike Reeb fields) is pseudoharmonic if and only if f is Levi harmonic. We give a variational interpretation of Levi harmonicity. Any Levi harmonic morphism is shown to be a Levi harmonic map.     

带自由边界调和映射的Liouville型定理

本文给出一类带由边界的调和映射的Liouville型定理,这种类型的定理在微分几何的一些问题中有十分重要的应用.我们通过对调和映射的能量选取特殊的变分族,得到任意从半空间的简单流形到一黎曼流形的带自由边界的调和映射在如果满足适当的条件(见定理)必为常值映射的结果.     

A New Class of Harmonic Measure Distribution Functions

Let Ω be a planar domain containing 0. Let h _Ω (r) be the harmonic measure at 0 in Ω of the part of the boundary of Ω within distance r of 0. The resulting function h _Ω is called the harmonic measure distribution function of Ω. In this paper we address the inverse problem by establishing several sets of sufficient conditions on a function f for f to arise as a harmonic measure distribution function. In particular, earlier work of Snipes and Ward shows that for each function f that increases from zero to one, there is a sequence of multiply connected domains X _n such that \(h_{X_{n}}\) converges to f pointwise almost everywhere. We show that if f satisfies our sufficient conditions, then f=h _Ω , where Ω is a subsequential limit of bounded simply connected domains that approximate the domains X _n . Further, the limit domain is unique in a class of suitably symmetric domains. Thus f=h _Ω for a unique symmetric bounded simply connected domain Ω.     

Harmonic morphisms as unit normal bundles¶of minimal surfaces

Let be an isometric immersion between Riemannian manifolds and be the unit normal bundle of f. We discuss two natural Riemannian metrics on the total space and necessary and sufficient conditions on f for the projection map to be a harmonic morphism. We show that the projection map of the unit normal bundle of a minimal surface in a Riemannian manifold is a harmonic morphism with totally geodesic fibres. Received: 6 February 1999     

Harmonic manifolds with some specific volume densities

We show that a noncompact, complete, simply connected harmonic manifold (M ^d, g) with volume densityθ _m(r)=sinh^d-1 r is isometric to the real hyperbolic space and a noncompact, complete, simply connected Kähler harmonic manifold (M ^2d, g) with volume densityθ _m(r)=sinh^2d-1 r coshr is isometric to the complex hyperbolic space. A similar result is also proved for quaternionic Kähler manifolds. Using our methods we get an alternative proof, without appealing to the powerful Cheeger-Gromoll splitting theorem, of the fact that every Ricci flat harmonic manifold is flat. Finally a rigidity result for real hyperbolic space is presented.     

Schatten Class Hankel Operators on the Harmonic Bergman Space of the Unit Ball

We give a necessary and sufficient condition for Hankel operators H_f on the harmonic Bergman space of the unit ball to be in the Schatten p-class for 2 ≤ p < ∞. A special case when symbol f is a harmonic function is also considered.     

Seven Classes of Harmonic Diffeomorphisms

We deduce two necessary and sufficient conditions for a diffeomorphism $f : M \to \overline{M}$ of a Riemannian manifold (M,g) onto a Riemannian manifold $(\overline{M},\bar g)$ to be harmonic. Using the representation theory of groups, we define in an intrinsic way seven classes of such harmonic diffeomorphisms and partly describe the geometry of each class.     

Harmonic maps with singular boundary value from complex hyperbolic spaces into rank one symmetric spaces

In this paper we consider the Dirichlet problem at infinity of proper harmonic maps from noncompact complex hyperbolic space to a rank one symmetric space N of noncompact type with singular boundary data . Under some conditions on f, we show that the Dirichlet problem at infinity admits a harmonic map which assumes the boundary data f continuously. Received: March 11, 1999 / Accepted April 23, 1999     

Harmonic functions of subordinate killed Brownian motion

In this paper we study harmonic functions of subordinate killed Brownian motion in a domain D. We first prove that, when the killed Brownian semigroup in D is intrinsic ultracontractive, all nonnegative harmonic functions of the subordinate killed Brownian motion in D are continuous and then we establish a Harnack inequality for these harmonic functions. We then show that, when D is a bounded Lipschitz domain, both the Martin boundary and the minimal Martin boundary of the subordinate killed Brownian motion in D coincide with the Euclidean boundary ∂D. We also show that, when D is a bounded Lipschitz domain, a boundary Harnack principle holds for positive harmonic functions of the subordinate killed Brownian motion in D.     

Representations of Solutions of Laplacian Boundary Value Problems on Exterior Regions

This paper treats the well-posedness and representation of solutions of Poisson’s equation on exterior regions $U\subsetneq{\mathbb{R}}^{N}$ with N≥3. Solutions are sought in a space E ¹(U) of finite energy functions that decay at infinity. This space contains H ¹(U) and existence-uniqueness theorems are proved for the Dirichlet, Robin and Neumann problems using variational methods with natural conditions on the data. A decomposition result is used to reduce the problem to the evaluation of a standard potential and the solution of a harmonic boundary value problem. The exterior Steklov eigenproblems for the Laplacian on U are described. The exterior Steklov eigenfunctions are proved to generate an orthogonal basis for the subspace of harmonic functions and also of certain boundary trace spaces. Representations of solutions of the harmonic boundary value problem in terms of these bases are found, and estimates for the solutions are derived. When U is the region exterior to a 3-d ball, these Steklov representations reduce to the classical multi-pole expansions familiar in physics and engineering analysis.     

Boundary slopes and the logarithmic limit set

The A-polynomial of a manifold whose boundary consists of a single torus is generalised to an eigenvalue variety of a manifold whose boundary consists of a finite number of tori, and the set of strongly detected boundary curves is determined by Bergman's logarithmic limit set, which describes the exponential behaviour of the eigenvalue variety at infinity. This enables one to read off the detected boundary curves of a multi-cusped manifold in a similar way to the 1-cusped case, where the slopes are encoded in the Newton polygon of the A-polynomial.     

Harmonic mappings with hereditary starlikeness

We study a hereditary starlikeness property for planar harmonic mappings on a disk and on an annulus. While such a property is a common trait of conformal mappings, it may be absent in harmonic mappings. It turns out that a sufficient condition for a harmonic mapping f to possess this hereditary property is to have a harmonic argument — a striking feature of conformal mappings that does not extend to all harmonic mappings.     

A Sharp Comparison Theorem for Compact Manifolds with Mean Convex Boundary

Let M be a compact n-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary ?M. Assume that the mean curvature H of the boundary ?M satisfies H≥(n?1)k>0 for some positive constant k. In this paper, we prove that the distance function d to the boundary ?M is bounded from above by \(\frac{1}{k}\) and the upper bound is achieved if and only if M is isometric to an n-dimensional Euclidean ball of radius \(\frac{1}{k}\) .     

Harmonic maps and asymptotic Teichmüller space

In this paper, the asymptotic boundary behavior of a Hopf differential or the Beltrami coefficient of a harmonic map is investigated and certain compact properties of harmonic maps are established. It is shown that, if f is a quasiconformal harmonic diffeomorphism between two Riemann surfaces and is homotopic to an asymptotically conformal map modulo boundary, then f is asymptotically conformal itself. In addition, we prove that the harmonic embedding map from the Bers space B Q D (X) of an arbitrary hyperbolic Riemann surface X to the Teichmüller space T (X) induces an embedding map from the asymptotic Bers space A B Q D (X), a quotient space of B Q D (X), into the asymptotic Teichmüller space AT (X). The work was supported by a Foundation for the Author of National Excellent Doctoral Dissertation (Grant No. 200518) of PR China and the National Natural Science Foundation of China (Grant No. 10401036).     

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.     

Harmonic cohomology of symplectic manifolds

Mathieu (Math. Helv. 70 (1995) 1) introduced a canonic filtration in the de Rham cohomology of a symplectic manifold and proved, that the middle filtration space is the space of harmonic cohomology classes. We give an interpretation of the other filtration spaces, prove a Künneth theorem for harmonic cohomology, prove Poincaré duality for harmonic cohomology and show how surjectivity of certain Lefschetz type mappings is related to properties of the filtration. For a closed symplectic manifold M we also introduce symplectic invariants , and show if M is of dimension 2n with n even.     

The maximum principle and the Dirichlet problem for Dirac-harmonic maps

We establish a maximum principle and uniqueness for Dirac-harmonic maps from a Riemannian spin manifold with boundary into a regular ball in any Riemannian manifold N. Then we prove an existence theorem for a boundary value problem for Dirac-harmonic maps.     

A Global Existence Result For The Heat Flow of Harmonic Maps

We consider the heat flow for harmonic maps from R^m to a compact manifold N. When the L^m norm of the gradient of the initial data is small, we prove the existence of a global solution. We prove a similar result for the boundary value problem, when the boundary of the manifold M maps into a point.