Search

In the context of synthetic differential geometry, we study the Laplace operator an a Riemannian manifold. The main newaspect is a neighbourhood of the diagonal, smaller than the second neighbourhood usually required as support for second order differential operators. The newneighbourhood has the property that a function is affine on it if and only if it is harmonic.

A growth of harmonic functions in the whole space ℝn is examined. We found the estimate for a uniform norm of spherical harmonics in terms of the best approximation of harmonic function in the ball by harmonic polynomials. An approximation error of harmonic function in the ball is estimated by the maximum modulus of an entire harmonic function in space, as well as the maximum modulus of an entire harmonic function in space in terms of the maximum modulus of some entire function of one complex variable or the maximal term of its power series. These results allowed us to obtain the necessary and sufficient conditions under which a harmonic function in the ball of an n-dimensional space, n≥3, can be continued to the entire harmonic one. This result is formulated in terms of the best approximation of the given function by harmonic polynomials. In order to characterize growth of an entire harmonic function, we used the generalized and the lower generalized orders. Formulae for the generalized and the lower generalized orders of an entire harmonic function in space are expressed in terms of the approximation error by harmonic polynomials of the function that continues. We also investigated the growth of functions of slow increase. The obtained results are analogues to classical results, which are known for the entire functions of one complex variable.The conducted research is important due to the fact that the harmonic functions occupy a special place not only in many mathematical studies, but also when applying mathematical analysis to physics and mechanics, where these functions are often employed to describe various stationary processes

Vanishing theorems for harmonic and infinitesimal harmonic transformations of complete Riemannian manifolds are proved. The proof uses well-known Liouville theorems on subharmonic functions on noncompact complete Riemannian manifolds.

In this paper, we have studied harmonic maps on trans-Sasakian manifolds. First it is proved that if F: M1 → M2 is a Riemannian ϕ-holomorphic map between two trans-Sasakian manifolds such that ξ2 ∈ (Im dF)⊥, then F can not be harmonic provided that β2 ≠ 0. We have also found the necessary and sufficient condition for the harmonic map to be constant map from Kaehler to trans-Sasakian manifold. Finally, we prove the non-existence of harmonic map from locally conformal Kaehler manifold to trans-Sasakian manifold.

New elements are proposed in the present work based on the harmonic coordinates in which the shape functions satisfy the Laplace equation. The harmonic functions have some appealing characteristics that made it possible to define elements with arbitrary shape functions on the element boundaries and arbitrary node arrangement. In the present work, without loss of generality, we formulate a set of quadrilateral elements with midside nodes and piecewise linear shape functions along the element boundaries. The analytical solution of Laplace equation in rectangular domains are used here to represent the harmonic shape function as Fourier series. To derive the shape functions of different elements with different node arrangements a systematic approach is proposed to represent all of the shape functions using a small set of harmonic functions. Two model boundary value problems, Poisson equation and linear elasticity are used here to evaluate the proposed method. Patch test and convergence analysis are done for each model problems via some examples. The elements with midside nodes have great potential to be used in mesh adaptive solutions. Therefore, some adaptive mesh refinement examples are solved based on the ZZ error estimation and quadtree grid structure as the mesh refinement technique.

Let $(W,q, \mathcal{D})$ be a 4-dimensional Walker manifold. After providing a characterization and some examples for several special $(1,1)$-tensor fields on $(W,q, \mathcal{D})$, we prove that the proper almost complex structure $J$, introduced by Matsushita, is harmonic in the sense of Garcia-Rio et al. if and only if the almost Hermitian structure $(J,q)$ is almost Kahler. We classify all harmonic functions locally defined on $(W,q, \mathcal{D})$. We deal with the harmonicity of quadratic maps defined on $\mathbb{R}^4$ (endowed with a Walker metric $q$) to the $n$-dimensional semi-Euclidean space of index $r$, and then between local charts of two 4-dimensional Walker manifolds. We obtain here the necessary and sufficient conditions under which these maps are harmonic, horizontally weakly conformal, or harmonic morphisms with respect to $q$.

We define two transforms of non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct, from one such harmonic map, a sequence of harmonic maps. We show that there is a correspondence between harmonic maps into the 3-sphere, H-surfaces in Euclidean 3-space and almost complex surfaces in the nearly Kahler manifold . As a consequence we can construct sequences of H-surfaces and almost complex surfaces.

By using potential-theoretic methods on locally compact spaces, two harmonic sheaves (resembling the solutions of the Laplace and the Schrodinger operators) on an infinite network are compared.

For the harmonic function on two dimensional Riemannian manifolds with constant Gaussian curvature, some differential equalities and inequalities are established to study the curvature estimate of the steepest descents by the maximum principle.

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.

The notion of exponentially harmonic maps was introduced by Eells and Lemaire (Proceedings of the Banach Center Semester on PDE, pp. 1990–1991, 1990). In this note, by using the maximum principle we get the gradient estimate of exponentially harmonic functions, and then derive a Liouville type theorem for bounded exponentially harmonic functions on a complete Riemannian manifold with nonnegative Ricci curvature and sectional curvature bounded below.

Given an open hyperbolic Riemannian manifold, we show that various vector spaces of harmonic functions coincide if and only if they are finite dimensional.

The article introduces substitutes for differential 1-forms on self-adjoint harmonic spaces by means of tensor products and energy norms. A representation using indicators of open sets connects 1-forms and topology. Finally, an orthogonal decomposition into exact and harmonic forms is provided, along with a description of harmonic forms in terms of locally harmonic functions.

We study L-harmonic functions (solutions of the stationary Schrodinger equation) on arbitrary noncompact Riemannian manifolds with finitely many ends. We establish some existence and uniqueness results, and obtain sharp dimension estimates for L-harmonic functions on such manifolds.

We consider boundary value problems in a disk and in a ring for homogeneous equations with the Laplace operator of the first and second orders. Solutions are represented in terms of bases of harmonic wavelets in Hardy spaces of harmonic functions in a disk and in a ring, which were constructed earlier.

In this paper we study the distribution of values of harmonic functions in non-Euclidean disks. We introduce the notion of a P′-sequence, which enables us to characterize the class of normal harmonic functions defined in the unit disk. Moreover, we obtain sufficient conditions for the existence of such sequences and give examples which show that these conditions are essential in the stated theorems.

In this note, we introduce an approximation of harmonic maps via a sequence of exponentially harmonic maps. We then reestablish the existence theorem of harmonic maps due to Eells and Sampson.

We describe the asymptotic behavior of functions of the Royden p-algebra in terms of p-extremal length. We also prove that each bounded -harmonic function with finite energy on a complete Riemannian manifold is uniquely determined by the behavior of the function along p-almost every curve.

Let (M,ℱ) be a compact codimension-one foliated manifold whose leaves are endowed with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of ℱ. If every such function is constant on leaves, we say that (M,ℱ) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. A related result for ℝ-covered foliations is also established.

This paper investigates positive harmonic functions on domains that are complementary to a subset of a cylindrical surface. It characterizes, both in terms of harmonic measure and of a Wiener-type criterion, those domains that admit minimal harmonic functions with exponential growth. Illustrative examples are provided. Two applications are also given. The first of these concerns minimal harmonic functions associated with an irregular boundary point, and amplifies a recent construction of Gardiner and Hansen. The second concerns the possible non-approximability of positive harmonic functions by integrable positive harmonic functions.

We study the problem of solvability of some boundary value problems on noncompact Riemannian manifolds with ends. We obtain the conditions for existence and uniqueness of solutions to the problems as well as the conditions for the fulfillment of Liouville-type theorems for harmonic functions on the manifolds.

Let $(M^{n},g)$ be a compact Riemannian manifold with $Ric\geq-(n-1) $. It is well known that the bottom of spectrum $\lambda_{0}$ of its unverversal covering satisfies $\lambda_{0}\leq(n-1) ^{2}/4 $. We prove that equality holds iff $M$ is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy., Comment: to appear in J. Geom. Anal

This paper describes the solvability of Dirichlet problems for Laplace's equation when the boundary data is not smooth enough for the existence of a weak solution in H1Ω. Scales of spaces of harmonic functions and of boundary traces are defined and the solutions are characterized as limits of classical harmonic functions in special norms. The generalized harmonic functions, and their norms, are defined using series expansions involving harmonic Steklov eigenfunctions on the domain. It is shown that the usual trace operator has a continuous extension to an isometric isomorphism of specific spaces. This provides a characterization of the generalized solutions of harmonic Dirichlet problems. Numerical simulations of a model problem are described. This problem is related to the dewetting of thin films and the associated phenomenology is described.

Let S Ĥ be the family of all planar harmonic, univalent and sense-preserving mappings f=h+g¯ where h and g are analytic funtions in the open unit disk. The purpose of this article is to investigate connections between the theory of harmonic mappings in the plane and hypergeometric functions by applying certain planar harmonic convolution operators on various subclasses of S Ĥ . Dedicated to Professor Eveyln M. Silvia, late Professor of Mathematics, UC Davis February 8, 1948–January 21, 2006

We present two transforms of non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct from one non-conformal harmonic map a sequence of non-conformal harmonic maps. We also discuss the correspondence between non-conformal harmonic maps into the 3-sphere, H-surfaces in Euclidean 3-space and almost complex surfaces in the nearly Kähler manifold S3 × S3. ispartof: International Journal of Geometric Methods in Modern Physics vol:12 issue:8 pages:1560012-1560017 ispartof: location:SPAIN, Univ Granada, Granada status: published

The aim of this paper is to extend the notion of pseudo harmonic morphism introduced by Loubeau in [15] (see also [7, 4]) to the case when the source manifold is an admissible Riemannian polyhedron. We define these maps to be harmonic, as in [9], and pseudo-horizontally weakly conformal, see Sec. 3. We characterize them by means of germs of harmonic functions on the source polyhedron (see [13] for a precise definition) and germs of holomorphic functions on the Kähler target manifold, similarly to [15, 7].

We integrate the Lame equation and find new solutions in the case of three-dimensional elasticity theory, which are expressed in terms of harmonic functions. We prove that the solution obtained involves only three independent functions. In a curvilinear orthogonal coordinate system, a general solution of the Lame equation is expressed in terms of three harmonic functions.

In this paper, we describe the behavior of bounded energy finite solutions for certain nonlinear elliptic operators on a complete Riemannian manifold in terms of its p-harmonic boundary. We also prove that if two complete Riemannian manifolds are roughly isometric to each other, then their p-harmonic boundaries are homeomorphic to each other. In the case, there is a one to one correspondence between the sets of bounded energy finite solutions on such manifolds. In particular, in the case of the Laplacian, it becomes a linear isomorphism between the spaces of bounded harmonic functions with finite Dirichlet integral on the manifolds.

A necessary and sufficient condition is given for a planar harmonic mapping f to be locally decomposable as a univalent harmonic mapping F of an analytic function φ.

In spite of the abundance of publications on harmonic mappings of man- ifolds, at present there exists neither a theory of harmonic diffeomorphisms, nor a definition of infinitesimal harmonic transformation of a Riemannian manifold, to say nothing of the theory of groups of such transformations. In the paper, this gap is partially filled, and a new subject of investigations is announced.

In this paper we study some algebraic properties of harmonic forms on Poisson manifolds. It is well known that in the classical case (on Riemannian manifolds) the product of harmonic forms is not harmonic. Here we describe the algebraic and analytical mechanisms explaining this fact. We also obtain a condition under which the product of de Rham cohomology classes, which includes harmonic representatives, can be represented by a harmonic form.

A Riemannian manifold is called {\it harmonic}, if for any point $x$ it admits a nonconstant harmonic function depending only on the distance to $x$. A.Lichnerowicz conjectured that any harmonic manifold is two-point homogeneous. This conjecture is proved in dimension $n \le 4$ and also for some classes of manifolds, but disproved in general, with the first counterexample of dimension $7$. We prove the Lichnerowicz Conjecture in dimension $5$: a five-dimensional harmonic manifold has constant sectional curvature. We also obtain a functional equation for the volume density function $\T(r)$ of a harmonic manifold and show that $\T(r)$ is an exponential polynomial, a finite linear combination of the terms of the form $\Re (c e^{\la r} r^m)$, with $c, \la$ complex constants.

In this paper, we give a sharp estimate on the cardi- nality of the set generating the convex hull containing the image of harmonic maps with polynomial growth rate on a certain class of manifolds into a Cartan-Hadamard manifold with sectional cur- vature bounded by two negative constants. We also describe the asymptotic behavior of harmonic maps on a complete Riemannian manifold into a regular ball in terms of massive subsets, in the case when the space of bounded harmonic functions on the manifold is flnite dimensional.

This paper introduces a family of area-type integrals over cones. These are used to investigate non-tangential boundary behaviour of harmonic functions on a half-space, extending results of Stein and Brossard.AMS 2000 Mathematics subject classification: Primary 31B25

If M and N are Riemannian manifolds, a harmonic morphism f : M → N is a map which pulls back local harmonic functions on N to local harmonic functions on M. If M is an Einstein 4-manifold and N is a Riemann surface, John Wood showed that such an f is holomorphic w.r.t. some integrable complex Hermitian structure defined on M away from the singular points of f. In this paper we extend this complex structure to the entire manifold M. It follows that there are no non-constant harmonic morphisms from [Formula: see text] or [Formula: see text] to a Riemann surface. The proof relies heavily on the real analyticity of the whole situation. We conclude by an example of a non-constant harmonic morphism from [Formula: see text] to [Formula: see text].

One of the present authors defined the infinitesimal harmonictransformation in a Riemannian manifold. This paperis devoted to the study of the local and global geometryinfinitesimal harmonic transformations.

For a non-compact harmonic manifold M, we establish an integral formula for the derivative of a harmonic function on M. As an application we show that for the harmonic spaces having minimal horospheres, bounded harmonic functions are constant. The main result of this article states that the harmonic spaces having polynomial volume growth are flat. In other words, if the volume density function Θ of M has polynomial growth, then M is flat. This partially answers a question of Szabo namely, which density functions determine the metric of a harmonic manifold. Finally, we give some natural conditions which ensure polynomial growth of the volume function.

In this paper we prove that when the Ricci curvature of a Riemannian manifoldM n is almost nonnegative, and a ballB L (p)⊂M n is close in Gromov-Hausdorff distance to a Euclidean ball, then the gradient of the harmonic functionb defined in [ChCo1] does not vanish. In particular, these functions can serve as harmonic coordinates on balls sufficiently close to an Euclidean ball. The proof, is based on a monotonicity theorem that generalizes monotonicity of the frequency for harmonic functions onR n .

unavailable at this time... Mathematics Subject Classification (2000): Primary 31B05; Secondary 47A16 Quaestiones Mathematicae 25 (2002), 527-530

Let M be a Riemannian manifold with asymptotically non-negative curvature. We study the asymptotic behavior of the energy densities of a harmonic map and an exponentially harmonic function on M. We prove that the energy density of a bounded harmonic map vanishes at infinity when the target is a Cartan- Hadamard manifold. Also we prove that the energy density of a bounded exponentially harmonic function vanishes at infinity.

We study spectral convergence of compact Riemannian manifolds or more generally certain Dirichlet spaces, obtaining some compactness results on harmonic functions and harmonic maps.

We show that any rotationally symmetric Riemannian manifold has the L1-Liouville property for harmonic functions, i.e., any integrable harmonic function on it must be identically constant. We also give a characterization of a manifold which carries a non-constant L1 nonnegative subharmonic function.

We consider three-dimensional contact metric manifolds whose unit characteristic vector field is harmonic or minimal.