Study Of Harmonic Functions Research Articles

Doob equivalence and non-commutative peaking for Markov chains

In this paper, we show how questions about operator algebras constructed from stochastic matrices motivate new results in the study of harmonic functions on Markov chains. More precisely, we characterize the coincidence of conditional probabilities in terms of (generalized) Doob transforms, which then leads to a stronger classification result for the associated operator algebras in terms of spectral radius and strong Liouville property. Furthermore, we characterize the non-commutative peak points of the associated operator algebra in a way that allows one to determine them from inspecting the matrix. This leads to a concrete analogue of the maximum modulus principle for computing the norm of operators in the ampliated operator algebras.

Harmonic maps on Sasakian manifolds

The theory of harmonic maps on Riemannian and Kahlerian manifolds is very rich in interesting results and continues to be one of the most important areas of differential geometry. A map between Riemannian manifolds is called harmonic if the divergence of its differential vanishes. For Riemannian manifolds with a differential structure, it is known that a smooth holomorphic map between Kaehler manifolds is harmonic. The study of harmonic maps on contact metric manifolds was initiated by Ianus et. al.. In this paper we consider harmonic maps on Sasakian manifolds and also discuss the non-existence of such maps.

The Berger-Ebin theorem and harmonic maps and flows

The purpose of this paper is to present a geometrization of the Berger-Ebin theorem. We use this theorem for the study of harmonic maps and flows, in particular, Ricci solitons. We also clarify the role of a vector field in the corresponding decompositions.

True or False? A Case in the Study of Harmonic Functions

Recent mathematical results, obtained by the author, in collaboration with Alexander Stokolos, Olof Svensson, and Tomasz Weiss, in the study of harmonic functions, have prompted the following reflections, intertwined with views on some turning points in the history of mathematics and accompanied by an interpretive key that could perhaps shed some light on other aspects of (the development of) mathematics.

Harmonic operators: the dual perspective

The study of harmonic functions on a locally compactgroupGhas recently been transferred to a non-commutative setting in two different directions: Chu and Lau replaced the algebra L ∞ (G) by the group von Neumann algebra VN(G) and the convolution action of a probability measure μ on L ∞ (G) by the canonical action of a positive definite function σ on VN(G); on the other hand, Jaworski and the first author replaced L ∞ (G) by B(L 2 (G)) to which the convolution action by μ can be extended in a natural way. We establish a link between both approaches. The action of σ on VN(G) can be extended to B(L 2 (G)). We study the corresponding space ˜ Hσ

Area Comparison, Superharmonic Functions, and Applications

In this paper, we study some geometrical and analytic properties of manifolds with non-negative sectional curvature at infinity. Then, we apply these results to the study of harmonic maps.

Heat equation for harmonic maps of the compactification of complete manifolds

for i = 1 ..... n, where f : M -+ N is a given initial map which is C 1 with bounded energy density. (This assumption on the initial value f is a standing one throughout this paper.) This harmonic map heat equation is a nonlinear parabolic system which proved to be useful in the study of harmonic maps. The nonlinear terms, which are due to the curvature of the target manifold, give distinct geometric flavor to this problem. The natural question is then how to understand the interplay between these terms and the target geometry. It is universally accepted that a harmonic map behaves much nicer when the target manifold has negative curvature. It is usually due to the geometric inequalities that come from the negativity of curvature of the target manifold. We like to go one step further in this direction. Namely, we would like to propose the thesis that a harmonic map heat equation behaves like a bunch of uncoupled linear equations when the target manifold has nonpositive sectional curvature, even in the presence of topology in the target manifold. In particular, we will show that each component function u i of the solution u = (u 1 ..... u n) of (1.1) can be well

A density function and the structure of singularities of the mean curvature flow

We study singularity formation in the mean curvature flow of smooth, compact, embedded hypersurfaces of non-negative mean curvature in ℝ n+1, primarily in the boundaryless setting. We concentrate on the so-called “Type I” case, studied by Huisken in [Hu 90], and extend and refine his results. In particular, we show that a certain restriction on the singular points covered by his analysis may be removed, and also establish results relating to the uniqueness of limit rescalings about singular points, and to the existence of “slow-forming singularities” of the flow. The main new ingredient introduced, to address these issues, is a certain “density function”, analogous to the usual density function in the study of harmonic maps in the stationary setting. The definition of this function is based on Huisken's important monotonicity formula for mean curvature flow.

Global solutions of the elliptic 2D periodic Toda lattice

This paper demonstrates how to use a dressing construction to produce globally analytic solutions to an elliptic version of the two-dimensional periodic Toda lattice associated with any simple Lie algebra. These equations are seen to occur in the study of harmonic maps and this construction also produces harmonic maps of R2 into a semisimple Lie group or its full flag manifold. In particular, it is shown how to extract Krichever's (1984) theta -function solutions (to the Toda lattice for SLn+1) from the dressing construction. There are harmonic planes in SUn+1 (and its full flag manifold) corresponding to these solutions and their construction is described fairly explicitly.

Irreducibility and monodromy of some families of linear series

— Let g, r, and d be positive integers such that g=(r-{-1) (g—d-\-r), so that the general curve of genus g has only finitely many g^s. We will show in this paper that for suitable families of curves ^ -> B, the family of all cffs on all fibers of ^ -> B is irreducible. We do this by analyzing the monodromy action on the set of 97 on a fibre, using a degeneration to reducible curves and our technique of limit series [198?^]. In the case r = 1 we prove the sharper statement that the monodromy is the full symmetric group, a result motivated by a problem posed by Verdier, and applied by him in the study of harmonic maps from 2 to S (Verdier [198?]). If we take ^ to be the universal curve over a suitable open set B of the moduli space My then the family of c^'s is a finite cover of B, and the branch locus of this cover (in the case r=l), analyzed through the tools developed in this paper, plays a fundamental role in the even-genus case in our proof [198?^] that Jig has general type for all g 24.

Positive Harmonic Functions on Complete Manifolds of Negative Curvature

This paper studies the interaction between the geometry of complete Riemannian manifolds of negative curvature and some aspects of function theory on these spaces. The study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction; we recall some aspects of this below. There is a well-known representation of positive harmonic functions on the unit disc U. due to Herglotz [13], in terms of positive Borel measures tt on the circle S1:

The Topology of the Level Curves of Harmonic Functions with Critical Points

Introduction. In a previous paper,2 of which this is a continuation, topological properties of curve families which filled the Euclidean plane 7r, or a simply connected domain in r, were investigated. The families were assumed regular (i. e. locally homeomorphic to parallel lines) except at a possibly infinite collection of isolated singularities at each of which the family had the structure of a multiple saddle point; such families were called branched regular curve families. Further investigation of these families, in particular their relation to harmonic functions, is the aim of this paper. In what follows the definitions and theorems in [I] will be assumed, and the same notation will be used. In particular F, G will denote branched regular curve families filling the plane 7r, B will denote the set of singular points, R the domain 7r B in which F is regular, and so on. The Euclidean plane will be taken as a model for all simply connected domains. The principal result of [I] was to prove that any branched regular curve family F filling Xr can be given as the family of level curves of a function f (p) which is continuous on all of 7r and has no relative extrema. This generalizes a portion of [II] in which the same theorem is proved for a curve family without singularities in 7r. In this paper there are two main results: the first, proved in Section 1, is that F is actually homeomorphic to the level curves of a harmonic function; the second, proved in Section 2, asserts the existence of a decomposition of F into a countable collection of subfamilies of curves, each of which has the structure of the parallel lines y=constant of the upper half-plane. Such subfamilies will be called half-parallel, and this decomposition has consequences for the study of harmonic functions and analytic functions which will be mentioned below. These two results generalize