Nonconstant Harmonic Functions Research Articles

Recurrent and (strongly) resolvable graphs

We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs. The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics. The main tool is a connection between polar sets in such boundaries and null sets of paths. This connection relies on suitably diverging functions of finite energy.

Harmonic functions on compactly generated groups

A compactly generated group is noncompact if and only if it admits a nonconstant harmonic function (for some, equivalently for every, reasonable measure). This generalizes the known fact that a finitely generated group is infinite if and only if it admits a nonconstant harmonic function (for some, equivalently every, reasonable measure).

A Liouville theorem on asymptotically Calabi spaces

In this paper, we will study harmonic functions on the complete and incomplete spaces with nonnegative Ricci curvature which exhibit inhomogeneous collapsing behaviors at infinity. The main result states that any nonconstant harmonic function on such spaces yields a definite exponential growth rate which depends explicitly on the geometric data at infinity.

Fractal Interpolation Using Harmonic Functions on the Koch Curve

The Koch curve was first described by the Swedish mathematician Helge von Koch in 1904 as an example of a continuous but nowhere differentiable curve. Such functions are now characterised as fractal since their graphs are in general fractal sets. Furthermore, it can be obtained as the graph of an appropriately chosen iterated function system. On the other hand, a fractal interpolation function can be seen as a special case of an iterated function system thus maintaining all of its characteristics. Fractal interpolation functions are continuous functions that can be used to model continuous signals. An in-depth discussion on the theory of affine fractal interpolation functions generating the Koch Curve by using fractal analysis as well as its recent development including some of the research made by the authors is provided. We ensure that the graph of fractal interpolation functions on the Koch Curve are attractors of an iterated function system constructed by non-constant harmonic functions.

Essential Self-Adjointness and the $$L^2$$-Liouville Property

We discuss connections between the essential self-adjointness of a symmetric operator and the constancy of functions which are in the kernel of the adjoint of the operator. We then illustrate this relationship in the case of Laplacians on both manifolds and graphs. Furthermore, we discuss the Green’s function and when it gives a non-constant harmonic function which is square integrable.

Fractal functions on the Sierpinski Gasket

In this paper, we ensure for the first time that graphs of fractal interpolation functions generated on the Sierpinski Gasket by nonconstant harmonic functions of fractal analysis are attractors of some iterated function systems, and at the same time, we give new nonlinear fractal interpolation functions.

Hausdorff Dimensions for Graph-Directed Measures Driven by Infinite Rooted Trees

We give upper and lower bounds for the Hausdorff dimensions for a class of graph-directed measures when its underlying directed graph is the infinite N-ary tree. These measures are different from graph-directed self-similar measures driven by finite directed graphs and are not necessarily Gibbs measures. However our class contains several measures appearing in fractal geometry and functional equations, specifically, measures defined by restrictions of non-constant harmonic functions on the two-dimensional Sierp\'inski gasket, the Kusuoka energy measures on it, and, measures defined by solutions of de Rham's functional equations driven by linear fractional transformations.

Harmonic Dirichlet Functions on Planar Graphs

Benjamini and Schramm (Invent Math 126(3):565–587, 1996) used circle packing to prove that every transient, bounded degree planar graph admits non-constant harmonic functions of finite Dirichlet energy. We refine their result, showing in particular that for every transient, bounded degree, simple planar triangulation T and every circle packing of T in a domain D, there is a canonical, explicit bounded linear isomorphism between the space of harmonic Dirichlet functions on T and the space of harmonic Dirichlet functions on D.

An almost rigidity theorem and its applications to noncompact RCD(0,N) spaces with linear volume growth

The main results of this paper consist of two parts. First, we obtain an almost rigidity theorem which roughly says that on an [Formula: see text] space, when a domain between two level sets of a distance function has almost maximal volume compared to that of a cylinder, then this portion is close to a cylinder as a metric space. Second, we apply this almost rigidity theorem to study noncompact [Formula: see text] spaces with linear volume growth. More precisely, we obtain the sublinear growth of diameter of geodesic spheres, and study the non-existence problem of nonconstant harmonic functions with polynomial growth on such [Formula: see text] spaces.

Diffusive estimates for random walks on stationary random graphs of polynomial growth

Let \({(G,\rho)}\) be a stationary random graph, and use \({B^G_{\rho}(r)}\) to denote the ball of radius r about \({\rho}\) in G. Suppose that \({(G,\rho)}\) has annealed polynomial growth, in the sense that \({\mathbb{E}[|B^G_{\rho}(r)|] \leq O(r^k)}\) for some \({k > 0}\) and every \({r \geq 1}\). Then there is an infinite sequence of times \({\{t_n\}}\) at which the random walk \({\{X_t\}}\) on \({(G,\rho)}\) is at most diffusive: almost surely (over the choice of \({(G,\rho)}\)), there is a number \({C > 0}\) such that $$\mathbb{E} \left[\mathrm{dist}_G(X_0, X_{t_n})^2 \mid X_0 = \rho, (G,\rho)\right]\leq C t_n\qquad \forall n \geq 1\,.$$ This result is new even in the case when G is a stationary random subgraph of \({\mathbb{Z}^d}\). Combined with the work of Benjamini et al. (Ann Probab 43(5):2332–2373, 2015), it implies that G almost surely does not admit a non-constant harmonic function of sublinear growth. To complement this, we argue that passing to a subsequence of times \({\{t_n\}}\) is necessary, as there are stationary random graphs of (almost sure) polynomial growth where the random walk is almost surely superdiffusive at an infinite subset of times.

Connective constants and height functions for Cayley graphs

The connective constant μ ( G ) \mu (G) of an infinite transitive graph G G is the exponential growth rate of the number of self-avoiding walks from a given origin. In earlier work of Grimmett and Li, a locality theorem was proved for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin. A condition of the theorem was that the graphs support so-called “unimodular graph height functions”. When the graphs are Cayley graphs of infinite, finitely generated groups, there is a special type of unimodular graph height function termed here a “group height function”. A necessary and sufficient condition for the existence of a group height function is presented, and may be applied in the context of the bridge constant, and of the locality of connective constants for Cayley graphs. Locality may thereby be established for a variety of infinite groups including those with strictly positive deficiency. It is proved that a large class of Cayley graphs support unimodular graph height functions, that are in addition harmonic on the graph. This implies, for example, the existence of unimodular graph height functions for the Cayley graphs of finitely generated solvable groups. It turns out that graphs with non-unimodular automorphism subgroups also possess graph height functions, but the resulting graph height functions need not be harmonic. Group height functions, as well as the graph height functions of the previous paragraph, are non-constant harmonic functions with linear growth and an additional property of having periodic differences. The existence of such functions on Cayley graphs is a topic of interest beyond their applications in the theory of self-avoiding walks.

Characterisations of algebraic properties of groups in terms of harmonic functions

We prove various results connecting structural or algebraic properties of graphs and groups to conditions on their spaces of harmonic functions. In particular: we show that a group with a finitely supported symmetric measure has a finite-dimensional space of harmonic functions if and only if it is virtually cyclic; we present a new proof of a result of V. Trofimov that an infinite vertex-transitive graph admits a non-constant harmonic function; we give a new proof of a result of T. Ceccherini-Silberstein, M. Coornaert and J. Dodziuk that the Laplacian on an infinite, connected, locally finite graph is surjective; and we show that the positive harmonic functions on a non-virtually nilpotent linear group span an infinite-dimensional space.

Absence of Non-Constant Harmonic Functions with ℓ p -gradient in some Semi-Direct Products

To obtain groups with bounded harmonic functions (which are amenable), one of the most frequent way is to look at some semi-direct products (e.g. lamplighter groups). The aim here is to show that many of these semi-direct products do not admit harmonic functions with gradient in l p , for \(p\in [1,\infty [\).

Three circles theorems for harmonic functions

We proved two Three Circles Theorems for harmonic functions on manifolds in integral sense. As one application, on manifold with nonnegative Ricci curvature, whose tangent cone at infinity is the unique metric cone with unique conic measure, we showed the existence of nonconstant harmonic functions with polynomial growth. This existence result recovered and generalized the former result of Y. Ding, and led to a complete answer of L. Ni's conjecture. Furthermore in similar context, combining the techniques of estimating the frequency of harmonic functions with polynomial growth, which were developed by Colding and Minicozzi, we confirmed their conjecture about the uniform bound of frequency.

Boundary values, random walks, and $\ell^p$-cohomology in degree one

The vanishing of reduced \ell^2 -cohomology for amenable groups can be traced to the work of Cheeger and Gromov in [10]. The subject matter here is reduced \ell^p -cohomology for p \in ]1,\infty[ , particularly its vanishing. Results for the triviality of \underline {\ell^pH}^1(G) are obtained, for example: when p \in ]1,2] and G is amenable; when p \in ]1,\infty[ and G is Liouville (e.g. of intermediate growth). This is done by answering a question of Pansu in [34, §1.9] for graphs satisfying certain isoperimetric profile. Namely, the triviality of the reduced \ell^p -cohomology is equivalent to the absence of non-constant harmonic functions with gradient in \ell^q ( q depends on the profile). In particular, one reduces questions of non-linear analysis ( p -harmonic functions) to linear ones (harmonic functions with a very restrictive growth condition).

STRUCTURE OF STABLE MINIMAL HYPERSURFACES IN A RIEMANNIAN MANIFOLD OF NONNEGATIVE RICCI CURVATURE

Let N be a complete Riemannian manifold with nonnegative Ricci curvature and let M be a complete noncompact oriented stable minimal hypersurface in N. We prove that if M has at least two ends and <TEX>${\int}_M{\mid}A{\mid}^2\;dv={\infty}$</TEX>, then M admits a nonconstant harmonic function with finite Dirichlet integral, where A is the second fundamental form of M. We also show that the space of <TEX>$L^2$</TEX> harmonic 1-forms on such a stable minimal hypersurface is not trivial. Our result is a generalization of one of main results in [12] because if N has nonnegative sectional curvature, then M admits no nonconstant harmonic functions with finite Dirichlet integral. And our result recovers a main theorem in [3] as a corollary.

Complete minimal surfaces and harmonic functions

We prove that for any open Riemann surface \mathcal{N} and any non-constant harmonic function h\colon \mathcal{N} \to \mathbb{R} , there exists a complete conformal minimal immersion X\colon \mathcal{N} \to \mathbb{R}^3 whose third coordinate function coincides with h . As a consequence, complete minimal surfaces with arbitrary conformal structure and whose Gauss map misses two points are constructed.

A Characterization of the Locally Finite Networks Admitting Non-Constant Harmonic Functions of Finite Energy

We characterize the locally finite networks admitting non-constant harmonic functions of finite energy. Our characterization unifies the necessary existence criteria of Thomassen (J Comb Theory, Ser B 49:87–102, 1990) and of Lyons and Peres (2011) with the sufficient criterion of Soardi (1991). We also extend a necessary existence criterion for non-elusive non-constant harmonic functions of finite energy due to Georgakopoulos (J Lond Math Soc, 2010).

Lamplighter graphs do not admit harmonic functions of finite energy

We prove that a lamplighter graph of a locally finite graph over a finite graph does not admit a non-constant harmonic function of finite Dirichlet energy.

Geometric remarks on the level curves of harmonic functions

Suppose that u is a nonconstant harmonic function on the plane. By the maximum principle, its zero set Z does not contain any simple closed curve. This paper provides bounds on the curvature of Z, and other conditions on Z, which imply that u is of some special type, such as polynomial, linear, or even constant. These theorems resemble unique continuation results, which often apply to various classes of nonharmonic functions. For such functions, Z may contain a closed curve, but a lower bound is established for the area of the enclosed region.