Related papers: Harmonic functions on minimal submanifold with cyl…

Suppose $(M,g)$ is a Riemannian manifold having dimension $n$, nonnegative Ricci curvature, maximal volume growth and unique tangent cone at infinity. In this case, the tangent cone at infinity $C(X)$ is an Euclidean cone over the…

We extend a theorem by Kleiner, stating that on a group with polynomial growth, the space of harmonic functions of polynomial of at most $k$ is finite dimensional, to the settings of locally compact groups equipped with measures with…

This paper investigates positive harmonic functions on a domain which contains an infinite cylinder, and whose boundary is contained in the union of parallel hyperplanes. (In the plane its boundary consists of two sets of vertical…

We study the connection between the dimension of certain spaces of harmonic functions on a group and its geometric and algebraic properties. Our main result shows that (for sufficiently "nice" random walk measures) a connected, compactly…

We show that, in dimension three and higher, the space of harmonic functions vanishing on the cone defined by a generically chosen harmonic quadratic polynomial is two-dimensional. This phenomenon is surprisingly robust, generalizing to…

We study the growth rate of harmonic functions in two aspects: gradient estimate and frequency. We obtain the sharp gradient estimate of positive harmonic function in geodesic ball of complete surface with nonnegative curvature. On complete…

In this article we have studied some properties of subharmonic functions in a strongly symmetric Riemannian manifold with a pole. As a generalization of polynomial growth of a function we have introduced the notion of polynomial growth of…

We give a definition of convergence of differential of Lipschitz functions with respect to measured Gromov-Hausdorff topology. As their applications, we give a characterization of harmonic functions with polynomial growth on asymptotic…

We study the growth of harmonic functions on complete Riemann-ian manifolds where the extrinsic diameter of geodesic spheres is sublinear. It is an generalization of a result of A. Kazue. We also get a Cheng and Yau estimates for the…

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space R^{m+1} was constructed recently, including a higher dimensional analogue of the logarithmic function in the complex…

We consider harmonic functions of polynomial growth of some order $d$ on Cayley graphs of groups of polynomial volume growth of order $D$ w.r.t. the word metric and prove the optimal estimate for the dimension of the space of such harmonic…

In this note we investigate the behavior of harmonic functions at singular points of $\mathsf{RCD}(K,N)$ spaces. In particular we show that their gradient vanishes at all points where the tangent cone is isometric to a cone over a metric…

These notes are concerned with harmonic and holomorphic functions on Euclidean spaces, using quaternions and Clifford algebras in higher dimensions. The main themes are weak solutions, the mean-value property, and subharmonicity.

In this paper, we study discrete harmonic functions on infinite penny graphs. For an infinite penny graph with bounded facial degree, we prove that the volume doubling property and the Poincar\'e inequality hold, which yields the Harnack…

Suppose $(M^{n},g)$ is a Riemannian manifold with nonnegative Ricci curvature, and let $h_{d}(M)$ be the dimension of the space of harmonic functions with polynomial growth of growth order at most $d$. Colding and Minicozzi proved that…

For any manifold with polynomial volume growth, we show: The dimension of the space of ancient caloric functions with polynomial growth is bounded by the degree of growth times the dimension of harmonic functions with the same growth. As a…

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…

For a proper immersed minimal disk in $\bf{R}^N$ with quadratic area growth, we show that any harmonic function whose negative part grows at a slow sub-linear rate is constant. This leads to a higher codimensional Bernstein theorem for…

Harmonic morphisms, maps which preserve Laplace's equation, are intimately connected to the topic of minimal submanifolds. In this article we first characterise harmonic morphisms between Riemannian manifolds as the weakly horizontally…

Lower bounds on Ricci curvature limit the volumes of sets and the existence of harmonic functions on Riemannian manifolds. In 1975, Shing Tung Yau proved that a complete noncompact manifold with nonnegative Ricci curvature has no…