Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, II: The mixed Dirichlet-Neumann Problem

In this paper we continue the study started in part I (posted). We consider a planar, bounded, $m$-connected region $\Omega$, and let \bord\Omega be its boundary. Let $\mathcal{T}$ be a cellular decomposition of \Omega\cup\bord\Omega, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair $(S,f)$ where $S$ is a special type of a (possibly immersed) genus $(m-1)$ singular flat surface, tiled by rectangles and $f$ is an energy preserving mapping from ${\mathcal T}^{(1)}$ onto $S$. In part I the solution of a Dirichlet problem defined on ${\mathcal T}^{(0)}$ was utilized, in this paper we employ the solution of a mixed Dirichlet-Neumann problem.Comment: 26 pages, 16 figures (color

Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, I: The Dirichlet Problem

Consider a planar, bounded, $m$-connected region $\Omega$, and let \bord\Omega be its boundary. Let $\mathcal{T}$ be a cellular decomposition of \Omega\cup\bord\Omega, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair $(S,f)$ where $S$ is a genus $(m-1)$ singular flat surface tiled by rectangles and $f$ is an energy preserving mapping from ${\mathcal T}^{(1)}$ onto $S$.Comment: 27 pages, 11 figures; v2 - revised definition (now denoted by the flux-gradient metric (1.9)) in section 1 and minor modifications of proofs; corrected typo

Electrical networks and Stephenson's conjecture

In this paper, we consider a planar annulus, i.e., a bounded, two-connected, Jordan domain, endowed with a sequence of triangulations exhausting it. We then construct a corresponding sequence of maps which converge uniformly on compact subsets of the domain, to a conformal homeomorphism onto the interior of a Euclidean annulus bounded by two concentric circles. As an application, we will affirm a conjecture raised by Ken Stephenson in the 90's which predicts that the Riemann mapping can be approximated by a sequence of electrical networks.Comment: Comments are welcome

Poincar\'e inequality on complete Riemannian manifolds with Ricci curvature bounded below

We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincar\'e inequalities. A global, uniform Poincar\'e inequality for horospheres in the universal cover of a closed, $n$-dimensional Riemannian manifold with pinched negative sectional curvature follows as a corollary.Comment: 20 pages, 2 fugure