15 research outputs found
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, -connected region , and let \bord\Omega be its
boundary. Let 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 where is a special type of a (possibly immersed)
genus singular flat surface, tiled by rectangles and is an energy
preserving mapping from onto . In part I the solution
of a Dirichlet problem defined on 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, -connected region , and let
\bord\Omega be its boundary. Let 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 where is a genus singular flat surface tiled
by rectangles and is an energy preserving mapping from
onto .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, -dimensional Riemannian manifold with pinched negative sectional
curvature follows as a corollary.Comment: 20 pages, 2 fugure