Abstract
We prove that for every simply connected graph $\Gamma$ embedded
in a compact surface ${\mathcal{R}}$ of genus $g\ge 0$, whose
edges $e_{kj}^i$ carry positive weights $w_{kj}^i$, there exist a
complex structure on ${\mathcal{R}}$ and a Jenkins-Strebel
quadratic differential $Q(z)\,dz^2$, whose critical graph $\Phi_Q$
complemented, if necessary, by second degree vertices on its
edges, is homeomorphic to $\Gamma$ on ${\mathcal{R}}$ and carries
the same set of weights. In other words, {\it every positive
simply connected graph on ${\mathcal{R}}$ can be analytically
embedded in} ${\mathcal{R}}$. As a consequence, we establish the
existence of systems of disjoint simply connected domains on
${\mathcal{R}}$ with a prescribed combinatorics of their
boundaries, which carry proportional harmonic measures on their
boundary arcs.
Original language | English |
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Title of host publication | Quadratic differentials and weighted graphs on compact surfaces |
Pages | 469-501 |
State | Published - 2009 |