Quadratic differentials and weighted graphs on compact surfaces

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Abstract

We prove that for every simply connected graph Γ embedded in a compact surface R of genus g ≥ 0, whose edges ekji carry positive weights wkji, there exist a complex structure on R and a Jenkins-Strebel quadratic differential Q(z) dz2, whose critical graph ΦQ complemented, if necessary, by second degree vertices on its edges, is homeomorphic to Γ on R and carries the same set of weights. In other words, every positive simply connected graph on R can be analytically embedded in R. We also discuss a problem on the extremal partition of R relative to such analytical embedding. As a consequence, we establish the existence of systems of disjoint simply connected domains on R with a prescribed combinatorics of their boundaries, which carry proportional harmonic measures on their boundary arcs.

Original languageEnglish
Title of host publicationAnalysis and Mathematical Physics
EditorsBjörn Gustafsson, Alexander Vasilev
PublisherSpringer International Publishing
Pages473-505
Number of pages33
ISBN (Print)9783764399054
DOIs
StatePublished - 2009
EventInternational Conference on New trends in harmonic and complex analysis, 2007 - Trondheim, Norway
Duration: May 7 2007May 12 2007

Publication series

NameTrends in Mathematics
Volume46
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X

Conference

ConferenceInternational Conference on New trends in harmonic and complex analysis, 2007
CountryNorway
CityTrondheim
Period05/7/0705/12/07

Keywords

  • Boundary combinatorics
  • Embedded graph
  • Extremal partition
  • Harmonic measure
  • Quadratic differential
  • Riemann surface

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