Quadratic Differentials and Weighted Graphs on Compact Surfaces

  • Alexander Yu. Solynin
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

We prove that for every simply connected graph Γ embedded in a compact surface R of genus g≥o, whose edges e i kj carry positive weights w i kj , there exist a complex structure on R and a Jenkins-Strebel quadratic differential Q(z) dz 2 , 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.

Keywords

Quadratic differential Riemann surface embedded graph boundary combinatorics harmonic measure extremal partition 
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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Alexander Yu. Solynin
    • 1
  1. 1.Texas Tech UniversityLubbockUSA

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