TY - GEN

T1 - Quadratic differentials and weighted graphs on compact surfaces

AU - Solynin, Alexander Yu

N1 - Funding Information:
This research was supported in part by NSF grant DMS-0525339.
Publisher Copyright:
© 2009 Birkhäuser Verlag Basel/Switzerland.

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

KW - Boundary combinatorics

KW - Embedded graph

KW - Extremal partition

KW - Harmonic measure

KW - Quadratic differential

KW - Riemann surface

UR - http://www.scopus.com/inward/record.url?scp=84927581954&partnerID=8YFLogxK

U2 - 10.1007/978-3-7643-9906-1_25

DO - 10.1007/978-3-7643-9906-1_25

M3 - Conference contribution

AN - SCOPUS:84927581954

SN - 9783764399054

T3 - Trends in Mathematics

SP - 473

EP - 505

BT - Analysis and Mathematical Physics

A2 - Gustafsson, Björn

A2 - Vasilev, Alexander

PB - Springer International Publishing

Y2 - 7 May 2007 through 12 May 2007

ER -