Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta

@article{Amini2013LiftingHM,
  title={Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta},
  author={Omid Amini and Matthew Baker and Erwan Brugall{\'e} and Joseph Rabinoff},
  journal={Research in the Mathematical Sciences},
  year={2013},
  volume={2},
  pages={1-67}
}
Let K be an algebraically closed, complete non-Archimedean field. The purpose of this paper is to carefully study the extent to which finite morphisms of algebraic K-curves are controlled by certain combinatorial objects, called skeleta. A skeleton is a metric graph embedded in the Berkovich analytification of X. A skeleton has the natural structure of a metrized complex of curves. We prove that a finite morphism of K-curves gives rise to a finite harmonic morphism of a suitable choice of… 

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