Tom
Yau Heng Wan, CUHK,
Prescribing trees for holomorphic quadratic differentials and harmonic
maps
A sufficient condition for the
existence of holomorphic quadratic differential on a
non-compact simply-connected Riemann surface with prescribed horizontal and
vertical trees is obtained. In particular, for any pair of complete R-trees of finite vertices with n+2 infinite edges, there exists a
polynomial quadratic differential on C of degree n such that the associated vertical and
horizontal trees are isometric to the given pair. Relationship to a problem in
harmonic maps will also be discussed.