2,461 research outputs found
Critical points of inner functions, nonlinear partial differential equations, and an extension of Liouville's theorem
We establish an extension of Liouville's classical representation theorem for
solutions of the partial differential equation and combine
this result with methods from nonlinear elliptic PDE to construct holomorphic
maps with prescribed critical points and specified boundary behaviour. For
instance, we show that for every Blaschke sequence in the unit disk
there is always a Blaschke product with as its set of critical
points. Our work is closely related to the Berger-Nirenberg problem in
differential geometry.Comment: 21 page
Support points and the Bieberbach conjecture in higher dimension
We describe some open questions related to support points in the class
and introduce some useful techniques toward a higher dimensional Bieberbach
conjecture
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