arXiv subjects

Daniela Kraus

Publications and source records attributed to Daniela Kraus.

20 records · Page 2Linked to original sources

arXiv2008

The behaviour of solutions of the Gaussian curvature equation near an isolated boundary point

A classical result of Nitsche \cite{Nit57} about the behaviour of the solutions to the Liouville equation $Δu=4 e^{2u}$ near isolated singularities is generalized to solutions of the Gaussian curvature equation $Δu=- κ(z) e^{2u}$ where $κ$ is a negative Hölder continuous function. As an application a higher--order version of the Yau--Ahlfors--Schwarz lemma for complete conformal Riemannian metrics is obtained.

math.AP↗

arXiv2007

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 $Δu=4 e^{2u}$ 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 $\{z_j\}$ in the unit disk there is always a Blaschke product with $\{z_j\}$ as its set of critical points. Our work is closely related to the Berger-Nirenberg problem in differential geometry.

math.CV↗