Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 915–930

ON THE RANGE OF HARMONIC MAPS IN THE PLANE

José G. Llorente

Universitat Autònoma de Barcelona, Departament de Matemàtiques
08193 Bellaterra, Barcelona, Spain; jgllorente 'at' mat.uab.cat

Abstract. We give geometrical conditions on the range of a harmonic map in the plane implying that it is constant. This approach is motivated and generalizes Lewis' proof of the classical Little Picard Theorem. We also provide results extending the harmonic Liouville Theorem and the planar version of a result by Murdoch and Kuran.

2010 Mathematics Subject Classification: Primary 30D20, 30D35, 31A05.

Key words: Picard theorem, Liouville theorem, harmonic map, harmonic function, harmonic polynomial.

Reference to this article: J. G. Llorente: On the range of harmonic maps in the plane. Ann. Acad. Sci. Fenn. Math. 45 (2020), 915–930.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4550