Year
2020
Season
Summer
Paper Type
Master's Thesis
College
College of Arts and Sciences
Degree Name
Master of Science in Mathematical Sciences (MS)
Department
Mathematics & Statistics
NACO controlled Corporate Body
University of North Florida. Department of Mathematics and Statistics
First Advisor
Daniel Dreibelbis
Second Advisor
Raimundo Araujo Dos Santos
Third Advisor
Denis Bell
Department Chair
Richard Patterson
Abstract
We study a problem at the intersection of harmonic morphisms and real analytic Milnor fibrations. Baird and Ou establish that a harmonic morphism from G: \mathbb{R}^m \setminus V_G \rightarrow \mathbb{R}^n\setminus \{0\} defined by homogeneous polynomials of order p retracts to a harmonic morphism \psi|: S^{m-1} \setminus K_\epsilon \rightarrow S^{n-1} that induces a Milnor fibration over the sphere. In seeking to relax the homogeneity assumption on the map G, we determine that the only harmonic morphism $\varphi: \mathbb{R}^m \setminus V_G \rightarrow S^{m-1}\K_\epsilon$ that preserves \arg G is radial projection. Due to this limitation, we confirm Baird and Ou's result, yet establish further that in fact only homogeneous polynomial harmonic morphisms retract to harmonic-morphism Milnor maps over the sphere.
Suggested Citation
Griffin, Murphy, "Harmonic Morphisms with One-dimensional Fibres and Milnor Fibrations" (2020). UNF Graduate Theses and Dissertations. 977.
https://digitalcommons.unf.edu/etd/977