2013/10/17 by Gaven Martin, Martin, Gaven J
Computer Science · Mathematics · #30C #Advanced Mathematical Modeling in Engineering #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1310.4871
openalex publication_date 2013/10/17 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28
The tension equation for a mapping f: mathbb C\→ mathbb C is the\nnonlinear second order equation \
Delta f +
varphi(f) fz f
bar z = 0 nSolutions are "harmonic" mappings. Here we give a complete description of the\nsolution space of mappings of degree 1 to this equation when \φ is\nentire. Each solution is a quasiconformal surjection and when the set of\nnormalised solutions is endowed with the Teichm "uller metric, the solution\nspace is isometric to the hyperbolic plane. More generally, for harmonic\nmappings f:\Ω \→ (\\Ω,\ρ) between domains in mathbb C,\nwith \ρ(w)|dw| defining a flat metric we stablish a very strong maximum\nprinciple for the distortion - up to multiplicative factor eiv, v real\nand harmonic, the Beltrami coefficient of f-1 is quasiregular - and thus\nopen and discrete when nonconstant. This follows from the remarkable fact that\nthe Beltrami coefficient of the inverse of a harmonic mapping itself satisfies\na nonlinear homogeneous Beltrami equation.\n