Working with the Hopf differential
Frédéric Hélein · Birkhäuser Basel eBooks · 2001
Let Ω be an open subset of ℂ and a Riemannian manifold as before. Let be any map. Define the function Recall that f is holomorphic if u is harmonic. Since this property of u is invariant by conformal changes of variables, one might be interested in the behaviour of f by such a transformation. So let’s choose a conformal map where Ω 1 is the domain of a map u : and let’s check how the corresponding function f transforms. We write for the coordinates of Ω 2 and Ω 1 , respectively. Set Compute These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.