On J. C. C. Nitsche's type inequality for hyperbolic space ${\mathbf{H}}^3$

Abstract

Let ${\mathbf{H}}^3$ be the hyperbolic space identified with the unit ball ${\mathbf{B}}^3=\{x\in {\mathbf{R}}^3:|x|<1\}$ with the Poincaré metric $d_h$ and assume that $\mathcal{A}(x_0,p,q):=\{x:p<d_h(x,x_0)<q\}\subset {\mathbf{H}}^3$ is an hyperbolic annulus with the inner and outer radii $0<p<q<\infty$. We prove that if there exists a proper hyperbolic harmonic mapping between annuli $\mathcal{A}(x_0,a,b)$ and $\mathcal{A}(y_0,\alpha ,\beta )$ in the hyperbolic space ${\mathbf{H}}^3$, then $\beta /\alpha >1+\psi (a,b)$, where $\psi$ is a positive function.

11 pages

References in corpus (1)

• Harmonic maps between annuli on Riemann surfaces

Cited by in corpus (1)

• A nonexistence result on harmonic diffeomorphisms between punctured spaces