Energy-minimal diffeomorphisms between doubly connected Riemann surfaces

Abstract

Let $M$ and $N$ be doubly connected Riemann surfaces with boundaries and with nonvanishing conformal metrics $\sigma$ and $\rho$ respectively, and assume that $\rho$ is a smooth metric with bounded Gauss curvature ${\mathcal {K}}$ and finite area. The paper establishes the existence of homeomorphisms between $M$ and $N$ that minimize the Dirichlet energy. Among all homeomorphisms $f :M{\overset{{}_{ \tiny {\mathrm{onto}} }}{\longrightarrow }} N$ between doubly connected Riemann surfaces such that ${{\mathrm{Mod\,}}}M \leqslant {{\mathrm{Mod\,}}}N$ there exists, unique up to conformal automorphisms of M, an energy-minimal diffeomorphism which is a harmonic diffeomorphism. The results improve and extend some recent results of Iwaniec et al. (Invent Math 186(3):667–707, 2011), where the authors considered bounded doubly connected domains in the complex plane w.r. to Euclidean metric.

Appendix

Here we give two important metrics for which the results of this section can be stated in a more explicit way.

Example 10.1

Let $\rho$ be the Riemann metric $\rho =\frac{2}{1+|z|^2}.$ Equation (1.1) becomes

$$\begin{aligned} u_{z\bar{z}}-\frac{2\bar{u}}{1+|u|^2}u_z\cdot u_{\bar{z}}=0. \end{aligned}$$

(10.1)

Notice this important example. The Gauss map of a surface $\Sigma$ in $\mathbb {R}^3$ sends a point on the surface to the corresponding unit normal vector $\mathbf {n}\in \overline{\mathbb {C}} \cong S^2$. In terms of a conformal coordinate $z$ on the surface, if the surface has constant mean curvature, its Gauss map $\mathbf {n}: \Sigma \mapsto \overline{\mathbb {C}}$, is a Riemann harmonic map [33]. Since

$$\begin{aligned} \int \limits _{{\mathbb {C}}}\rho ^2(w)dudv=4\pi , \end{aligned}$$

it follows that the Riemann metric is allowable for every double connected domain (bounded or unbounded).

Example 10.2

If $u:{\mathbb {U}}\mapsto {\mathbb {U}}$ is a harmonic mapping with respect to the hyperbolic metric $\lambda =\dfrac{2}{1-|z|^2}$ then Euler-Lagrange equation of $u$ is

$$\begin{aligned} u_{z\bar{z}}+\frac{2\bar{u}}{1-|u|^2}u_z\cdot u_{\bar{z}}=0. \end{aligned}$$

(10.2)

An important example of hyperbolic harmonic mapping is the Gauss map of a space-like surfaces with constant mean curvature $H$ in the Minkowski $3$-space $M^{2,1}$ (see [4]). This metric is allowable in compact bounded domains in ${\mathbb {U}}$ but for every $r<1$, the integral $\int _{A(r,1)}\lambda ^2(w)dudv$ diverges.