Existence and uniqueness of complete constant mean curvature surfaces at infinity of \({\mathbb{H}}^3\) (Q1589350)

The author introduces in the paper the notion of ``a surface patch at infinity'' of \(\mathbb H^3,\) denoted by \(\Sigma_\infty,\) that is a conformally embedded image in the asymptotic boundary \(\partial_\infty H^3\) of a metric surface patch. The author calls a ``surface at infinity'' the conformally immersed image in \(\partial_\infty H^3\) of a metric surface. He also gives the notion of ``surface of constant mean curvature at infinity of \(\mathbb H^3\)''. He infers the following results: Theorem A. Given a holomorphic quadratic differential \(A\) on either the complex plane \(\mathbb C\) or the unit disc \({\mathcal D} =\{|z |< 1 \},\) there exists a unique complete surface \(\Sigma_\infty\) of constant mean curvature \(\pm 1\) at infinity of \(\mathbb H^3\) with \(A\) as the holomorphic part of its second fundamental form. In the first case, \(\Sigma_\infty\) is flat and in the second it is hyperbolic. Thus the surfaces of constant mean curvature at \(\partial_\infty H^3\) are parametrized up to isometry by holomorphic quadratic differentials on either \(\mathbb C\) or \(\mathcal D,\) with the exception of the sphere \(\partial_\infty H^3\) itself. Theorem B. The complete constant mean curvature surfaces at infinity of \(\mathbb H^3\) are in a natural \(1-1\) correspondence with quadratic differentials on either \(\mathbb C\cup\{\infty\}\), \(\mathbb C,\) or \(\mathcal D.\) The map from quadratic differentials to surfaces (up to ambient isometry) is given by solving the Schwarzian differential equation, and the inverse is via the map from surfaces to the \(zz\)-part of their second fundamental form.