Another construction of a CMC-1 surface in \(H^3\) (Q2782795)

Minimal surfaces \(M\) in \({\mathbb R}^3\) in conformal parametrization can be represented as real parts of minimal curves \(\Phi\) in \({\mathbb C}^3\), i.e., of complex curves with infinitesimal arc length 0. The classical Weierstrass formula establishes (roughly speaking) a 1-1 correspondence between pairs \((g,\omega)\) consisting of a function \(g\) and a meromorphic differential form \(\omega\) on a Riemann surface \({\mathcal X}\) (the Weierstrass data of the surface) and classes of such minimal surfaces in \(M\in{\mathbb R}^3\), where two minimal surfaces \(M_1\) and \(M_2\) are regarded as equivalent, if the corresponding minimal curves \(\Phi_1\) and \(\Phi_2\) differ by a complex-orthogonal transformation and a translation of \({\mathbb C}^3\). NEWLINENEWLINENEWLINEAn analogous correspondence between such pairs \((g,\omega)\) and surfaces of constant mean curvature \(c=1\) (CMC-surfaces) in the 3-dimensional hyperbolic space \(H^3\) of sectional curvature \(-1\) was established by \textit{R. Bryant} in [Astérisque 154/155, 321-347 (1988; Zbl 0635.53047)]. CMC-surfaces of \(H^3\) and minimal surfaces of \({\mathbb R}^3\) corresponding to the same data are called cousins of each other. NEWLINENEWLINENEWLINEThe Costa-surface \(C\subset{\mathbb R}^3\) is an example of a complete, embedded minimal surface of finite total curvature. A conformal parametrization of \(C\) and the corresponding data \((g_C,\omega_C)\) are defined on the punctured torus \({\mathcal T}=({\mathbb C}\setminus\{0,(1+i)/2\})/L\), where \(L\) is the square lattice \(L={\mathbb Z}\oplus i {\mathbb Z}\). NEWLINENEWLINENEWLINEThe paper under review is concerned with a new technique for the construction of CMC-surfaces in \(H^3\) by R. Bryant's formula. A cousin of the Costa-surface \(C\) is constructed by means of this method. The conformal immersion of \({\mathcal T}\) into \(H^3\) corresponding to this cousin induces a pseudometric on \({\mathcal T}\) of curvature 1 and with conical singularities of non-integral order.