The Minding formula and its applications
The author defines \(M\)-surfaces to be complete open 2-dimensional Riemannian manifolds that are conformal to compact Riemann surfaces with a finite number of points removed, and are allowed to have isolated branch points, and must satisfy certain asymptoticity conditions at its ends and branch points. The author shows that examples of \(M\)-surfaces include: 1) meromorphic surfaces (those whose metric \(ds^2\) is the finite sum of terms of the form \(|\phi_j|^2,\) where \(\phi_j\) is a meromorphic 1-form); 2) complete minimal surfaces with finite total curvature in Euclidean \(n\)-space; 3) complete constant mean curvature 1 surfaces with finite total curvature in hyperbolic 3-space. The author provides a proof of the Minding formula, and then combines the Minding formula with the Gauss-Bonnet formula to get a formula for the total Gaussian curvature of an \(M\)-surface that depends only on the branching orders of the \(M\)-surface at the ends and branch points. The author then shows the applicability of this formula by using it to reprove the Chern-Osserman inequality and the Jorge-Meeks equality for the total curvature of minimal surfaces. The author also uses this formula to reprove two inequalities of Umehara and Yamada for constant mean curvature 1 surfaces in hyperbolic 3-space: one is the strict inequality in the Cohn-Vossen inequality, and the other is the analogue of the Chern-Osserman inequality for constant mean curvature 1 surfaces in hyperbolic 3-space.
scientific article; zbMATH DE number 1333337