Spinor representation of Bryant surfaces with catenoidal and smooth ends

In this thesis we consider surfaces of constant mean curvature 1 (CMC-1) in a hyperbolic 3-space (Bryant surfaces). For these surfaces we present a global representation in terms of holomorphic spinors which are defined on the same Riemann surface as the immersion they represent. In contrast to Bryant’s representation preferred by many researchers, our spinor representation appears to be much more useful because the holomorphic spinors are globally defined on the Riemann surface of the immersed surface. The solutions of these equations are determined in the closed form with the help of special functions which allows us to apply various tools of the complex function theory for their analysis. Among other things, by means of this approach we succeed in solving the monodromy problem, in obtaining different examples of well-defined CMC-1 surfaces and in their visualization in Mathematica software package. At a starting point of our analysis we use the global spinor representation to derive in generic case explicit formulas for the so-called CMC-1 trinoids, i.e. Bryant’s surfaces of genus zero with three regular ends which are asymptotic to the ends of catenoid cousins. These formulas involve the hypergeometric functions appearing as solutions of a Fuchsian system of differential equations with three singularities. The crux of this part of the work is the proof of the global gauge equivalence of the system of differential equations describing a CMC-1 immersion with three regular embedded ends to a Fuchsian system with three singular points. Further we solve the monodromy problem for trinoids and describe thereby all closed (well-defined) CMC-1 surfaces with three catenoidal ends in generic case. The description of non-generic CMC-1 trinoids is restricted to those with smooth catenoidal ends which constitute a family of closed surfaces and can be expressed in terms of rational functions. The solution of the case of surfaces with three regular embedded ends opens us a way to construct and describe surfaces with n, n > 3, catenoidal ends. In particular if a surface possesses additional symmetries it is possible to get explicit formulas for the immersions: we apply methods developed for trinoids to the fundamental piece (triangle) of a surface with symmetries and translate this piece according to the action of the symmetry group. Thus we obtain closed form formulas for all n-noids, n > 3, with rotational symmetries (stars, pyramidal stars, bipyramidal stars etc.), and for n-noids with ends located at the vertices of Platonic polyhedra and enjoying corresponding symmetries (tetranoid, octanoid, cubonoid, dodecanoid, icosanoid). Another special and probably more interesting class of CMC-1 surfaces are surfaces with smooth catenoidal ends, i.e. surfaces which can be smoothly extended through the ideal boundary. Here we prove that a regular embedded end of a well-defined CMC-1 immersion is always smooth and that a smooth trinoid has a meromorphic Bryant representation, i.e. it is expressed in rational functions. We establish a relationship between smooth and reducible n-noids; in particular, we show that all smooth n-noids are reducible. At the end of this part of the thesis, we prove that the Willmore energy for CMC-1 surfaces with smooth catenoidal ends is quantized just as for Euclidean minimal surfaces with planar ends. In the final part of this thesis we treat smooth CMC-1 surfaces of genus one with two catenoidal ends, the so-called smooth genus-one catenoid cousins. As in the case of Bryant’s trinoids, we prove that the matrix differential equation describing the immersion is gauge equivalent to a Schrödinger differential equation with Treibich–Verdier–Lamé potential. Using the explicit formulas for its solution we solve the monodromy problem and show that there are no smooth genus-one catenoid cousins for n = 1 and 2. We also give examples of smooth cylinders with two catenoidal ends.