Uniformization of branched surfaces and Higgs bundles

Given a compact connected Riemann surface [Formula: see text] of genus [Formula: see text], and an effective divisor [Formula: see text] on [Formula: see text] with [Formula: see text], there is a unique cone metric on [Formula: see text] of constant negative curvature [Formula: see text] such that the cone angle at each point [Formula: see text] is [Formula: see text] [R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988) 222–224; M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793–821]. We describe the Higgs bundle on [Formula: see text] corresponding to the uniformization associated to this conical metric. We also give a family of Higgs bundles on [Formula: see text] parametrized by a nonempty open subset of [Formula: see text] that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin’s results in [N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59–126] for the case [Formula: see text].

Analytical continuation of two-dimensional wave fields

Wave fields obeying the two-dimensional Helmholtz equation on branched surfaces (Sommerfeld surfaces) are studied. Such surfaces appear naturally as a result of applying the reflection method to diffraction problems with straight scatterers bearing ideal boundary conditions. This is for example the case for the classical canonical problems of diffraction by a half-line or a segment. In the present work, it is shown that such wave fields admit an analytical continuation into the domain of two complex coordinates. The branch sets of such continuation are given and studied in detail. For a generic scattering problem, it is shown that the set of all branches of the multi-valued analytical continuation of the field has a finite basis. Each basis function is expressed explicitly as a Green’s integral along so-called double-eight contours. The finite basis property is important in the context of coordinate equations, introduced and used by the authors previously, as illustrated in this article for the particular case of diffraction by a segment.

Repeated boundary slopes for 2-bridge knots

We investigate the question of when distinct branched surfaces in the complement of a 2-bridge knot support essential surfaces with identical boundary slopes. We determine all instances in which this occurs and identify an infinite family of knots for which no boundary slopes are repeated.