Crystallography and Riemann surfaces

The level set of an elliptic function is a doubly periodic point set in ℂ. To obtain a wider spectrum of point sets, we consider, more generally, a Riemann surface S immersed in ℂ² and its sections (“cuts”) by ℂ More specifically, we consider surfaces S defined in terms of a fundamental surface element obtained as a conformai map of triangular domains in ℂ. The discrete group of isometries of ℂ² generated by reflections in the triangle edges leaves S invariant and generalizes double-periodicity. Our main result concerns the special case of maps of right triangles, with the right angle being a regular point of the map. For this class of maps we show that only seven Riemann surfaces, when cut, form point sets that are discrete in ℂ. Their isometry groups all have a rank 4 lattice subgroup, but only three of the corresponding point sets are doubly periodic in ℂ. The remaining surfaces form quasiperiodic point sets closely related to the vertex sets of quasiperiodic tilings. In fact, vertex sets of familiar tilings are recovered in all cases by applying the construction to a piecewise flat approximation of the corresponding Riemann surface. The geometry of point sets formed by cuts of Riemann surfaces is no less “rigid” than the geometry determined by a tiling, and has the distinct advantage in having a regular behavior with respect to the complex parameter which specifies the cut.     

Circle packing and Riemann surfaces

Research of both authors was supported in part by the N.S.F.     

Poisson varieties from Riemann surfaces

Short survey based on talk at the Poisson 2012 conference. The main aim is to describe and give some examples of wild character varieties (naturally generalising the character varieties of Riemann surfaces by allowing more complicated behaviour at the boundary), their Poisson/symplectic structures (generalising both the Atiyah–Bott approach and the quasi-Hamiltonian approach), and the wild mapping class groups.     

p-Groups acting on Riemann surfaces

Let p be a prime integer and let $r\ge 3$ be an integer so that $p\ge 5r?7$. We show that a closed Riemann surface S of genus $g\ge 2$ has at most one p-group H of conformal automorphisms so that $S/H$ has genus zero and exactly r cone points. This, in particular, asserts that, for $r=3$ and $p\ge 11$, the minimal field of definition of S coincides with that of $(S,H)$. Another application of this fact, for the case that S is pseudo-real, is that $\mathrm{Aut}(S)/H$ must be either trivial or a cyclic group and that r is necessarily even. This generalizes a result due to Bujalance–Costa for the case of pseudo-real cyclic p-gonal Riemann surfaces.     

Uniformization of Riemann surfaces revisited

Annals of Global Analysis and Geometry - We give an elementary and self-contained proof of the uniformization theorem for noncompact simply connected Riemann surfaces.     

Limit surfaces of Riemann examples

The family of embedded, singly periodic minimal surfaces of Riemann have as limit-surfaces the helicoid, the catenoid, a single plane, or an infinite set of equally-spaced parallel planes.