Laplacian Spectra on Nearly Nodal Riemann Surfaces

Consider a family of complex curves C→D such that the central fibre is a nodal Riemann surface while other fibres are smooth Riemann surfaces. We choose a family of conformal metrics by restricting a smooth metric on C. So near the nodes (with local models xy=t, where t is the coordinate on D), the metric … Read more

Uniformization of Riemann surfaces by iso-classical Schottky groups

Let Γ=<g1,…,gn> ⊂PGL2(C) be a Schottky group of rank n. The group Γ is called classical if there exists a set of 2n pairwise disjoint closed balls {B1,C1,…,Bn,Cn} such that gi(P1C∖Bi)=Ci˚ and g_i(\mathbb{P}^1_{\mathbb{C}} \setminus \mathring{B_i})= C_i for every i=1, \dots, n. The group \Gamma is called iso-classical if the balls B_i, C_i can be taken … Read more

Generalising definition of Hurwitz number of compactified moduli space of curve

I am asking mostly for reference if such a definition exists in the literature. I am also interested in the count if it appears somewhere. Let μ:=(μ1,…,μn)⊢d for positive integer d then simple Hurwitz number Hg(μ) is defined to be the number of degree d covering map say f counted with weight 1m!Aut(f) from smooth … Read more

Questions related to compact complex curves, symmetric products and linear independence

Let X be a compact complex curve and let L be a very ample line bundle over X. Denote by Cn(X) the configuration space of n (ordered) distinct points on X. Given distinct points z1, …, zn on X, so that they form an element in Cn(X), let p1=[(z2,…,zn)], p2=[(z1,z3,…,zn)], ⋮ pn=[(z1,z2,…,zn−1)] in the symmetric … Read more

Global analysis on punctured surfaces

Global analysis on open manifolds seems pretty hard. For one, the space of Cn,α functions on an open manifold need not be a tame Fréchet space (see the post Are smooth functions tame? for the case C∞(R)), so one cannot generally apply the implicit function theorem. I am interested in doing global analysis on a … Read more

Products of eigenfunctions on compact Riemann surfaces

Let M be a compact Riemann surface with genus g≥2, endowed with the Riemannian metric with constant sectional curvature −1. Let f1,f2 be two (global) eigenfunctions for the Laplace-Beltrami operator. Clearly the product f1⋅f2 admits a series expansion in terms of other eigenfunctions. My questions are the following: Is the series, by any chance, involves … Read more

Can every hyperelliptic genus 3 surface be minimally immersed in flat $T^3$

Every minimally immersed genus 3 surface in flat $T^3$ must be hyperelliptic, as the Gauss map gives the degree 2 covering map. How about the converse of this problem? The only thing I can find is that in Meeks’s paper he constructed a real 5-dimensional family of minimal genus 3 surface in $T^3$. Thanks! Answer … Read more

Conformally immersed Riemann surfaces and foliations

I want to show that conformally immersed Riemann surfaces in R4 are leaves of a 2-foliation F. I start with the generalized Weierstrass representation of the surfaces: take 4 holomorphic functions {ϕ(z)α,ψ(z)α}, α=1,2 that satisfy a Dirac equation ∂zϕα=pψα, ∂ˉzψα=−pϕα with real-valued p(z,ˉz). These define a conformal immersion into R4 with coordinates Xa(z,ˉz),a=1,2,3,4 that satisfy dX1=i2(ˉϕ1ˉϕ2+ψ1ψ2)dz+c.c. dX2=12(ˉϕ1ˉϕ2−ψ1ψ2)dz+c.c. … Read more

Zeroes of global sections killed by differential operators

I asked this question some two weeks ago on StackExchange, but received no feedback of any sort … Let X be a compact connected Riemann surface and let Φ:M→N be an elliptic differential operator where M and N are two complex line bundles on X. Let f be a C∞-global section of M (meaning that … Read more

multivalued holomorphic function on Riemann surfaces

Let M be an open Riemann surface and f a multivalued holomorphic function from M to H, where H is the upper half plane. Suppose that the monodromy of f lies in the two-dimensional Lie subgroup A of PSL(2,R), i.e. A={(ab01a):a>0,b∈R}. I conjecture that M must be a hyperbolic Riemann surface (The surface which is … Read more