[Paper Review] Metrics of constant positive curvature with conic singularities. A survey

Alexandre Erëmenko|arXiv (Cornell University)|Mar 24, 2021

Geometric Analysis and Curvature Flows19 references4 citations

TL;DR

This survey investigates conformal metrics of constant positive curvature 1 on compact Riemann surfaces with finitely many conic singularities, focusing on classification and existence via developing maps and monodromy. It establishes that for the sphere with four singularities of angles (1/2,1/2,1/2,3/2), at most one such metric exists per torus parameter τ, determined by explicit inequalities on the period ratio τ.

ABSTRACT

We consider conformal metrics of constant curvature 1 on a Riemann surface, with finitely many prescribed conic singularities and prescribed angles at these singularities. Especially interesting case which was studied by C. L. Chai, C. S Lin and C. L. Wang is described in some detail, with simplified proofs.

Motivation & Objective

• To classify and understand the existence of conformal metrics of constant curvature 1 on compact Riemann surfaces with prescribed conic singularities and angles.
• To address the open problem of non-uniqueness in spherical geometry, contrasting it with the well-understood non-positive curvature case.
• To provide simplified proofs and detailed analysis of the case studied by Chai, Lin, and Wang involving four conic singularities on the sphere.
• To establish a criterion for existence of such metrics using the monodromy of linear differential equations and the geometry of the torus double cover.

Proposed method

• Uses the developing map f: S\A → Ĉ (Riemann sphere) as a multivalued holomorphic function with PSU(2) monodromy and prescribed local behavior f(z) ~ c(z−a_j)^{α_j} near singularities.
• Reconstructs the metric via ρ(z) = 2|f′(z)| / (1 + |f(z)|²), with equivalence defined by post-composition with Möbius transformations.
• Reduces the existence problem to the unitarizability of the projective monodromy of the Lamé equation on a torus double cover.
• Analyzes the accessory parameter problem via the Lamé equation w′′ − (2℘(z) + λ)w = 0 and its monodromy, using Weierstrass elliptic functions.
• Applies Hermite’s solution formula involving σ and ζ functions to derive conditions for unitarizable monodromy in terms of ω_jζ(a) − η_ja being purely imaginary.
• Derives explicit inequalities in the τ-plane (modular parameter) that characterize the existence region for such metrics.

Experimental results

Research questions

• RQ1 For a given compact Riemann surface and prescribed conic singularities with specified angles, how many equivalence classes of constant curvature 1 metrics exist?
• RQ2 What conditions on the positions of singularities and angles ensure the existence of such a metric on the sphere?
• RQ3 How does the monodromy of the associated linear differential equation determine the existence and uniqueness of the metric?
• RQ4 In the case of four singularities with angles (1/2,1/2,1/2,3/2), how many metrics exist, and how does this depend on the torus parameter τ?
• RQ5 What is the geometric and analytic characterization of the region in the τ-plane where a solution with unitarizable monodromy exists?

Key findings

• For the sphere with four conic singularities of angles (1/2,1/2,1/2,3/2), there exists at most one equivalence class of constant curvature 1 metric for each torus parameter τ.
• The existence of such a metric is equivalent to the projective monodromy of the Lamé equation being unitarizable, which occurs precisely when two complex equations involving ω_jζ(a) − η_ja are purely imaginary.
• The region in the τ-plane for which such a metric exists is characterized by the inequalities Im( (πi)/(e_j ω_1² + η_1 ω_1) − 2τ ) < 0 for j = 1,2,3.
• The accessory parameter λ = ℘(a) is uniquely determined by the solution a of a linear equation Aa + Bā + ζ(a) = 0, which has either zero or two solutions ±a.
• Trivial solutions a = ω_j (j=1,2,3) do not yield linearly independent solutions and are therefore excluded from defining valid metrics.
• The metric on the sphere lifts to an even metric on the torus via a 2-sheeted cover, and the unique even solution corresponds to the desired spherical metric.