Asymptotic Properties of the Hyperbolic Metric on the Sphere With Three Conical Singularities

Abstract

Abstract. The explicit formula for the hyperbolic metric λ α, β, γ (z)|dz| on the thrice-punctured sphere P\{0, 1, ∞} with singularities of order 0 < α, β < 1, γ ≤ 1, α + β + γ > 2 at 0, 1, ∞ was given by Kraus, Roth and Sugawa in [9]. In this article we investigate the asymptotic properties of the higher order derivatives of λ α, β, γ (z) near the origin and give more precise descriptions for the asymptotic behavior of λ α, β, γ (z).

On Asymptotically Locally Hyperbolic Metrics with Negative Mass

“…, n; when M 2 is a sphere one needs n ≥ 3. There exists on M 2 \ {p k } n k=1 a smooth function ω, with puncture singularities, or cusps, at the points p k , such that the metric e ω h is complete, has constant Gauss curvature equal to −1, and such that M 2 , e ω h has finite total area; compare [8, Proposition 2.3], [17], and references therein. An artist's impression of a punctured torus can be seen in Figure 2.…”

Bounded Projective Functions and Hyperbolic Metrics with Isolated Singularities

“…After Heins' work [7], both McOwen [11] and Troyanov [17], who were unaware of M. Heins' work [7] apparently, proved the theorem for the case of θ j > 0 by using different PDE methods. Moreover, the hyperbolic metrics on the Riemann sphere with three singularities were expressed explicitly in [1,10,19] by using the Gauss hypergeometric functions and some refined properties of the metrics were also studied there.…”