Extension of a key identity

Abstract

In this article, we extend a certain key identity proved by J. Jorgenson and J. Kramer in [6] to noncompact hyperbolic Riemann orbisurfaces of finite volume. This identity relates the two natural metrics, namely the hyperbolic metric and the canonical metric defined on a Riemann orbisurface.The extended version of the key identity enables us to extend the work of J. Jorgenson and J. Kramer to noncompact hyperbolic Riemann orbisurfaces of finite volume. In an upcoming article [2], using the key identity, we extend the bounds derived in [6].Acknowledgements This article is part of the PhD thesis of the author, which was completed under the supervision of J. Kramer at Humboldt Universität zu Berlin. The author would like to express his gratitude to J. Kramer for his support and many valuable scientific discussions. The author would also like to extend his gratitude to J. Jorgenson for sharing new scientific ideas, and to R. S. de Jong for many interesting scientific discussions and for pointing out a mistake in the first proof. Background materialLet Γ ⊂ PSL 2 (R) be a Fuchsian subgroup of the first kind acting by fractional linear transformations on the upper half-plane H. Let X be the quotient space Γ\H, and let g denote the genus of X. The quotient space X admits the structure of a Riemann orbisurface. Let E, P be the finite set of elliptic fixed points and cusps of X, respectively; put S = E ∪ P. For e ∈ E, let m e denote the order of e; for p ∈ P, put m p = ∞; for z ∈ X\E, put m z = 1. Let X denote X = X ∪ P. Locally, away from the elliptic fixed points and cusps, we identity X with its universal cover H, and hence, denote the points on X\S by the same letter as the points on H.Structure of X as a Riemann surface The quotient space X admits the structure of a compact Riemann surface. We refer the reader to section 1.8 in [10], for the details regarding the structure of X as a compact Riemann surface. For the convenience of the reader, we recall the coordinate functions for the neighborhoods of elliptic fixed points and cusps. Let w ∈ U r (e) denote a coordinate disk of radius r around an elliptic fixed point e ∈ E. Then, the coordinate function ϑ e (w) for the coordinate disk U r (e) is given by ϑ e (w) =w − e w − e me .

Bounds for Green’s functions on noncompact hyperbolic Riemann orbisurfaces of finite volume

“…In this article, using the key-identity from [2] and by extending the methods used in [10], we study the difference (2) on compact subsets of X, and as an application, we derive upper bounds for the canonical Green's function g X ,can (z, w) on X. Our bounds are similar to the ones derived in [10].…”

“…(4) From Proposition 2.1 in [2], (or from Proposition 2.4.1 in [3]) for a fixed w ∈ X, and for z ∈ X with Im(σ −1 p z) > Im(σ −1 p w), and Im(σ…”

“…(5) For any f ∈ C ℓ,ℓℓ (X) and for any fixed w ∈ X\Sing(f ), from Corollary 2.5 in [2] (or from Corollary 3.1.8 in [3]), we have the equality of integrals…”

“…Using the key-identity, they expressed the difference (2) in terms of integrals which involve only the hyperbolic heat kernel and the hyperbolic metric. This allowed them to derive bounds for the difference (2) in terms of invariants coming from the hyperbolic geometry of X, namely, the injectivity radius of X and the first non-zero eigenvalue λ X,1 of the hyperbolic Laplacian ∆ hyp acting on smooth functions defined on X.…”

“…In [2], we extend the key-identity from [10] to cusps and elliptic fixed points at the level of currents. This relation serves as a starting point for extending the bounds for the canonical and the hyperbolic Green's function from [10] to noncompact hyperbolic Riemann orbisurfaces of finite volume.…”