Non-completeness of the Arakelov-induced metric on moduli space of curves

Jorgenson, Jay; Kramer, Jürg

doi:10.1007/s00229-006-0625-2

“…Further results concerning eigenvalue and eigenfunction convergence have been obtained by Judge in [17] and [18], and Wolpert used degenerating techniques to study the problem of existence of L 2 eigenfunctions on general finite volume hyperbolic Riemann surfaces. More recently, in [11], it was shown that one can use the results from [16] to prove results for other metrics, namely it was shown that the metric on Teichmüller space induced from the canonical metric is not complete.…”

Section: Introductionmentioning

confidence: 99%

On the appearance of Eisenstein series through degeneration

Garbin¹,

Jorgenson²,

Munn³

2008

Comment. Math. Helv.

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Let Γ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane H, and let M = Γ\H be the associated finite volume hyperbolic Riemann surface. If γ is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If γ is hyperbolic, then, following ideas due to Kudla-Millson, there is a corresponding hyperbolic Eisenstein series. In this article, we study the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. If γ ∈ Γ corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface.

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“…More recently, in [11], it was shown that one can use the results from [16] to prove results for other metrics, namely it was shown that the metric on Teichmüller space induced from the canonical metric is not complete.…”

Section: Introductionmentioning

confidence: 99%

On the Behavior of Eisenstein Series Through Elliptic Degeneration

Garbin

¹

,

Pippich

²

2009

Commun. Math. Phys.

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Let Γ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane H, and let M = Γ\H be the associated finite volume hyperbolic Riemann surface. If γ is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If γ is hyperbolic, then, following ideas due to Kudla-Millson, there is a corresponding hyperbolic Eisenstein series. In this article, we study the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. If γ ∈ Γ corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface.

show abstract