Quasiconformal Embeddings of Y-Pieces

We study the energy distribution of harmonic 1-forms on a compact hyperbolic Riemann surface S where a short closed geodesic is pinched. If the geodesic separates the surface into two parts, then the Jacobian variety of S develops into a variety that splits. If the geodesic is nonseparating then the Jacobian degenerates. The aim of this work is to get insight into this process and give estimates in terms of geometric data of both the initial surface S and the final surface, such as its injectivity radius and the lengths of geodesics that form a homology basis. The Jacobians in this paper are represented by Gram period matrices. As an invariant we introduce new families of symplectic matrices that compensate for the lack of full dimensional Gram-period matrices in the noncompact case.

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“…The results of this section are presented in [3]. We summarize them for convenience and slightly extend them for the present needs.…”

Section: Mappings Of Y-piecesmentioning

confidence: 95%

“…In particular, the boundary geodesic γ 3 of length goes to the horocycle hˆ of lengthˆ , where asymptoticallyˆ ∼ 2 π , as → 0. With φ thus extended we have the following variant of Theorem 5.1 in [3]:…”

Section: Mappings Of Y-piecesmentioning

confidence: 99%

Energy distribution of harmonic 1-forms and Jacobians of Riemann surfaces with a short closed geodesic

et al. 2020

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show abstract

“…The results of this subsection are presented in [BMMS14]. We summarize them for convenience and slightly extend them for the present needs.…”

Section: Mappings Of Y-piecesmentioning

confidence: 95%

“…In particular, the boundary geodesic γ 3 of length ǫ goes to the horocycle h ǫ of length ǫ, where asymptotically ǫ ∼ 2 π ǫ, as ǫ → 0. With φ thus extended we have the following variant of Theorem 5.1 in [BMMS14]:…”

Section: Mappings Of Y-piecesmentioning

confidence: 99%

Energy distribution of harmonic 1-forms and Jacobians of Riemann surfaces with a short closed geodesic

Buser¹,

Makover²,

Muetzel³

et al. 2018

Preprint

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We study the energy distribution of harmonic 1-forms on a compact hyperbolic Riemann surface S where a short closed geodesic is pinched. If the geodesic separates the surface into two parts, then the Jacobian variety of S develops into a variety that splits. If the geodesic is nonseparating then the Jacobian degenerates. The aim of this work is to get insight into this process and give estimates in terms of geometric data of both the initial surface S and the final surface, such as its injectivity radius and the lengths of geodesics that form a homology basis. The Jacobians in this paper are represented by Gram period matrices.As an invariant we introduce new families of symplectic matrices that compensate for the lack of full dimensional Gram-period matrices in the noncompact case.

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“…The last step of the proof is to quasiconfromally embed X ∈ T g,n ( ) into Φ(X) ∈ T g,n (0) in some nice way. We need the following theorem due to Buser-Makover-Muetzel-Silhol ( [5]).…”

Section: Proof Of Theorem 15mentioning

confidence: 99%