Complete flat cone metrics on punctured surfaces

Sağlam, İsmail

doi:10.3906/mat-1806-23

Cited by 6 publications

(12 citation statements)

References 20 publications

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“…But our approach is significantly different then their approach. See [5], [7] and [8] for more information about flat metrics on surfaces.…”

Section: İ Smail Saglammentioning

confidence: 99%

On the Moduli Space of Flat Tori Having Unit Area

Sağlam

2021

International Electronic Journal of Geometry

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Inspiring from Thurston's asymmetric metric on Teichmüller spaces, we define and study a natural (weak) metric on the Teichmüller space of the torus. We prove that this weak metric is indeed a metric: it separates points and it is symmetric. Our main strategy to do this is to compute the metric explicitly. We relate this metric with the hyperbolic metric on the upper half-plane. We define another metric which measures how much length of a closed geodesic changes when we deform a flat structure on the torus. We show that these two metrics coincide.

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“…But our approach is significantly different then their approach. See [5], [7] and [8] for more information about flat metrics on surfaces.…”

Section: İ Smail Saglammentioning

confidence: 99%

On the Moduli Space of Flat Tori Having Unit Area

Sağlam

2021

International Electronic Journal of Geometry

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“…That is, it may not be true that the set of vertices of the triangulation coincides with the set of singular points of the surface. In [9], it was proven that any complete singular flat metric in a noncompact surface of finite type can be triangulated by finitely many isometry types of Euclidean triangles. Note that a surface is called finite type if it is obtained from a compact surface by removing finitely many points.…”

Section: Singular Flat Surfacementioning

confidence: 99%

Ideal triangulation and disc unfolding of a singular flat surface

Sağlam¹

2021

Turk J Math

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An ideal triangulation of a singular flat surface is a geodesic triangulation such that its vertex set is equal to the set of singular points of the surface. Using the fact that each pair of points in a surface has a finite number of geodesics having length ≤ L connecting them, where L is any positive number, we prove that each singular flat surface has an ideal triangulation provided that the surface has singular points when it has no boundary components, or each of its boundary components has a singular point. Also, we prove that such a surface contains a finite number of geodesics which connect its singular points so that when we cut the surface through these arcs we get a flat disc with a nonsingular interior.

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“…In this paper, we use the notation in [12]. A doubly labeled surface is a compact surface together with labeled points.…”

Section: Doubly Labeled Surfacesmentioning

confidence: 99%

“…By a regular puncture on a flat surface, we mean a puncture which has a neighborhood isometric to that of the point at infinity of a cone. Flat surfaces with possibly irregular punctures have been studied in [12]. We now state the main results of [12].…”

Section: Introductionmentioning

confidence: 99%