Circle packings on surfaces with projective structures and uniformization

Kojima, Shozo; Mizushima, Shigeru; Tan, Ser Peow

doi:10.2140/pjm.2006.225.287

Cited by 6 publications

(4 citation statements)

References 10 publications

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“…This result was previously shown for surfaces of genus at least 2 when τ has only one vertex, see [20,Theorem 1.1]. For us, Theorem 1.7 is a special case of Theorem 1.14 below, as explained there.…”

Section: A Properness Results For Circle Packingssupporting

confidence: 52%

Properness for circle packings and Delaunay circle patterns on complex projective structures

Schlenker¹,

Yarmola²

2018

Preprint

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We consider circle packings and, more generally, Delaunay circle patternsarrangements of circles arising from a Delaunay decomposition of a finite set of points -on surfaces equipped with a complex projective structure. Motivated by a conjecture of Kojima, Mizushima and Tan, we prove that the forgetful map sending a complex projective structure admitting a circle packing with given nerve (resp. a Delaunay circle pattern with given nerve and intersection angles) to the underlying complex structure is proper.

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Section: A Properness Results For Circle Packingssupporting

confidence: 52%

Properness for circle packings and Delaunay circle patterns on complex projective structures

Schlenker¹,

Yarmola²

2018

Preprint

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“…Convex surfaces in hyperbolic ends and circle packings. One motivation for Question W * imm H 3 is a beautiful conjecture of Kojima, Mizushima and Tan [KMT03,KMT06b,KMT06a]. They considered circle packings on closed surfaces equipped with a complex projective structure, which are more flexible than circle packings on closed hyperbolic or Euclidean surfaces.…”

Section: Circle Patterns and Circle Packingsmentioning

confidence: 99%

On the Weyl problem for complete surfaces in the hyperbolic and anti-de Sitter spaces

Schlenker

2020

Preprint

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The classical Weyl problem (solved by Lewy, Alexandrov, Pogorelov, and others) asks whether any metric of curvature K ≥ 0 on the sphere is induced on the boundary of a unique convex body in R 3 . The answer was extended to surfaces in hyperbolic space by Alexandrov in the 1950s, and a "dual" statement, describing convex bodies in terms of the third fundamental form of their boundary (e.g. their dihedral angles, for an ideal polyhedron) was later proved.We describe three conjectural generalizations of the Weyl problem in H 3 and its dual to unbounded convex subsets and convex surfaces, in ways that are relevant to contemporary geometry since a number of recent results and well-known open problems can be considered as special cases. One focus is on convex domain having a "thin" asymptotic boundary, for instance a quasicircle -this part of the problem is strongly related to the theory of Kleinian groups. A second direction is towards convex subsets with a "thick" ideal boundary, for instance a disjoint union of disks -here one find connections to problems in complex analysis, such as the Koebe circle domain conjecture. A third direction is towards complete, convex disks of infinite area in H 3 and surfaces in hyperbolic ends -with connections to questions on circle packings or grafting on the hyperbolic disk. Similar statements are proposed in anti-de Sitter geometry, a Lorentzian cousin of hyperbolic geometry where interesting new phenomena can occur, and in Minkowski and Half-pipe geometry. We also collect some partial new results mostly based on recent works.

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“…Question 1.4 also has a positive answer in the "limit" case where h * is replaced by a measured lamination, this is the main result of [DW08]. In another limit case where h * corresponds to the pleating pattern associated to a circle packing or circle pattern, Question 1.4 corresponds to a well-known conjecture of Kojima, Mizushima and Tan, see [KMT03,KMT06b,KMT06a,SY18].…”

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confidence: 90%