CMC-1 SURFACES VIA OSCULATING MÖBIUS TRANSFORMATIONS BETWEEN CIRCLE PATTERNS

[en] Given two circle patterns of the same combinatorics in the plane, the Möbius transformations mapping circumdisks of one to the other induce a PSL(2,C)-valued function on the dual graph. Such a function plays the role of an osculating Möbius transformation and induces a realization of the dual graph in hyperbolic space. We characterize the realizations and obtain a one-to-one correspondence in the cases that the two circle patterns share the same discrete conformal structure. These correspondences are analogous to the Weierstrass representation for surfaces with constant mean curvature H ≡ 1 in hyperbolic space. We further establish convergence on triangular lattices.

Disciplines :

Mathematics

Author, co-author :

LAM, Wai Yeung ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

External co-authors :

Language :

English

Title :

CMC-1 SURFACES VIA OSCULATING MÖBIUS TRANSFORMATIONS BETWEEN CIRCLE PATTERNS

Publication date :

May 2024

Journal title :

Transactions of the American Mathematical Society

ISSN :

0002-9947

eISSN :

1088-6850

Publisher :

American Mathematical Society

Volume :

377

Issue :

Pages :

3657 - 3690

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://www.ams.org/tran/0000-000-00/S0002-9947-2024-09121-3/tran9121_AM.pdf

FnR Project :

FNR11554412 - SoS - Structures On Surfaces, 2016 (01/04/2018-30/09/2022) - Hugo Parlier
FNR14766753 - CoSH - Convex Surfaces In Hyperbolic Geometry, 2020 (01/09/2021-31/08/2024) - Jean-marc Schlenker

Funders :

Fonds National de la Recherche Luxembourg

Funding text :

This work was partially supported by the ANR/FNR project SoS, INTER/ANR/16/11554412/SoS, ANR-17-CE40-0033, FNR grant CoSH O20/14766753.

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since 02 April 2025

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