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Discrete Weierstrass-Type Representations

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Abstract

Discrete Weierstrass-type representations yield a construction method in discrete differential geometry for certain classes of discrete surfaces. We show that the known discrete Weierstrass-type representations of certain surface classes can be viewed as applications of the \(\varOmega \)-dual transform to lightlike Gauss maps in Laguerre geometry. From this construction, further Weierstrass-type representations arise. As an application of the techniques we develop, we show that all discrete linear Weingarten surfaces of Bryant or Bianchi type locally arise via Weierstrass-type representations from discrete holomorphic maps.

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Notes

  1. In this paper, the term “Weierstrass-type representation” refers to representations that are built from discrete holomorphic functions. However, this term can also refer to other representations, such as in [24] where a discrete analogue of the Kenmotsu representation is developed for discrete nonzero cmc surfaces in \({\mathbb {R}}^3\) using discrete harmonic maps.

  2. For \(B\in O(3,1)\), the same Möbius transformation is induced by \(\pm B\), precisely one of which preserves time orientation.

  3. Note that complex Möbius transformations are even in the sense that each complex Möbius transformation is the product of an even number of reflections in a circle. Therefore, complex Möbius transformations in the light cone model correspond to \(\text {SO}(3,1)\) transformations and induce isometries on \({\mathbb {H}}^3\).

  4. Choosing \(\infty \) may sound surprising, but in terms of a complex Möbius transformation, \(\infty \mapsto a/c\).

  5. This terminology is solely motivated by the smooth result in [21]. Developement of a discrete curvature theory for discrete spacelike surfaces in the lightcone could be the subject of future research.

  6. Note that many publications, such as [10], use the cross ratio factorizing function \(a_{ij}={1}/{m_{ij}}\).

  7. One can think of T(t) as a trivialisation of \(\varGamma \) in the sense that \(T(t)_i\varGamma (t)_{ij}T(t)_j^{-1} = \text {id}_{ij}\), where \(\text {id}_{ij}\) denotes the trivial map \(\{j\}\times {\mathbb {R}}^{3,1} \rightarrow \{i\}\times {\mathbb {R}}^{3,1}\).

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Acknowledgements

The authors would like to thank Joseph Cho for fruitful discussions during an impromptu stay in Kobe that sparked many ideas in this paper. We also express our gratitude to Udo Hertrich-Jeromin for his input which added significant results. Furthermore, we gratefully acknowledge financial support from the FWF research project P28427-N35 “Non-rigidity and symmetry breaking” and the JSPS/FWF Joint Project I3809-N32 “Geometric shape generation”. The first author was also supported by the GNSAGA of INdAM and the MIUR grant “Dipartimenti di Eccellenza” 2018–2022, CUP: E11G18000350001, DISMA, Politecnico di Torino and gratefully acknowledges support from the JSPS Grant-in-Aid for JSPS fellows 19J10679. The third author was partly supported by JSPS KAKENHI Grant Numbers JP18H04489, JP19J02034, JP20K14314, JP20H01801, JP20K03585, JST CREST Grant Number JPMJCR1911, and Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849). Finally, the authors would like to thank the anonymous referees for their careful reading and insightful suggestions.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK

    Mason Pember

  2. Department of Mathematics, Kobe University, 1-1 Rokkodai-cho, Nada-Ku, Kobe, 657-8501, Japan

    Denis Polly

  3. Graduate School of Technology, Industrial and Social Sciences, Tokushima University, 2-1 Minamijyousanjima-cho, Tokushima, 770-8506, Japan

    Masashi Yasumoto

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  1. Mason Pember
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Correspondence to Denis Polly.

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Pember, M., Polly, D. & Yasumoto, M. Discrete Weierstrass-Type Representations. Discrete Comput Geom 70, 816–844 (2023). https://doi.org/10.1007/s00454-022-00439-z

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