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Discrete Minimal Nets with Symmetries

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Minimal Surfaces: Integrable Systems and Visualisation (m:iv 2017, m:iv 2018, m:iv 2018, m:iv 2019)

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Abstract

In this paper, we extend the notion of Schwarz reflection principle for smooth minimal surfaces to the discrete analogues for minimal surfaces, and use it to create global examples of discrete minimal nets with high degree of symmetry.

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References

  1. Agafonov, S.I., Bobenko, A.I.: Discrete \(Z^\gamma \) and Painlevé equations. Int. Math. Res. Notices 2000(4), 165–193 (2000). https://doi.org/10.1155/S1073792800000118

    Article  MATH  Google Scholar 

  2. Ando, H., Hay, M., Kajiwara, K., Masuda, T.: An explicit formula for the discrete power function associated with circle patterns of Schramm type. Funkcial. Ekvac. 57(1), 1–41 (2014). https://doi.org/10.1619/fesi.57.1

    Article  MathSciNet  MATH  Google Scholar 

  3. Berglund, J., Rossman, W.: Minimal surfaces with catenoid ends. Pac. J. Math. 171(2), 353–371 (1995). https://doi.org/10.2140/pjm.1995.171.353

    Article  MathSciNet  MATH  Google Scholar 

  4. Bobenko, A.I., Bücking, U., Sechelmann, S.: Discrete minimal surfaces of Koebe type. In: Najman, L., Romon, P. (eds.) Modern Approaches to Discrete Curvature, Lecture Notes in Mathematics, vol. 2184, pp. 259–291. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-58002-9_8

  5. Bobenko, A.I., Eitner, U.: Painlevé Equations in the Differential Geometry of Surfaces, Lecture Notes in Mathematics, vol. 1753. Springer, Berlin (2000). https://doi.org/10.1007/b76883

  6. Bobenko, A.I., Pinkall, U.: Discrete isothermic surfaces. J. Reine Angew. Math. 475, 187–208 (1996). https://doi.org/10.1515/crll.1996.475.187

    Article  MathSciNet  MATH  Google Scholar 

  7. Bobenko, A.I., Pottmann, H., Wallner, J.: A curvature theory for discrete surfaces based on mesh parallelity. Math. Ann. 348(1), 1–24 (2010). https://doi.org/10.1007/s00208-009-0467-9

    Article  MathSciNet  MATH  Google Scholar 

  8. Bobenko, A.I., Suris, Y.B.: Discrete Differential Geometry. Graduate Studies in Mathematics, vol. 98. American Mathematical Society, Providence, RI (2008)

    Google Scholar 

  9. Bücking, U.: Approximation of conformal mappings by circle patterns and discrete minimal surfaces. Ph.D. Thesis, Technische Universität Berlin (2007)

    Google Scholar 

  10. Bücking, U.: Minimal surfaces from circle patterns: Boundary value problems, examples. In: Bobenko, A.I.,  Schröder, P., Sullivan, J.M., Ziegler, G.M. (eds.) Discrete Differential Geometry, Oberwolfach Semin., vol. 38, pp. 37–56. Birkhäuser, Basel (2008). https://doi.org/10.1007/978-3-7643-8621-4_2

  11. Burstall, F.E., Hertrich-Jeromin, U., Rossman, W.: Discrete linear Weingarten surfaces. Nagoya Math. J. 231, 55–88 (2018). https://doi.org/10.1017/nmj.2017.11

    Article  MathSciNet  MATH  Google Scholar 

  12. Costa, C.J.: Example of a complete minimal immersion in \( {R}^3\) of genus one and three embedded ends. Bol. Soc. Brasil. Mat. 15(1–2), 47–54 (1984). https://doi.org/10.1007/BF02584707

    Article  MathSciNet  MATH  Google Scholar 

  13. Doliwa, A., Santini, P.M., Mañas, M.: Transformations of quadrilateral lattices. J. Math. Phys. 41(2), 944–990 (2000). https://doi.org/10.1063/1.533175

    Article  MathSciNet  MATH  Google Scholar 

  14. Ejiri, N., Fujimori, S., Shoda, T.: A remark on limits of triply periodic minimal surfaces of genus 3. Topology Appl. 196(part B), 880–903 (2015). https://doi.org/10.1016/j.topol.2015.05.014

  15. Hoffman, D.A., Meeks III, W.H.: A complete embedded minimal surface in \( {R}^3\) with genus one and three ends. J. Differ. Geom. 21(1), 109–127 (1985). https://doi.org/10.4310/jdg/1214439467

    Article  MATH  Google Scholar 

  16. Hoffman, D.A., Meeks III, W.H.: Embedded minimal surfaces of finite topology. Ann. Math. (2) 131(1), 1–34 (1990). https://doi.org/10.2307/1971506

  17. Hoffmann, T.: Discrete curves and surfaces. Ph.D. Thesis, Technische Universität Berlin (2000)

    Google Scholar 

  18. Hoffmann, T., Rossman, W., Sasaki, T., Yoshida, M.: Discrete flat surfaces and linear Weingarten surfaces in hyperbolic 3-space. Trans. Am. Math. Soc. 364(11), 5605–5644 (2012). https://doi.org/10.1090/S0002-9947-2012-05698-4

    Article  MathSciNet  MATH  Google Scholar 

  19. Hoffmann, T., Sageman-Furnas, A.O., Wardetzky, M.: A discrete parametrized surface theory in \(\mathbb{R}^3\). Int. Math. Res. Not. IMRN 2017(14), 4217–4258 (2017). https://doi.org/10.1093/imrn/rnw015

    Article  MATH  Google Scholar 

  20. Jorge, L.P., Meeks III, W.H.: The topology of complete minimal surfaces of finite total Gaussian curvature. Topology 22(2), 203–221 (1983). https://doi.org/10.1016/0040-9383(83)90032-0

    Article  MathSciNet  MATH  Google Scholar 

  21. Karcher, H.: Embedded minimal surfaces derived from Scherk’s examples. Manuscripta Math. 62(1), 83–114 (1988). https://doi.org/10.1007/BF01258269

    Article  MathSciNet  MATH  Google Scholar 

  22. Karcher, H.: Construction of minimal surfaces. Preprint No. 12, SFB 256, Universität Bonn (1989)

    Google Scholar 

  23. Karcher, H.: The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions. Manuscripta Math. 64(3), 291–357 (1989). https://doi.org/10.1007/BF01165824

    Article  MathSciNet  MATH  Google Scholar 

  24. Karcher, H., Polthier, K.: Construction of triply periodic minimal surfaces. Philos. Trans. Roy. Soc. Lond. Ser. A 354(1715), 2077–2104 (1996). https://doi.org/10.1098/rsta.1996.0093

  25. Koch, E., Fischer, W.: On \(3\)-periodic minimal surfaces with noncubic symmetry. Z. Krist. 183(1–4), 129–152 (1988). https://doi.org/10.1524/zkri.1988.183.14.129

    Article  MathSciNet  MATH  Google Scholar 

  26. Konopelchenko, B.G., Schief, W.K.: Three-dimensional integrable lattices in Euclidean spaces: Conjugacy and orthogonality. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454(1980), 3075–3104 (1998). https://doi.org/10.1098/rspa.1998.0292

  27. Nutbourne, A.W., Martin, R.R.: Differential Geometry Applied to Curve and Surface Design, vol. 1. Ellis Horwood Ltd., Chichester (1988)

    Google Scholar 

  28. Pottmann, H., Liu, Y., Wallner, J., Bobenko, A.I., Wang, W.: Geometry of multi-layer freeform structures for architecture. ACM Trans. Graph. (TOG) 26(3), 65–1–65–11 (2007). https://doi.org/10.1145/1276377.1276458

  29. Rossman, W.: Minimal surfaces in \(\bf {R}^3\) with dihedral symmetry. Tohoku Math. J. (2) 47(1), 31–54 (1995). https://doi.org/10.2748/tmj/1178225634

  30. Rossman, W., Umehara, M., Yamada, K.: Irreducible constant mean curvature \(1\) surfaces in hyperbolic space with positive genus. Tohoku Math. J. (2) 49(4), 449–484 (1997). https://doi.org/10.2748/tmj/1178225055

  31. Rossman, W., Yasumoto, M.: Discrete linear Weingarten surfaces with singularities in Riemannian and Lorentzian spaceforms. In:  Izumiya, S., Ishikawa, G., Yamamoto, M., Saji, K., Yamamoto, T., Takahashi, M. (eds.) Singularities in Generic Geometry, Advanced in Studied Pure Mathematics, vol. 78, pp. 383–410. Math. Soc. Japan, Tokyo (2018). https://doi.org/10.2969/aspm/07810000

  32. Sauer, R.: Parallelogrammgitter als Modelle pseudosphärischer Flächen. Math. Z. 52, 611–622 (1950). https://doi.org/10.1007/BF02230715

    Article  MathSciNet  MATH  Google Scholar 

  33. Sauer, R.: Differenzengeometrie. Springer, Berlin-New York (1970)

    Book  Google Scholar 

  34. Sauer, R., Graf, H.: Über Flächenverbiegung in Analogie zur Verknickung offener Facettenflache. Math. Ann. 105(1), 499–535 (1931). https://doi.org/10.1007/BF01455828

    Article  MathSciNet  MATH  Google Scholar 

  35. Schief, W.K.: On the unification of classical and novel integrable surfaces. II. Differ. Geom. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459(2030), 373–391 (2003). https://doi.org/10.1098/rspa.2002.1008

  36. Schoen, A.H.: Infinite periodic minimal surfaces without self-intersections. Technical Report NASA-TN-D-5541, C-98, NASA Electronics Research Center, Cambridge, MA (1970)

    Google Scholar 

  37. Wunderlich, W.: Zur Differenzengeometrie der Flächen konstanter negativer Krümmung. Österreich. Akad. Wiss. Math.-Nat. Kl. S.-B. IIa. 160, 39–77 (1951)

    MathSciNet  MATH  Google Scholar 

  38. Xu, Y.: Symmetric minimal surfaces in \( {R}^3\). Pac. J. Math. 171(1), 275–296 (1995). https://doi.org/10.2140/pjm.1995.171.275

    Article  Google Scholar 

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Acknowledgements

The authors would like to express their gratitude to Professor Masashi Yasumoto for fruitful discussions, and the referee for valuable comments. The first author was partially supported by JSPS/FWF Bilateral Joint Project I3809-N32 “Geometric shape generation” and Grant-in-Aid for JSPS Fellows No. 19J10679; the second author was partially supported by two JSPS grants, Grant-in-Aid for Scientific Research (C) 15K04845 and (S) 17H06127 (P.I.: M.-H. Saito); the third author was partially supported by NRF 2017 R1E1A1A 03070929.

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Authors and Affiliations

  1. Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstrasse 8-10/104, 1040, Wien, Austria

    Joseph Cho

  2. Department of Mathematics, Kobe University, Kobe-shi, Hyogo-ken, 657-8501, Japan

    Wayne Rossman

  3. Department of Mathematics, Korea University, Seoul, 02841, Korea

    Seong-Deog Yang

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  1. Joseph Cho
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  2. Wayne Rossman
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  3. Seong-Deog Yang
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Correspondence to Joseph Cho .

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Editors and Affiliations

  1. Fakultät für Mathematik, Technical University Munich, Garching, Germany

    Tim Hoffmann

  2. School of Mathematical Sciences, University College Cork, Cork, Ireland

    Martin Kilian

  3. School of Mathematics & Actuarial Science, University of Leicester, Leicester, UK

    Katrin Leschke

  4. Department of Geometry & Topology, University of Granada, Granada, Spain

    Francisco Martin

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Cho, J., Rossman, W., Yang, SD. (2021). Discrete Minimal Nets with Symmetries. In: Hoffmann, T., Kilian, M., Leschke, K., Martin, F. (eds) Minimal Surfaces: Integrable Systems and Visualisation. m:iv m:iv m:iv m:iv 2017 2018 2018 2019. Springer Proceedings in Mathematics & Statistics, vol 349. Springer, Cham. https://doi.org/10.1007/978-3-030-68541-6_3

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