Abstract
Flat fronts with symmetry similar to regular polyhedrons are studied. A flat front is a flat surface in hyperbolic three-space admitting a certain kind of singularities. Explicit descriptions of regular-polyhedral flat fronts (tetrahedral, cubic, octahedral, dodecahedral and icosahedral flat fronts) are given and a characterization of them is shown.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Gálvez, J.A., Martínez, A., Milán, F.: Flat surfaces in hyperbolic 3-space. Math. Ann. 316, 419–435 (2000)
Kokubu, M., Rossman, W., Saji, K., Umehara, M., Yamada, K.: Singularities of flat fronts in hyperbolic space. Pacific J. Math. 221(2), 303–351 (2005)
Kokubu, M., Rossman, W., Umehara, M., Yamada, K.: Flat fronts in hyperbolic 3-space and their caustics. J. Math. Soc. Jpn. 59(1), 265–299 (2007)
Kokubu, M., Rossman, W., Umehara, M., Yamada, K.: Asymptotic behavior of flat surfaces in hyperbolic 3-space. J. Math. Soc. Jpn. 61(3), 799–852 (2009)
Kokubu, M., Umehara, M., Yamada, K.: An elementary proof of Small’s formula and an analogue for Legendrian curves in\({\rm PSL(2},{\mathbf{C}})\). Osaka J. Math. 40(3), 697–715 (2003)
Kokubu, M., Umehara, M., Yamada, K.: Flat fronts in hyperbolic 3-space. Pacific J. Math. 216(1), 149–175 (2004)
Martínez, A., dos Santos, J.P., Tenenblat, K.: Helicoidal flat surfaces in hyperbolic 3-space. Pacific J. Math. 264(1), 195–211 (2013)
Rossman, W., Umehara, M., Yamada, K.: Irreducible constant mean curvature 1 surfaces in hyperbolic space with positive genus. Tôhoku Math. J. 49, 449–484 (1997)
Sasaki, S.: On complete flat surfaces in hyperbolic 3-space. Kodai Math. Sem. Rep. 25, 449–457 (1973)
Umehara, M., Yamada, K.: Surfaces of constant mean curvature-\(c\)in\(H^3(-c^2)\)with prescribed hyperbolic Gauss map. Math. Ann. 304, 203–224 (1996)
Volkov, Yu. A., Vladimirova, S. M.: Isometric immersions of a Euclidean plane in Lobachevskii space, Math. Notices, Acad. Sci. USSR 10 (1972), 619–622, Russian Original (1971)
Xu, Y.: Symmetric minimal surfaces in\({\mathbb{R}}^3\). Pacific J. Math. 171(1), 275–296 (1995)
Acknowledgements
The author would like to thank the reviewer for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author states that there is no conflict of interest.
Additional information
Dedicated to Professor Masaaki Umehara and Professor Kotaro Yamada on the occasion of their sixtieth birthdays.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The author was supported by Grant-in-Aid for Scientific Research (C) No. 20K03617 from the Japan Society for the Promotion of Science.
Rights and permissions
About this article
Cite this article
Kokubu, M. Flat fronts with polyhedral symmetry in hyperbolic three-space. J. Geom. 113, 15 (2022). https://doi.org/10.1007/s00022-022-00633-7
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00022-022-00633-7