Abstract
We describe the construction of CMC surfaces with symmetries in \(\mathbb {S}^{3}\) and \(\mathbb {R}^{3}\) using a CMC quadrilateral in a fundamental tetrahedron of a tessellation of the space. The fundamental piece is constructed by the generalized Weierstrass representation using a geometric flow on the space of potentials.
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Anosov, D. V., Bolibruch, A. A.: Aspects Math, vol. E22. Braunschweig, Vieweg and Sohn (1994)
Biswas, I.: Parabolic bundles as orbifold bundles. Duke Math. J. 88(2), 305–325 (1997)
Biswas, I., Dumitrescu, S., Heller, S.: Irreducible flat sl(2,r)-connections on the trivial holomorphic bundle. J. Math. Pures Appl. 149, 28–46 (2021)
Bobenko, A. I.: Constant mean curvature surfaces and integrable equations. Uspekhi Mat. Nauk. Russian Math. Surv. 46(4), 3–42 (1991)
DGD-Gallery, https://www.discretization.de/gallery/
Dorfmeister, J., Pedit, F., Wu, H.: Weierstrass type representation of harmonic maps into symmetric spaces. Comm. Anal. Geom. 6(4), 633–668 (1998)
Fokas, A. S., Its, A. R., Kapaev, A. A., Novokshenov, V. Y.: Painlevé Transcendents: The Riemann-Hilbert Approach, Math Surveys and Monographs, vol. 128. AMS, Providence (2006)
Große-Brauckmann, K.: New surfaces of constant mean curvature. Math Z. 214(4), 527–565 (1993)
Grosse-Brauckmann, K.: Triply periodic minimal and constant mean curvature surfaces. Interface Focus 2, 582–588 (2012)
Große-Brauckmann, K., Polthier, K.: Constant mean curvature surfaces derived from Delaunay’s and Wente’s examples. Vis. Math., 119–134 (1997)
Heller, L., Heller, S.: Abelianization of Fuchsian systems on a 4-punctured sphere and applications. J. Symplectic Geom. 14(4), 1059–1088 (2016)
Heller, L., Heller, S., Schmitt, N.: Navigating the space of symmetric CMC surfaces. J. Differ. Geom. 110(3), 413–455 (2018)
Heller, L., Heller, S., Traizet, M.: Area estimates for high genus Lawson surfaces via dpw, arXiv:1907.07139 (2019)
Heller, S.: Lawson’s genus two surface and meromorphic connections. Math. Z. 274(3-4), 745–760 (2013)
Heller, S., Schmitt, N.: Deformations of symmetric cmc surfaces in the 3-sphere. Exper. Math. 24(01) (2015)
Karcher, H., Pinkall, U., Sterling, I.: New minimal surfaces in S3. J. Differ. Geom. 28(2), 169–185 (1988)
Kilian, M., Rossman, W., Schmitt, N.: Delaunay ends of constant mean curvature surfaces. Compos. Math. 144(1), 186–220 (2008)
Lawson, H., Jr.: Complete minimal surfaces in S3. Ann. Math. (2) 92, 335–374 (1970)
Manca, B.: Dpw potentials for compact symmetric cmc surfaces in. J. Geom. Phys. 156(103791), 16 (2020)
McIntosh, I.: Global solutions of the elliptic 2D periodic Toda lattice. Nonlinearity 7(1), 85–108 (1994)
Mehta, V., Seshadri, C.: Moduli of vector bundles on curves with parabolic structures. Math. Ann. 248(3), 205–239 (1980)
Oberknapp, B., Polthier, K.: An algorithm for discrete constant mean curvature surfaces. In: Hege, H.C., Polthier, K. (eds.) Visualization and Mathematics, pp 141–161 (1997)
Pirola, G.: Monodromy of constant mean curvature surface in hyperbolic space. Asian J. Math. 11(4), 651–669 (2007)
Pressley, A., Segal, G.: Oxford Mathematical Monographs, the Clarendon Press, Oxford University Press. Oxford Science Publications, New York (1986)
Schmitt, N., Kilian, M., Kobayashi, S., Rossman, W.: Unitarization of monodromy representations and constant mean curvature trinoids in 3-dimensional space forms. J. Lond. Math. Soc. (2) 75(3), 563–581 (2007)
Simpson, C.: Harmonic bundles on noncompact curves. J. Amer. Math. Soc. 3(3), 713–770 (1990)
Traizet, M.: Construction of constant mean curvature n-noids using the DPW method. J. Reine Angew. Math. 763, 223–249, arXiv:1709.00924(2020)
Traizet, M.: Gluing Delaunay ends to minimal n-noids using the dpw method. Math. Ann. 377(3), 1481–1508 (2020)
Wu, H.: A simple way for determining the normalized potentials for harmonic maps. Ann. Glob. Anal. Geom. 17, 189–199 (1999)
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The first author is partially supported by the DFG Collaborative Research Center TRR 109 Discretization in Geometry and Dynamics. The second author is supported by the DFG grant HE 6829/3-1 of the DFG priority program SPP 2026 Geometry at Infinity. The third author is supported by the DFG Collaborative Research Center TRR 109 Discretization in Geometry and Dynamics.
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Communicated by:Alexander P. Veselov
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Bobenko, A.I., Heller, S. & Schmitt, N. Constant Mean Curvature Surfaces Based on Fundamental Quadrilaterals. Math Phys Anal Geom 24, 37 (2021). https://doi.org/10.1007/s11040-021-09397-z
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DOI: https://doi.org/10.1007/s11040-021-09397-z