Opening nodes on horosphere packings

Traizet, Martin

doi:10.1090/tran/6550

Opening nodes on horosphere packings
HTML articles powered by AMS MathViewer

by Martin Traizet PDF

Trans. Amer. Math. Soc. 368 (2016), 5701-5725 Request permission

Abstract:

We use Bryant representation to construct constant mean curvature-1 surfaces in hyperbolic space by desingularisation of a horosphere packing.

References

Robert L. Bryant, Surfaces of mean curvature one in hyperbolic space, Astérisque 154-155 (1987), 12, 321–347, 353 (1988) (English, with French summary). Théorie des variétés minimales et applications (Palaiseau, 1983–1984). MR 955072
Pascal Collin, Laurent Hauswirth, and Harold Rosenberg, The geometry of finite topology Bryant surfaces, Ann. of Math. (2) 153 (2001), no. 3, 623–659. MR 1836284, DOI 10.2307/2661364
Benoît Daniel, Surfaces de Bryant dans $\Bbb H^3$ de type fini, Bull. Sci. Math. 126 (2002), no. 7, 581–594 (French, with English and French summaries). MR 1931187, DOI 10.1016/S0007-4497(02)01130-2
Benoît Daniel, Flux for Bryant surfaces and applications to embedded ends of finite total curvature, Illinois J. Math. 47 (2003), no. 3, 667–698. MR 2007230
Benoît Daniel, Minimal disks bounded by three straight lines in Euclidean space and trinoids in hyperbolic space, J. Differential Geom. 72 (2006), no. 3, 467–508. MR 2219941
J. Dorfmeister, F. Pedit, and H. Wu, Weierstrass type representation of harmonic maps into symmetric spaces, Comm. Anal. Geom. 6 (1998), no. 4, 633–668. MR 1664887, DOI 10.4310/CAG.1998.v6.n4.a1
John D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973. MR 0335789
H. Blaine Lawson Jr., Complete minimal surfaces in $S^{3}$, Ann. of Math. (2) 92 (1970), 335–374. MR 270280, DOI 10.2307/1970625
Howard Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J. 43 (1976), no. 3, 623–635. MR 417456
Frank Pacard and Fernando A. A. Pimentel, Attaching handles to constant-mean-curvature-1 surfaces in hyperbolic 3-space, J. Inst. Math. Jussieu 3 (2004), no. 3, 421–459. MR 2074431, DOI 10.1017/S147474800400012X
Harold Rosenberg, Bryant surfaces, The global theory of minimal surfaces in flat spaces (Martina Franca, 1999) Lecture Notes in Math., vol. 1775, Springer, Berlin, 2002, pp. 67–111. MR 1901614, DOI 10.1007/978-3-540-45609-4_{3}
Wayne Rossman, Masaaki Umehara, and Kotaro Yamada, Irreducible constant mean curvature $1$ surfaces in hyperbolic space with positive genus, Tohoku Math. J. (2) 49 (1997), no. 4, 449–484. MR 1478909, DOI 10.2748/tmj/1178225055
Ricardo Sa Earp and Eric Toubiana, On the geometry of constant mean curvature one surfaces in hyperbolic space, Illinois J. Math. 45 (2001), no. 2, 371–401. MR 1878610
Michael Taylor, Introduction to Differential Equations. Pure and Applied Undergraduate Texts 14, American Mathematical Society (2011).
Gerald Teschl, Ordinary differential equations and dynamical systems, Graduate Studies in Mathematics, vol. 140, American Mathematical Society, Providence, RI, 2012. MR 2961944, DOI 10.1090/gsm/140
Martin Traizet, Adding handles to Riemann’s minimal surfaces, J. Inst. Math. Jussieu 1 (2002), no. 1, 145–174. MR 1954942, DOI 10.1017/S147474800200004X
Martin Traizet, An embedded minimal surface with no symmetries, J. Differential Geom. 60 (2002), no. 1, 103–153. MR 1924593
Martin Traizet, On the genus of triply periodic minimal surfaces, J. Differential Geom. 79 (2008), no. 2, 243–275. MR 2420019
Martin Traizet, Opening infinitely many nodes, J. Reine Angew. Math. 684 (2013), 165–186. MR 3181559, DOI 10.1515/crelle-2011-0007
Masaaki Umehara and Kotaro Yamada, A parametrization of the Weierstrass formulae and perturbation of complete minimal surfaces in $\mathbf R^3$ into the hyperbolic $3$-space, J. Reine Angew. Math. 432 (1992), 93–116. MR 1184761, DOI 10.1515/crll.1992.432.93
Masaaki Umehara and Kotaro Yamada, Complete surfaces of constant mean curvature $1$ in the hyperbolic $3$-space, Ann. of Math. (2) 137 (1993), no. 3, 611–638. MR 1217349, DOI 10.2307/2946533
Masaaki Umehara and Kotaro Yamada, A duality on CMC-$1$ surfaces in hyperbolic space, and a hyperbolic analogue of the Osserman inequality, Tsukuba J. Math. 21 (1997), no. 1, 229–237. MR 1467234, DOI 10.21099/tkbjm/1496163174