Minimal area surfaces and fibered hyperbolic 3-manifolds

Farre, James; Vargas Pallete, Franco

doi:10.1090/proc/15998

Minimal area surfaces and fibered hyperbolic $3$-manifolds
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by James Farre and Franco Vargas Pallete PDF

Proc. Amer. Math. Soc. 150 (2022), 4931-4946 Request permission

Abstract:

By work of Uhlenbeck, the largest principal curvature of any least area fiber of a hyperbolic $3$-manifold fibering over the circle is bounded below by one. We give a short argument to show that, along certain families of fibered hyperbolic $3$-manifolds, there is a uniform lower bound for the maximum principal curvatures of a least area minimal surface which is greater than one.

References

Michael T. Anderson, Complete minimal varieties in hyperbolic space, Invent. Math. 69 (1982), no. 3, 477–494. MR 679768, DOI 10.1007/BF01389365
Michael T. Anderson, Complete minimal hypersurfaces in hyperbolic $n$-manifolds, Comment. Math. Helv. 58 (1983), no. 2, 264–290. MR 705537, DOI 10.1007/BF02564636
Hubert Bray, Simon Brendle, and Andre Neves, Rigidity of area-minimizing two-spheres in three-manifolds, Comm. Anal. Geom. 18 (2010), no. 4, 821–830. MR 2765731, DOI 10.4310/CAG.2010.v18.n4.a6
Robert Brooks and J. Peter Matelski, Collars in Kleinian groups, Duke Math. J. 49 (1982), no. 1, 163–182. MR 650375
Danny Calegari and David Gabai, Shrinkwrapping and the taming of hyperbolic 3-manifolds, J. Amer. Math. Soc. 19 (2006), no. 2, 385–446. MR 2188131, DOI 10.1090/S0894-0347-05-00513-8
Baris Coskunuzer, Minimal surfaces in hyperbolic 3-manifolds, Comm. Pure Appl. Math. 74 (2021), no. 1, 114–139. MR 4178181, DOI 10.1002/cpa.21961
Michael Freedman, Joel Hass, and Peter Scott, Least area incompressible surfaces in $3$-manifolds, Invent. Math. 71 (1983), no. 3, 609–642. MR 695910, DOI 10.1007/BF02095997
Joel Hass, Minimal fibrations of hyperbolic 3-manifolds, arXiv:1512.04145 [math.DG], 2015.
Zheng Huang and Ben Lowe, Beyond almost Fuchsian space, arXiv:2104.11284 [math.DG], 2022.
Zheng Huang and Biao Wang, Complex length of short curves and minimal fibrations of hyperbolic three-manifolds fibering over the circle, Proc. Lond. Math. Soc. (3) 118 (2019), no. 6, 1305–1327. MR 3957822, DOI 10.1112/plms.12216
Yair Minsky, The classification of Kleinian surface groups. I. Models and bounds, Ann. of Math. (2) 171 (2010), no. 1, 1–107. MR 2630036, DOI 10.4007/annals.2010.171.1
Jean-Pierre Otal, Les géodésiques fermées d’une variété hyperbolique en tant que nœuds, Kleinian groups and hyperbolic 3-manifolds (Warwick, 2001) London Math. Soc. Lecture Note Ser., vol. 299, Cambridge Univ. Press, Cambridge, 2003, pp. 95–104 (French, with English and French summaries). MR 2044546, DOI 10.1017/CBO9780511542817.005
Antoine Song, Existence of infinitely many minimal hypersurfaces in closed manifolds, arXiv:1806.08816 [math.DG], 2018.
R. Schoen, L. Simon, and S. T. Yau, Curvature estimates for minimal hypersurfaces, Acta Math. 134 (1975), no. 3-4, 275–288. MR 423263, DOI 10.1007/BF02392104
Clifford Henry Taubes, Minimal surfaces in germs of hyperbolic 3-manifolds, Proceedings of the Casson Fest, Geom. Topol. Monogr., vol. 7, Geom. Topol. Publ., Coventry, 2004, pp. 69–100. MR 2172479, DOI 10.2140/gtm.2004.7.69
Karen K. Uhlenbeck, Closed minimal surfaces in hyperbolic $3$-manifolds, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 147–168. MR 795233