Abstract
We study the space of Hopf differentials of almost fuchsian minimal immersions of compact Riemann surfaces. We show that the extrinsic curvature of the immersion at any given point is a concave function of the Hopf differential. As a consequence, we show that the set of all such Hopf differentials is a convex subset of the space of holomorphic quadratic differentials of the surface. In addition, we address the non-equivariant case, and obtain lower and upper bounds for the size of this set.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
References
Abresch, U., Rosenberg, H.: A Hopf differential for constant mean curvature surfaces in s 2\(\times \) r and h 2\(\times \) r. Acta Math. 193, 141–174 (2004)
Donaldson, S.K.: Moment maps in differential geometry. Surv. Differ. Geom. 8(1), 171–189 (2003)
El Emam, C., Seppi, A.: On the Gauss map of equivariant immersions in hyperbolic space. J. Topol. 15(1), 238–301 (2022)
Epstein, C.L.: Envelopes of horospheres and Weingarten surfaces in hyperbolic 3-space. Unpublished (1984)
Gehring, F.W.: Rings and quasiconformal mappings in space. Trans. Am. Math. Soc. 103(3), 353–393 (1962)
Huang, Z., Lucia, M., Tarantello, G.: Bifurcation for minimal surface equation in hyperbolic 3-manifolds. In: Annales de l’Institut Henri Poincaré C, Analyse non linéaire, vol. 38, pp. 243–279. Elsevier (2021)
Stephen Hodge, T.W.: Hyperkähler geometry and Teichmüller space. PhD thesis, Imperial College London (2005)
Huang, Z., Wang, B.: On almost-Fuchsian manifolds. Trans. Am. Math. Soc. 365(9), 4679–4698 (2013)
Krasnov, K., Schlenker, J.-M.: Minimal surfaces and particles in 3-manifolds. Geom. Dedicata 126(1), 187–254 (2007)
Labourie, F.: Cyclic surfaces and Hitchin components in rank 2. Ann. Math. 185(1), 1–58 (2017)
Li, O., Mochizuki, T.: Complete solutions of Toda equations and cyclic Higgs bundles over non-compact surfaces. arXiv preprint arXiv:2010.05401 (2020)
Mostow, G.D.: Quasi-conformal mappings in \( n \)-space and the rigidity of hyperbolic space forms. Publ. Math. l’IHÉS 34, 53–104 (1968)
Omori, H.: Isometric immersions of Riemannian manifolds. J. Math. Soc. Jpn. 19(2), 205–214 (1967)
Seppi, A.: Minimal discs in hyperbolic space bounded by a quasicircle at infinity. Comment. Math. Helv. 91(4), 807–839 (2016)
Trautwein, S.: The hyperkähler metric on the almost-fuchsian moduli space. EMS Surv. Math. Sci. 6(1), 83–131 (2019)
Uhlenbeck, K.K.: Closed minimal surfaces in hyperbolic 3-manifolds. In: Seminar on Minimal Submanifolds, vol. 103, pp. 147–168 (1983)
Wan, T.: Constant mean curvature surface, harmonic maps, and universal Teichmuller space. J. Differ. Geom. 35, 643–657 (1992)
Yau, S.-T.: Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28(2), 201–228 (1975)
Author information
Authors and Affiliations
Contributions
Both authors worked on the whole proof of the paper. Both authors reviewed the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Bronstein, S., Smith, G.A. On a convexity property of the space of almost fuchsian immersions. Geom Dedicata 218, 19 (2024). https://doi.org/10.1007/s10711-023-00865-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10711-023-00865-0