Duke Mathematical Journal

Effective finiteness of irreducible Heegaard splittings of non-Haken 3-manifolds

Tobias Holck Colding and David Gabai

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Abstract

The main result is a short effective proof of Tao Li’s theorem that a closed non-Haken hyperbolic 3-manifold N has at most finitely many irreducible Heegaard splittings. Along the way we show that N has finitely many branched surfaces of pinched negative sectional curvature carrying all closed index-1 minimal surfaces. This effective result, together with the sequel with Daniel Ketover, solves the classification problem for Heegaard splittings of non-Haken hyperbolic 3-manifolds.

Article information

Source
Duke Math. J., Volume 167, Number 15 (2018), 2793-2832.

Dates
Received: 4 October 2015
Revised: 10 January 2018
First available in Project Euclid: 3 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1538532049

Digital Object Identifier
doi:10.1215/00127094-2018-0022

Mathematical Reviews number (MathSciNet)
MR3865652

Zentralblatt MATH identifier
06982207

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds

Keywords
Heegaard splitting 3-manifold lamination hyperbolic minimal surface

Citation

Colding, Tobias Holck; Gabai, David. Effective finiteness of irreducible Heegaard splittings of non-Haken $3$ -manifolds. Duke Math. J. 167 (2018), no. 15, 2793--2832. doi:10.1215/00127094-2018-0022. https://projecteuclid.org/euclid.dmj/1538532049


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References

  • [1] W. Breslin, Curvature bounds for surfaces in hyperbolic $3$-manifolds, Canad. J. Math. 62 (2010), 994–1010.
  • [2] A. Casson and C. Gordon, Reducing Heegaard splittings, Topol. Appl. 27 (1987), 275–283.
  • [3] S. Y. Cheng and J. Tysk, An index characterization of the catenoid and index bounds for minimal surfaces in ${\mathbf{R}}^{4}$, Pacific J. Math. 134 (1988), 251–260.
  • [4] T. H. Colding, D. Gabai, and D. Ketover, On the classification of Heegaard splittings, Duke Math. J. 167 (2018), 2833–2856.
  • [5] T. H. Colding and W. P. Minicozzi, II, “Embedded minimal surfaces without area bounds in $3$-manifolds” in Geometry and Topology (Aarhus, 1998), Contemp. Math. 258, Amer. Math. Soc., Providence, 2000, 107–120.
  • [6] T. H. Colding and W. P. Minicozzi, II, Estimates for parametric elliptic integrands, Int. Math. Res. Not. 2002, no. 6, 291–297.
  • [7] T. H. Colding and W. P. Minicozzi, II, The space of embedded minimal surfaces of fixed genus in a $3$-manifold, I: Estimates off the axis for disks, Ann. of Math. (2) 160 (2004), 27–68.
  • [8] T. H. Colding and W. P. Minicozzi, II, The space of embedded minimal surfaces of fixed genus in a $3$-manifold, III: Planar domains, Ann. of Math. (2) 160 (2004), 523–572.
  • [9] T. H. Colding and W. P. Minicozzi, II, The Calabi-Yau conjectures for embedded surfaces, Ann. of Math. (2) 167 (2008), 211–243.
  • [10] T. H. Colding and W. P. Minicozzi, II, The space of embedded minimal surfaces of fixed genus in a $3$-manifold, V: Fixed genus, Ann. of Math. (2) 181 (2015), 1–153.
  • [11] T. H. Colding and W. P. Minicozzi, II, A Course in Minimal Surfaces, Grad. Stud. Math. 121, Amer. Math. Soc., Providence, 2011.
  • [12] T. H. Colding and W. P. Minicozzi, II, The singular set of mean curvature flow with generic singularities, Invent. Math. 204 (2016), 443–471.
  • [13] P. Collin, L. Hauswirth, L. Mazet, and H. Rosenberg, Minimal surfaces in finite volume non compact hyperbolic $3$-manifolds, Trans. Amer. Math. Soc. 369 (2017), 4293–4309.
  • [14] M. Gromov, Hyperbolic Groups, Math. Sci. Res. Inst. Publ. 8, Springer, New York, 1987, 75–263.
  • [15] W. Haken, Theorie der Normalflächen, Acta Math. 105 (1961), 245–375.
  • [16] G. Huisken and A. Polden, “Geometric evolution equations for hypersurfaces” in Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996), Lecture Notes in Math. 1713, Springer, New York, 1999, 45–84.
  • [17] W. Jaco and U. Oertel, An algorithm to decide if a 3-manifold is a Haken manifold, Topology 23 (1984), 195–209.
  • [18] K. Johannson, Topology and Combinatorics of $3$-Manifolds, Lecture Notes in Math. 1599, Springer, Berlin, 1995.
  • [19] D. Ketover, Genus bounds for min-max minimal surfaces, to appear in J. Differential Geom., preprint, arXiv:1312.2666 [math.DG].
  • [20] D. Ketover and Y. Liokumovich, On the existence of unstable minimal Heegaard surfaces, preprint, arXiv:1709.09744 [math.DG].
  • [21] D. Ketover, Y. Liokumovich, and A. Song, On the existence of minimal Heegaard surfaces, in preparation.
  • [22] T. Li, Heegaard surfaces and measured laminations, I: The Waldhausen conjecture, Invent. Math. 167 (2007), 135–177.
  • [23] T. Li, Heegaard surfaces and measured laminations, II: Non-Haken $3$-manifolds, J. Amer. Math. Soc. 19 (2006), 625–657.
  • [24] T. Li, An algorithm to determine the Heegaard genus of a $3$-manifold, Geom. Topol. 15 (2011), 1029–1106.
  • [25] F. J. Lopez and A. Ros, Complete minimal surfaces with index one and stable constant mean curvature surfaces, Comment. Math. Helv. 64 (1989), 34–43.
  • [26] F. Luo, S. Tillmann, and T. Yang, Thurston’s spinning construction and solutions to the hyperbolic gluing equations for closed hyperbolic $3$-manifolds, Proc. Amer. Math. Soc. 141, 335–350.
  • [27] J. Manning, Algorithmic detection and description of hyperbolic structures on closed $3$-manifolds with solvable word problem, Geom. Topol. 6 (2002), 1–26.
  • [28] L. Mosher and U. Oertel, Two-dimensional measured laminations of positive Euler characteristic, Q. J. Math. 52 (2001), 195–216.
  • [29] U. Oertel, Measured laminations in $3$-manifolds, Trans. Amer. Math. Soc. 305 (1988), 531–573.
  • [30] J. Pitts and J. Rubinstein, “Applications of minimax to minimal surfaces and the topology of $3$-manifolds” in Microconference on Geometry and Partial Differential Equations, 2(Canberra, 1986), Proc. Centre Math. Anal. Austral. Nat. Univ. 12, Austral. Nat. Univ., Canberra, 1987, 137–170.
  • [31] J. F. Plante, Foliations with measure preserving holonomy, Ann. of Math. (2) 102 (1975), 327–361.
  • [32] J. H. Rubinstein, “Minimal surfaces in geometric $3$-manifolds” in Global Theory of Minimal Surfaces, Clay Math. Proc. 2, Amer. Math. Soc., Providence, 2005, 725–746.
  • [33] M. Scharlemann, Local detection of strongly irreducible Heegaard splittings, Topol. Appl. 90 (1998), 135–147.
  • [34] R. M. Schoen, “Estimates for stable minimal surfaces in three-dimensional manifolds” in Seminar on Minimal Submanifolds, Ann. of Math. Stud. 103, Princeton Univ. Press, Princeton, 1983, 111–126.
  • [35] A. Song, Local min-max surfaces and strongly irreducible minimal Heegaard splittings, preprint, arXiv:1706.01037 [math.DG].
  • [36] M. Stocking, Almost normal surfaces in $3$-manifolds, Trans. Amer. Math. Soc. 352 (2000), 171–207.
  • [37] F. Waldhausen, “Some problems on $3$-manifolds” in Algebraic and Geometric Topology (Stanford, 1976), Proc. Sympos. Pure Math. 32, Amer. Math. Soc., Providence, 1978, 313–322.
  • [38] B. White, The space of minimal submanifolds for varying Riemannian metrics, Indiana Univ. Math. J. 40 (1991), 161–200.