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Unstable minimal surfaces and Heegaard splittings. (English) Zbl 0678.57009

This paper refines a result of J. T. Pitts and J. H. Rubinstein on the existence of an unstable minimal surface obtained from a Heegaard surface for a 3-manifold by a minimax process [Proc. Cent. Math. Anal. Aust. Natl. Univ. 10, 163-176 (1986; Zbl 0602.49028)], and then uses this to prove that the 3-torus has a unique genus three Heegaard splitting and to give an example of a genus four minimal surface in the 3-torus.
Reviewer: J.Hempel

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Citations:

Zbl 0602.49028
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References:

[1] Allard, W.: On the first variation of a varifold. Ann. Math.95, 417-491 (1972) · Zbl 0252.49028 · doi:10.2307/1970868
[2] Almgren, Jr., F.J., Simon, L.: Existence of embedded solutions of Plateau’s Problem. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser.6, 447-495 (1979) · Zbl 0417.49051
[3] Bonahon, F.: Diffeotopies des espaces lenticulaires. Topology22, 305-314 (1983) · Zbl 0526.57009 · doi:10.1016/0040-9383(83)90016-2
[4] Bonahon, F., Otal, J.P.: Scindements de Heegaard des espaces lenticulaires. Ann. Sci. Ec. Norm. Super., IV. Ser.16, 451-466 (1983)
[5] Boileau, M., Collins, D.J., Zieschang, H.: Scindements de Heegaard des petites varietes de Seifert. Math. Ann. (in press) · Zbl 0651.57010
[6] Federer, H.: Geometric measure theory. Berlin-Heidelberg-New York: Springer 1969 · Zbl 0176.00801
[7] Frohman, C.: Minimal surfaces and Heegaard splittings of the three-torus. Pac. J. Math.124, 119-130 (1986) · Zbl 0604.57006
[8] Hass, J., Scott, G.P.: The existence of least area surfaces in 3-manifolds. Trans. Am. Math. Soc. (in press) · Zbl 0711.53008
[9] Hass, J.: Minimal Surfaces in Seifert Fiber Spaces. Topology Appl.18, 145-151 (1984) · Zbl 0559.57005 · doi:10.1016/0166-8641(84)90006-3
[10] Hass, J.: Surfaces minimizing area in their homology classes and group actions on 3-manifolds. Math. Z. (in press) · Zbl 0715.57005
[11] Hyde, S.: Triply periodic minimal surfaces and crystallography. Thesis, Canberra University 1986
[12] Hempel, J.: Three-manfolds. Ann. Math. Studies86, Princeton University Press (1976)
[13] Jaco, W.: Adding a two-handle to a three-manifold: An application to propertyR. Proc. Am. Math. Soc.92, 288-292 (1984) · Zbl 0564.57009
[14] Meeks, W.: Lectures on Plateau’s problem. I.M.P.A., Rio, Brazil (1978)
[15] Meeks, W., Simon, L., Yau, S.T.: Embedded minimal surfaces, exotic spheres and manifolds of positive Ricci curvature. Ann. Math.116, 621-659 (1982) · Zbl 0521.53007 · doi:10.2307/2007026
[16] Morgan, F.: Geometric measure theory: A beginners guide. New York: Academic Press 1988 · Zbl 0671.49043
[17] Moriah, Y.: Heegaard splittings of Seifert fibered spaces. Invent. Math. (in press) · Zbl 0651.57012
[18] Pitts, J.T.: Existence and regularity of minimal surfaces on Riemannian manifolds. Princeton, J: University Press 1981 · Zbl 0462.58003
[19] Pitts, J.T., Rubinstein, J.H.: Existence of minimal surfaces of bounded topological type in three-manifolds. Proc. Cent. Math. Anal. Aust. Natl. Univ. Canberra,10, 163-176 (1986) · Zbl 0602.49028
[20] Pitts, J.T., Rubinstein, J.H.: Applications of minimax to minimal surfaces and the topology of three-manifolds. Proc. Cent. Math. Anal. Aust. Natl. Univ. Canberra,12, 137-170 (1987) · Zbl 0639.49030
[21] Pitts, J.T., Rubinstein, J.H.: Equivariant minimax and minimal surfaces in geometric 3-manifolds, Bull. A.M.S.19, 303-309 (1988) · Zbl 0665.49034 · doi:10.1090/S0273-0979-1988-15652-2
[22] Przytycki, J.: Incompressibility of surfaces after Dehn surgery. Mich. Math. J.30, 289-308 (1983) · Zbl 0549.57007 · doi:10.1307/mmj/1029002906
[23] Scott, G.P.: There are no fake Seifert fiber spaces. Ann. Math.117, 35-70 (1983) · Zbl 0516.57006 · doi:10.2307/2006970
[24] Scharlemann, M.: Outermost forks and a theorem of Jaco. Proceedings of the Rochester conference, 1982 · Zbl 0589.57011
[25] Simon, L., Smith, F.: On the existence of embedded minimal two-spheres in the 3-sphere endowed with an arbitrary metric. Preprint, University of Melbourne
[26] Waldhausen, F.: Heegaard-Zerlugen der 3-sphäre. Topology7, 195-203 (1968) · Zbl 0157.54501 · doi:10.1016/0040-9383(68)90027-X
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