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Equivariant minimax and minimal surfaces in geometric three-manifolds. (English) Zbl 0665.49034

New infinite families of minimal surfaces in \(S^ 3\) are presented, which are constructed by a minimax procedure.
Reviewer: G.Dziuk

MSC:

49Q05 Minimal surfaces and optimization
49Q20 Variational problems in a geometric measure-theoretic setting
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49J35 Existence of solutions for minimax problems
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[1] J. Hass and P. Scott, The existence of least area surfaces, preprint. · Zbl 0711.53008
[2] Wu-yi Hsiang and H. Blaine Lawson Jr., Minimal submanifolds of low cohomogeneity, J. Differential Geometry 5 (1971), 1 – 38. · Zbl 0219.53045
[3] H. Karcher, U. Pinkall, and I. Sterling, New minimal surfaces in \?³, J. Differential Geom. 28 (1988), no. 2, 169 – 185. · Zbl 0653.53004
[4] H. B. Lawson, Complete minimal surfaces in S3, Ann. of Math. (2) 90 (1970), 335-374. · Zbl 0205.52001
[5] William Meeks III, Leon Simon, and Shing Tung Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621 – 659. · Zbl 0521.53007
[6] P. Orlik, Seifert fiber spaces, Lecture Notes in Math., vol. 291, Springer-Verlag, Berlin and New York, 1981.
[7] Jon T. Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, Mathematical Notes, vol. 27, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981. · Zbl 0462.58003
[8] Jon T. Pitts and J. H. Rubinstein, Existence of minimal surfaces of bounded topological type in three-manifolds, Miniconference on geometry and partial differential equations (Canberra, 1985) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 10, Austral. Nat. Univ., Canberra, 1986, pp. 163 – 176.
[9] J. Pitts and J. H. Rubinstein, Minimal surfaces of bounded topological type in three-manifolds, preprint. · Zbl 0602.49028
[10] Jon T. Pitts and J. H. Rubinstein, Applications of minimax to minimal surfaces and the topology of 3-manifolds, Miniconference on geometry and partial differential equations, 2 (Canberra, 1986) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 12, Austral. Nat. Univ., Canberra, 1987, pp. 137 – 170.
[11] R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127 – 142. · Zbl 0431.53051
[12] Peter Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401 – 487. · Zbl 0561.57001
[13] P. Scott, There are no false Seifert fibre spaces with infinite \pi 1, Ann. of Math. (2) 117 (1983), 35-70. · Zbl 0516.57006
[14] L. Simon and F. Smith, On the existence of embedded minimal 2-spheres in the 3-sphere, endowed with an arbitrary metric, preprint.
[15] W. Thurston, Geometry and topology of 3-manifolds, mimeographed lecture notes, Princeton Univ., 1978.
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