Celebratio Mathematica

J. T. Pitts and J. H. Rubinstein: “Existence of minimal surfaces of bounded topological type in three-manifolds,” pp. 163–176 in Miniconference on geometry and partial differential equations (Canberra, 1–3 August 1985). Edited by L. M. Simon and N. S. Trudinger. Proceedings of the Centre for Mathematical Analysis, Australian National University 10. Australian National University (Canberra), 1986. First miniconference. MR 857665 Zbl 0602.49028 incollection

Leon Melvyn Simon	Related
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Neil Sidney Trudinger	Related

J. T. Pitts and J. H. Rubinstein: “Applications of minimax to minimal surfaces and the topology of 3-manifolds,” pp. 137–170 in Miniconference on geometry and partial differential equations (Canberra, 26–27 August 1986). Edited by J. E. Hutchinson and L. M. Simon. Proceedings of the Centre for Mathematical Analysis, Australian National University 12. Australian National University (Canberra), 1987. Second miniconference. MR 924434 Zbl 0639.49030 incollection

Leon Melvyn Simon	Related
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John Edward Hutchinson	Related

J. T. Pitts and J. H. Rubinstein: “Equivariant minimax and minimal surfaces in geometric three-manifolds,” Bull. Am. Math. Soc., New Ser. 19 : 1 (1988), pp. 303–309. MR 940493 Zbl 0665.49034 article

J. Hass, J. T. Pitts, and J. H. Rubinstein: “Existence of unstable minimal surfaces in manifolds with homology and applications to triply periodic minimal surfaces,” pp. 147–162 in Differential geometry (Los Angeles, 8–28 July 1990), part 1: Partial differential equations on manifolds. Edited by R. E. Green and S.-T. Yau. Proceedings of Symposia in Pure Mathematics 54. American Mathematical Society (Providence, RI), 1993. MR 1216582 Zbl 0798.53009 incollection

Robert Everist Greene	Related
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Shing-Tung Yau	Related
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J. T. Pitts and J. H. Rubinstein: “The topology of minimal surfaces in Seifert fiber spaces,” Mich. Math. J. 42 : 3 (1995), pp. 525–535. MR 1357622 Zbl 0866.57011 article

A basic question in the theory of minimal surfaces in 3-dimensional manifolds is to decide which embeddings of surfaces can be realized by minimal surfaces. Fundamental results were obtained in the case of Riemannian metrics of positive curvature in [Frankel 1966], [Lawson 1970], and [Schoen and Yau 1979] for sectional, Ricci, and scalar curvatures, respectively. In [Rubinstein 1985] a fairly complete description was obtained of the topology of embeddings of minimal surfaces in 3-manifolds of positive scalar curvature.

Seifert fiber spaces are an important class of examples of 3-dimensional manifolds that admit 1-dimensional foliations by circles. Thurston [1978] has proposed a geometrization program for classifying closed 3-manifolds by decomposing them into pieces that admit eight geometries. Six of the eight geometries occur on Seifert fiber spaces. Moreover, the natural metrics are compatible with the Seifert fiber structure, in the sense that (possibly after passing to a double cover) the isometry group of the metric has an \( SO(2) \) component with orbits the circle fibers.

In [1984], Hass studied the topology of \( \pi_1 \)-injective minimal surfaces in Seifert fiber spaces. In Section 3 we obtain a topological classification of arbitrary embedded minimal surfaces in such 3-manifolds, extending [Hass 1984]. Finally in Section 4, using the minimax technique developed in [Pitts 1981; Pitts and Rubinstein 1986, 1987; Hass, et al. 1993], examples of interesting minimal surfaces in Seifert fiber spaces are constructed. Note that, throughout this paper, the only restriction on the Riemannian metric is that the \( SO(2) \) action of the previous paragraph be by isometries.

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J. Hyam Rubinstein

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