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Minimal spheres of arbitrarily high Morse index. (English) Zbl 1104.53055

This is a beautiful article related to a question of Pitts and Rubinstein, if embedded minimal surfaces of fixed genus in a three-manifold have bounded Morse index [see J. T. Pitts and J. H. Rubinstein, Applications of minimax to minimal surfaces and topology of \(3\)-manifolds, Geometry and partial differential equations, 2nd Miniconf., Canberra/Aust. 1986, Proc. Cent. Math. Anal. Aust. Natl. Univ. 12, 137–170 (1987; Zbl 0639.49030)]. The authors construct metrics for which there are embedded minimal spheres of arbitrarily large Morse index. The main result is the following.
Theorem 1. On any three-manifold there exists a metric for which there are embedded minimal spheres of arbitrarily large Morse index. On the three-sphere it can be chosen to have non-negative Ricci curvature.
The authors use techniques from W. Hsiang and H. B. Lawson [J. Differ. Geom. 5, 1–38 (1971; Zbl 0219.53045)]. They observe that their result is complementary to the result given in T. Colding and N. Hingston, [Duke Math. J. 119, No. 2, 345–365 (2003; Zbl 1059.53037)].

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C20 Global Riemannian geometry, including pinching
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