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Closed geodesics on incomplete surfaces. (English) Zbl 1096.53006

The authors study the question of existence of embedded closed geodesics for a class of incomplete, positive curvature metrics on the \(2\)-sphere minus a point. The metrics in question are asymptotic to \(dr^2+rd\theta^2\) near that point and, consequently, have unbounded curvature. In their main result the authors establish the existence of an embedded closed geodesic for such a metric in two cases, where the metric satisfies an additional technical requirement. Incomplete metrics, such as the ones under discussion, arise in the study of minimal submanifolds. The authors conlcude the paper by applying their main result to obtain results on embedded minimal tori in a class of metrics on the \(3\)-sphere and embedded minimal \(3\)-tori in a class of toric Kähler metrics on \({\mathbb C}P^2\).

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C22 Geodesics in global differential geometry
57N10 Topology of general \(3\)-manifolds (MSC2010)
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