On the lengths of closed geodesics on almost round spheres (Q1077739)

The following theorem is proved: For every \(\epsilon >0\) there is a \(\delta =(\epsilon,n)>0\) such that every prime closed geodesic on a Riemannian sphere \((S^ n,g)\) with curvature between 1-\(\delta\) and \(1+\delta\) has length between \(2\pi\)-\(\epsilon\) and \(2\pi +\epsilon\) or else larger than 1/\(\epsilon\). This was proved for \(n=2\) in [\textit{W. Ballmann}, Invent. Math. 71, 593-597 (1983; Zbl 0505.53020)] and for general n in the special case of ellipsoids [\textit{M. Morse}, ''The calculus of variations in the large'' (1934; Zbl 0011.02802)]. The proof is based on a result for general flows of independent interest.