Closed geodesics on Riemannian manifolds via the heat flow (Q1075610)

On a complete Riemannian manifold M the author considers the heat equation \(\partial_ t f_ t=\tau (f_ t)\), where \(f_ t: S^ 1\to M\) is the solution with \(t\geq 0\) and \(\tau (f_ t)=\nabla_{\partial_{\theta}} \partial_{\theta} f_ t\), \(\theta \in S^ 1\). The author points out a condition for boundedness of solutions and proves that under this condition the initial value problem of the heat equation admits a unique solution \(f_ t\) defined for all \(t\geq 0\). Moreover, the solution \(f_ t\) has the following basic property of subconvergence: there is a sequence \(t_ k\to \infty\) such that the curves \(f_{t_ k}\) converge uniformly to a closed geodesic of M. This paper follows closely the work of \textit{J. Eells} and \textit{J. H. Sampson} [Am. J. Math. 86, 109-160 (1964; Zbl 0122.401)]. A significant difference between the present paper and the cited work is that here, when the domain is the circle \(S^ 1\), it is not necessary to impose further conditions involving the curvature of M and the embedding of M in some Euclidean space.