Convergence of the heat flow for closed geodesics

On a closed Riemannian manifold pM, gq, the geodesic heat flow deforms loops through an energydecreasing homotopy uptq. It is known that there is a sequence tn Ñ 8 so that uptnq converges in C to a closed geodesic, but it is also known that convergence fails in general. We show that for a generic set of metrics on M , the heat flow converges for all initial loops. We also show that convergence, when it holds, is in C8. The proofs are based on a Morse theory approach using the Palais-Smale condition. Geometric heat flows have been extensively studied, yet one of the simplest examples is not yet fully understood. This note clarifies several issues concerning the convergence as t Ñ 8 for the heat flow associated with maps from S to a closed Riemannian manifold pM, gq. It is well-known that every map u0 : S 1 ÑM into a closed Riemannian manifold pM, gq is homotopic to a closed geodesic. Intuitively, this can be proven by deforming u0 along the flow of the downward gradient vector field of the energy function Epuq “ 1 2 ż S1 |du| dθ (0.1) on the space of maps u : S Ñ M . There are two standard ways of directly realizing this intuition. In the “Morse theory” approach, one works on the infinite-dimensional manifold LM of finite-energy loops in M and uses the gradient of E defined by the Sobolev W 1,2 metric; this approach provides techniques for showing that the flow paths converge (cf. Section 3). Alternatively, one can use the gradient of E defined by the L metric. The resulting downward gradient flow is a solution of the “geodesic heat flow equation” 9 u “ ∇ ̊du upθ, 0q “ u0pθq. (0.2) This equation, which is the 1-dimensional harmonic map heat flow equation, is typically studied using parabolic techniques, without reference to the manifold LM . Our main purpose here is to show that Morse theory methods can be productively applied to the heat flow. It is known that the geodesic heat flow exists and is smooth for all t ą 0. Thus central issue is convergence as t Ñ 8. Specifically, do all solutions of (0.2) converge to closed geodesics as t Ñ 8, and in which norms does one have convergence? The subtleties of this problem have been repeatedly underestimated. In 1965 Eells and Sampson [ES2] asserted that the geodesic heat flow converges to a closed geodesic at tÑ8 (see also [Sa] and ([J]). Ottarsson [O] rigorously proved the long-time existence of the heat flow with smooth initial data. Very recently, L. Lin and L. Wang [LW], building on arguments of Struwe [St], extended Ottarsson’s theorem to W 1,2 initial data, proving that the heat flow exists for all time and is unique. Ottarsson also showed that there is a sequence tn Ñ 8 so that uptnq converges in C to a closed geodesic. On the other hand, the flow itself does not necessarily converge: an elegant construction of Topping yields the following fact: ̊partially supported by the NSF. MSC codes: 53C22, 58J35.