Nancy Hingston

Loop products and closed geodesics

We show the Chas-Sullivan product (on the homology of the free loop space of a Riemannian manifold) is related to the Morse index of its closed geodesics. We construct related products in the cohomology of the free loop space and of the based loop space, and show they are nontrivial.

Geodesic laminations with closed ends on surfaces and Morse index; Kupka-Smale metrics

LetM2 be a closed orientable surface with curvatureK and γ ⊂ M a closed geodesic. The Morse index of γ is the index of the critical point γ for the length functional on the space of closed curves, i.e., the number of negative eigenvalues (counted with multiplicity) of the second derivative of length. Since the second derivative of length at γ in the direction of a normal variation un is − ∫ γ uLγ u where Lγ u = u′′ +Ku, the Morse index is the number of negative eigenvalues of Lγ . (By convention, an eigenfunction φ with eigenvalue λ of Lγ is a solution of Lγ φ + λφ = 0.) Note that if λ = 0, then φ (or φn) is a (normal) Jacobi field. γ is stable if the index is zero. The index of a noncompact geodesic is the dimension of a maximal vector space of compactly supported variations for which the second derivative of length is negative definite. We also say that such a geodesic is stable if the index is 0. We give in this paper bounds for the Morse indices of a large class of simple geodesics on a surface with a generic metric. To our knowledge these bounds are the first that use only the generic hypothesis on the metric.

Metrics without Morse index bounds

On any surface we give an example of a metric that contains simple closed geodesics with arbitrary high Morse index. Similarly, on any 3-manifold we give an example of a metric that contains embedded minimal tori with arbitrary high Morse index. Previously no such examples were known. We also discuss whether or not such bounds should hold for a generic metric and why bumpy does not seem to be the right generic notion. Finally, we mention briefly what such bounds might be used for.

On the equivariant Morse complex of the free loop space of a surface

We prove two theorems about the equivariant topology of the free loop space of a surface. The first deals with the nondegenerate case and says that the “ordinary” Morse complex can be given an O(2)-action in such a way that it carries the O(2)-homotopy type of the free loop space. The second says that, in terms of topology, the iterates of an isolated degenerate closed geodesic “look like” the continuous limit of the iterates of a finite, fixed number of nondegenerate closed geodesics. Let M be a compact Riemannian manifold, and Λ = Maps(S,M) the free loop space of M . The energy function E : Λ → R has as its critical points the closed geodesics on M . The group O(2) of isometries of S acts on Λ leaving the energy function invariant. Morse theory allows one to describe the equivariant homotopy type of Λ in terms of the closed geodesics on M and their indices as critical points. We show that, when M is a surface, this connection between topology and geometry is in some sense optimal, as direct as one could hope for. The first proposition deals with the nondegenerate case and gives an equivariant version of the Morse chain complex. The second says that, in terms of topology, the iterates of a degenerate closed geodesic can be resolved as the continuous limit of the iterates of a finite, fixed number of nondegenerate closed geodesics. These results are applied in [6], where we show that if M is a two-sphere, the number N(`) of closed geodesics of length ≤ ` grows at least like the prime numbers, that is, lim inf N(`) log(`) ` > 0. Proposition I is nice from a theoretical point of view (see the third remark below), but even more important from a practical point of view, as it allows one to do something which authors since Morse (see the last remark) have attempted unsuccessfully: to do Morse theory on the free loop space equivariantly without involving the lower strata (by the action of the stability group) of the negative bundle of a critical point. Proposition I is precisely what one needs in order to carry out equivariantly the usual geometric constructions (e.g. minimax) associated with Morse theory (see Corollaries 1, 2, 3). The effect of Proposition II in the above application is to reduce the degenerate to the nondegenerate case. The analog of Proposition II seems to be true for manifolds M of higher dimension; however, as explained at the very end, we think it very unlikely that Proposition I holds in higher dimensions. Received by the editors May 4, 1996. 1991 Mathematics Subject Classification. Primary 58E10; Secondary 57R91, 53C22. c ©1998 American Mathematical Society