Metrics without Morse Index Bounds

On any surface we give an example of a metric that contains simple closed geodesics with arbitrarily high Morse index. Similarly, on any 3-manifold we give an example of a metric that contains embedded minimal tori with arbitrarily 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. 0. Introduction Let M2 be a closed orientable surface with curvature K , and let γ ⊂ M be a closed geodesic. The Morse index of γ is the index of the critical point γ for the length functional, that is, the number of negative eigenvalues (counted with multiplicity) of the second derivative of length. (Throughout, curves are always in H1.) Since the second derivative of length at γ in the direction of a normal variation u n is − ∫ γ u Lγ u, where Lγ u = u + K u, 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 φ n) is a (normal) Jacobi field. The geodesic 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 zero. Our main result is the following. THEOREM 0.1 On any M2, there exists a metric with a geodesic lamination with infinitely many unstable leaves. Moreover, there is such a metric with simple closed geodesics of arbitrarily high Morse index. The first part of Theorem 0.1 is relatively easy to achieve, and below we do that first. DUKE MATHEMATICAL JOURNAL Vol. 119, No. 2, c © 2003 Received 22 March 2002. Revision received 26 September 2002. 2000 Mathematics Subject Classification. Primary 53C22, 49Q05, 58E05. Colding’s work partially supported by National Science Foundation grant number DMS-9803253.