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

Abstract

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.