Equivariant Morse theory for starshaped Hamiltonian systems

Viterbo, Claude

doi:10.1090/S0002-9947-1989-0978370-5

Equivariant Morse theory for starshaped Hamiltonian systems

Author: Claude Viterbo
Journal: Trans. Amer. Math. Soc. 311 (1989), 621-655
MSC: Primary 58F05; Secondary 58E05, 58F35
DOI: https://doi.org/10.1090/S0002-9947-1989-0978370-5
MathSciNet review: 978370
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Abstract: Let $\Sigma$ be a starshaped hypersurface in ${R^{2n}}$; the problem of finding closed characteristics of $\Sigma$ can be classically reduced to a variational problem. This leads to studying an ${S^1}$-equivariant functional on a Hilbert space. The equivariant Morse theory of this functional, together with the assumption that $\Sigma$ only has finitely many geometrically distinct characteristics, leads to a remarkable formula relating the average indices of the characteristics. Using this formula one can prove, at least for $n$ even, that genetically there are infinitely many characteristics (cf. [E1] for the convex case).

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