Mathematisches Institut Universität Tübingen
Preprints AB Analysis


Wilhelm Klingenberg, Frank Josellis

Structural Aspects of Closed Characteristics on S1-invariant Hypersurfaces

Preprint November 1995

Abstract:

Let the standard symplectic space (R^2n,w) be endowed with the S1-structure induced by the representation of the circle group S1->Sp(n) which is given by t->exp(2pi.t/tau.J). Supposing that Sigma is a compact, connected S1-hypersurface in R^2n\{0} of contact type with H^1_{dR}(Sigma)=0, we show that Sigma carries at least n distinct symmetric closed characteristics. Moreover, if the total number of closed characteristics on Sigma is finite, then all of them turn out to be symmetric. We also discuss the special case of star-shaped S1-hypersurfaces. In particular, there is a bijection f->Sigma_f between the functions f:CP^{n-1} -> R and the star-shaped S1-hypersurfaces in R^2m which allows one to reduce the investigation of symmetric closed characteristics on Sigma_f to the study of the critical points of f.

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