Pseudo-Orbits of Contact Forms

Abstract

This is a brief summary of a paper to appear, where I developed some tools in order to study the Weinstein conjecture [1]. This conjecture states that any contact vector-field on a compact contact manifold (M²ⁿ⁻¹, α) has a periodic orbit, provided H¹(M²ⁿ⁻¹, Z) = 0. This conjecture has been seen to hold in case M²ⁿ⁻¹ may be embedded in R ²ⁿ, with an embedding i such that i*ω) = dα, where ω is the standard symplectic form in R ²ⁿ. The proof in the R ²ⁿ-case, after the results of Birkhoff and Seifert, starts with the work of Paul H. Rabinowitz and A. Weinstein, who studied the case where M is a starshaped or convex hypersurface in R ²ⁿ. Recently, C. Viterbo proved the conjecture under the general assumption i* ω = dα. There is, at this precise moment, a gap between the full conjecture and the known results; hopefully, it will be filled soon.