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摘要
Let alpha be a contact form on S-3, let xi be its Reeb vector-field and let v be a non-singular vector-field in ker alpha. Let C-beta be the space of curves x on S-3 such x = a xi + bv,a = 0, a >= 0. Let L+, respectively L-, be the set of curves in C-beta such that b >= 0, respectively b <= 0. Let, for x is an element of C-beta, J(x) = integral(1)(0) alpha(x)(x)dt. The framework of the present paper has been introduced previously in eg [3]. We establish in this paper that some cycles (an infinite number of them, indexed by odd integers, tending to infinity) in the S-1-equivariant homology of C-beta, relative to L+ boolean OR L- and to some specially designed "bottom set", see section 4, are achieved in the Morse complex of (J, C-beta) by unions of unstable manifolds of critical points (at infinity) which must include periodic orbits of xi; ie unions of unstable manifolds of critical points at infinity alone cannot achieve these cycles. At the odd indexes (2k-1) = 1+(2k-2), 1 for the linking, (2k-2) for the S-1-equivariance, we find that the equivariant contributions of a critical point at infinity to L+ and to L- are fundamentally asymmetric when compared to those of a periodic orbit [5]. The topological argument of existence of a periodic orbit for xi turns out therefore to be surprisingly close, in spirit, to the linking/equivariant argument of P. Rabinowitz in [12]; e.g. the definition of the "bottom sets" of section 4 can be related in part to the linking part in the argument of [12]. The objects and the frameworks are strikingly different, but the original proof of [12] can be recognized in our proof, which uses degree theory, the Fadell-Rabinowitz index [8] and the fact that pi(n+1)(S-n) = Z(2), n >= 3. We need of course to prove, in our framework, that these topological classes cannot be achieved by critical points at infinity only, periodic orbits of xi excluded, and this is the fundamental difficulty. The arguments hold under the basic assumption that no periodic orbit of index 1 connects L+ and L-. It therefore follows from the present work that either a periodic orbit of index 1 connects L+ and L- (as is probably the case for all three dimensional over-twisted [8] contact forms, see the work of H. Hofer [10], the periodic orbit found in [10] should be of index 1 in the present framework); or (with a flavor of exclusion in either/or) a linking/ equivariant variational argument a la P. Rabinowitz [12] can be put to work. Existence of (possibly multiple) periodic orbits of xi, maybe of high Morse index, follows then. Therefore, to a certain extent, the present result runs, especially in the case of threedimensional over-twisted [8] contact forms, against the existence of non-trivial algebraic invariants defined by the periodic orbits of xi and independent of what ker alpha and/or alpha are.
- 出版日期2015-8
- 单位rutgers
