an:03885011
Zbl 0555.58013
Sikorav, J.-C.
Points fixes d'une application symplectique homologue ?? l'identit??
FR
J. Differ. Geom. 22, 49-79 (1985).
00150106
1985
j
37J99 57R70
symplectic manifold; Hamiltonian isotopy; fixed points; critical points; Arnold conjecture
Let (M,\(\sigma)\) be a closed symplectic manifold, and \((\phi_ t)\) a Hamiltonian isotopy of M, i.e. \({\dot \phi}{}_ t=X_ t\circ \phi_ t\) where \(X_ t\) is a Hamiltonian vectorfield. V. I. Arnold has conjectured that \(\phi_ 1\) has at least as many fixed points as a function on M has critical points. Using a method introduced by \textit{C. C. Conley} and \textit{E. Zehnder} for \(M=T^{2n}\) [Invent. Math. 73, 33-49 (1983; Zbl 0516.58017)], we prove the following result: Assume M admits a metric of nonpositive curvature such that: 1) \(\sigma (X,Y)=<JX,Y>\), where J is an almost complex structure on M; 2) \((\exp^*_ p\sigma)_ X(Y,JY)\geq \alpha | Y|^ 2\), with \(\alpha >0\), for \(p\in M\) and \(X,Y\in T_ pM\). Then \(\phi_ 1\) has at least \(CL(M)+1\) fixed points, where \(CL(M)=cuplength\) of \(M=\sup \{\ell: \exists\) a ring R and \(\omega_ 1,...,\omega_{\ell}\in \tilde H^*(M;R)\) with \(\omega_ 1\cup...\cup \omega_{\ell}\neq 0\}\). If the fixed points are nondegenerate then their number is at least \(SB(M)=\sup \{rk H^*(M;F)\), F a field\(\}\). In particular this proves the Arnold conjecture for surfaces of genus \(\geq 1\).
Zbl 0516.57018; Zbl 0516.58017