摘要
In this paper, we study the Hofer-Zehnder capacity and the Weinstein conjecture in symplectic manifold (M×R^(2n), ω(?)σ). Let us define l_1(M, ω)=inf{<ω, α>|>0, α∈π_2(M)}. Suppose l_1(M, ω)>O, O<πr^2<2/1 l_1(M, ω). Then C_(HZ)(M×B(r))=C_(HZ)(M×Z(r))=πr^2. In the case M is a point {P}, we obtain the well-known result at present. For n>1, consider on Cp^(n-1) the standard symplectic form co such that ω[u]=n for a generator u of H_2(CP^(n-1). Suppose O<πr^2<2/1 n. ThenC_(HZ)(M×B(r))=C_(HZ)(M×Z(r))=πr^2.As an application, we claim that the Weinstein conjecture in M×Z(r) is proved correct.
In this paper, we study the Hofer-Zehnder capacity and the Weinstein conjecture in symplectic manifold (M×R<sup>2n</sup>, ω（?）σ). Let us define l<sub>1</sub>（M, ω）=inf{<ω, α>|>0, α∈π<sub>2</sub>（M）}. Suppose l<sub>1</sub>（M, ω）>O, O<πr<sup>2</sup><2/1 l<sub>1</sub>（M, ω）. Then C<sub>HZ</sub>（M×B（r））=C<sub>HZ</sub>（M×Z（r））=πr<sup>2</sup>. In the case M is a point {P}, we obtain the well-known result at present. For n>1, consider on Cp<sup>n-1</sup> the standard symplectic form co such that ω[u]=n for a generator u of H<sub>2</sub>(CP<sup>n-1</sup>. Suppose O<πr<sup>2</sup><2/1 n. ThenC<sub>HZ</sub>（M×B（r））=C<sub>HZ</sub>（M×Z（r））=πr<sup>2</sup>.As an application, we claim that the Weinstein conjecture in M×Z（r） is proved correct.
基金
Project supported by the Science Foundation of Tsinghua University
