The method by McDuff is used to get the existence of the J holomorphic sphere in some symplectic Manifolds and then the J holomorphic sphere is perturbed to split a periodic solution of Hamiltonian systems in these sy...The method by McDuff is used to get the existence of the J holomorphic sphere in some symplectic Manifolds and then the J holomorphic sphere is perturbed to split a periodic solution of Hamiltonian systems in these sympletic manifolds. As a result,the Weinstein conjecture is proved in the asymptotically manifolds. 展开更多
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...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.展开更多
文摘The method by McDuff is used to get the existence of the J holomorphic sphere in some symplectic Manifolds and then the J holomorphic sphere is perturbed to split a periodic solution of Hamiltonian systems in these sympletic manifolds. As a result,the Weinstein conjecture is proved in the asymptotically manifolds.
基金Project supported by the Science Foundation of Tsinghua University
文摘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.
