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139, 661–702 (1994) 40. : Tiling spaces are Cantor fiber bundles Ergith and Dya. Systs. 23, 307– 316(2003) 41. , 1990 42. : Modules projectifs et espaces fibr´es a` fibre vectorielle, (1958) S´eminaire P. -L. Dubreil-Jacotin et C. Pisot, 1957/58, Fasc. 2, Expos´e 23, Paris: Secr´etariat math´ematique, 18 pp. Spaces of Tilings, Finite Telescopic Approximations and Gap-Labeling 41 43. : Algebraic Topology, New York: McGraw-Hill, 1966 44. : Cycles for the dynamical study of foliated manifolds and complex manifolds.
1 Generalized string equation and dilation equation . . . . 2 The WDVV equation . . . . . . . . . . . . 3 Quantum cohomology . . . . . . . . . . . G. Lu . . . 85 85 . . . . . 91 93 97 . . 100 . . 101 . . . . . 101 104 104 108 119 123 123 127 128 . . . . . . . . . 1. Introduction and Main Results This paper is a continuation of my paper [Lu1]. In [Lu1] we defined the Gromov-Witten invariants on semi-positive noncompact geometrically bounded symplectic manifolds by combining Ruan-Tian’s method([RT1]) with that of McDuff-Salamon ([McS]), and also used them to generalize work on the topological rigidity of Hamiltonian loops on semi-positive closed symplectic manifolds ([LMP]) to the Hamiltonian loops with compact support on this class of noncompact symplectic manifolds.
Let C0 >√0 be an upper bound of all sectional curvatures Kµ of µ and r0 = min{i(M, µ), π/ C0 }. Assume that h : S → M is a J -holomorphic map from a compact connected Riemannian surface S with boundary to M. Let p ∈ M and 0 < r ≤ r0 be such that p ∈ h(S) ⊂ Bµ (p, r) and h(∂S) ⊂ Bµ (p, r), then Areaµ (h(S)) ≥ πα0 2 r . 1 of [Sik] we may choose C1 = 1/π and C2 = β0 /α0 in the present case. 1 of [Sik] is equal 4β0 to πα 2r . 1]). Let (M, ω, µ, J ) be as above, and h : S → M be a J -holomorphic map from a connected compact Riemannian surface S to M.