Lagrangian Floer Homology and Fukaya Categories in Non-Compact Calabi-Yau Manifolds

We investigate the structural properties of the wrapped Fukaya category W(M) associated with a non-compact Calabi-Yau manifold M equipped with a Liouville structure. Unlike the compact case, the non-compact setting requires a rigorous treatment of behavior at infinity, necessitating the use of Hamiltonian perturbations that grow quadratically with respect to the radial coordinate. We establish the well-definedness of the A-structure maps by invoking the maximum principle to control the moduli spaces of pseudo-holomorphic discs with Lagrangian boundary conditions. Furthermore, we analyze the generation of the derived wrapped Fukaya category DW(M) by the cocores of the Weinstein handle decomposition. We provide a geometric criterion for split-generation, linking the symplectic cohomology SH*(M) to the Hochschild homology of W(M) via the open-closed string map. These results constitute a foundational step towards proving Homological Mirror Symmetry for non-compact log-Calabi-Yau varieties.