On the topology of Lagrangian fillings of the standard Legendrian sphere

Abstract: In this paper we study the uniqueness of Lagrangian fillings of the standard Legendrian sphere $\mathcal{L}0$ in the standard contact sphere $(S^{2n-1}, \xi{\text st})$. We show that every exact Maslov zero Lagrangian filling $L$ of $\mathcal{L}0$ in a Liouville filling of $(S^{2n-1}, \xi{\text st})$ is a homology ball. If we restrict ourselves to real Lagrangian fillings, then $L$ is diffeomorphic to the $n$-ball for $n \geq 6$.