Locally analytic vectors in the completed cohomology of unitary Shimura curves

We use the methods introduced by Lue Pan to study the locally analytic vectors in the completed cohomology of unitary Shimura curves. As an application, we prove a classicality result on two-dimensional regular σ-de Rham representations of Gal (L/L) appearing in the locally σ-analytic vectors of the completed cohomology, where L is a finite extension of Qₚ and σ: L E is an embedding of L into a sufficiently large finite extension E of Qₚ. We also prove that if a two-dimensional representation of Gal (L/L) appears in the locally σ-algebraic vectors of the completed cohomology then it is σ-de Rham. Finally, we give a geometric realization of some locally σ-analytic representations of GL₂ (L). This realization has some applications to the p-adic local Langlands program, including a locality theorem for Galois representations arising from classical automorphic forms, an admissibility result for coherent cohomology of Drinfeld curves, and some special cases of the Breuil's locally analytic Ext¹-conjecture for GL₂ (L).