The global Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups III: Proof of the main theorems

This is the third and the last of a series of three papers where we prove the Gan–Gross–Prasad conjecture for Fourier–Jacobi periods on unitary groups and an Ichino–Ikeda type refinement. Our strategy is based on the comparison of relative trace formulae formulated by Liu. In this paper, we compute the spectral expansions of these formulae and end the proof of the conjectures via a reduction to the corank zero case.