In this paper, we consider the Cauchy problem for the defocusing nonlinear Schr \(\ddot{\text {o}}\) dinger equation with a finite genus algebro-geometric background. Long-time asymptotics of the solution are derived in four space-time regions. It comes out that the leading-order term in the expansion is, up to a constant, given by the background solution with a shift of the parameter. The subleading term, however, decays at different rates for different regions. We particularly highlight that in the two transition regions, they are of order \(\mathcal {O}(t^{-1/3})\) and the coefficients involve an integral of the Painlevé XXXIV transcendent. We establish our results by applying a nonlinear steepest descent analysis to the associated Riemann–Hilbert problems.
