Comparison results for solutions of Poisson equations with Robin boundary on complete Riemannian manifolds

In this paper, by using Schwarz rearrangement and isoperimetric inequalities, we prove comparison results for the solutions of Poisson equations on complete Riemannian man-ifolds with Ric = (n- 1)?, ? = 0 or 1, which extends the results in [A. Alvino, C. Nitsch and C. Trombetti, A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions, Comm. Pure Appl. Math. 76(3) (2023) 585-603]. Further-more, as applications of our comparison results, we obtain the Saint-Venant inequality and Bossel-Daners inequality for Robin Laplacian.