Same sharp isoperimetric theorems for Riemannian manifolds

We prove that a region of small prescribed volume in a smooth, compact Riem annian manifold has at least as much perimeter as a round ball in the model space form, using differential inequalities and the Gauss-Bonnet-Chern the orem with boundary term. First we show that a minimizer is a nearly round s phere. We also provide some new isoperimetric inequalities in surfaces.