Willmore-type inequalities for closed hypersurfaces in weighted manifolds

Abstract: In this paper, we prove some Willmore-type inequalities for closed hypersurfaces in weighted manifolds with nonnegative Bakry-\'Emery Ricci curvature. In particular, we get a sharp Willmore type inequality steady gradient Ricci solitons. We also prove a sharp Willmore-like inequality in shrinking gradient Ricci solitons. Moreover, we characterize the equality cases of our Willmore-type inequalities. These results can be regarded as generalizations of Agostiniani-Fogagnolo-Mazzieri's Willmore-type inequality to weighted manifolds. As applications, we derive some isoperimetric type inequalities under certain existence assumptions of isoperimetric regions in weighted manifolds.