Alexandrov's theorem in the anti-Gaussian space: A Reilly-formula approach

We give an alternative proof of Alexandrov's theorem about constant weighted mean curvature hypersurfaces in the anti-Gaussian space, due to the authors and Johne-Silini independently, via a Reilly-type formula approach. Moreover, we establish a sharp Heintze-Karcher-type inequality for hypersurfaces in manifolds with a generalized substatic condition, together with its rigidity in Euclidean space with density.