A Remark On Case-Gursky-Vétois identity and its applications

Based on the works of Gursky (CMP, 1997), Vétois (Potential Anal., 2023) and Case (Crelle’s journal, 2024), we make use of an Obata type formula established in these works to obtain some Liouville type theorems on conformally Einstein manifolds. In particular, we solve Hang-Yang conjecture (IMRN, 2020) via an Obata-type argument and obtain optimal perturbation.