A Remark On Case-Gursky-V\'etois identity and its applications

TL;DR

This paper leverages an Obata-type formula to prove Liouville theorems on conformally Einstein manifolds, solving the Hang-Yang conjecture and establishing optimal perturbation results.

Contribution

It introduces new Liouville theorems on conformally Einstein manifolds and solves the Hang-Yang conjecture using an Obata-type approach, building on recent foundational works.

Findings

Proved Liouville type theorems for conformally Einstein manifolds.

Solved the Hang-Yang conjecture (IMRN, 2020).

Established optimal perturbation results.

Abstract

Based on the works of Gursky (CMP, 1997), V\'etois (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.