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

Mingxiang Li, Juncheng Wei

Abstract

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.

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

Abstract

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.

Paper Structure

This paper contains 3 sections, 12 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Case-Gursky-Vétois identity on conformal Einstein manifolds
  3. Applications

Key Result

Theorem 1.1

(Vétois' theorem in Vetois) Suppose that $(M^n,g_0)$ where $n\geq 3$ is a compact Einstein manifold with non-negative scalar curvature $R_0$. Consider a conformal metric $g=u^2g_0$ where $u>0$. Suppose that the Q-curvature of conformal metric $g$ is constant. Then $g$ is Einstein. Furthermore, if $(

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 9 more