Uniqueness Of Conformal Measures For Four-Dimensional Closed Einstein Manifolds

This article focuses on the work of JEROME VETOIS.The study of conformal metric has always been a hot and difficult problem in geometric analysis,which usually boils down to the study of some specific partial differential equations.There are many predetermined curvature problems on Riemann manifolds.JEROME VETOIS worked on the predetermined Q-curvature problem,namely whether there exists a metric on a Riemannian manifold that is conformal with the original metric such that the corresponding Q-curvature is equal to a constant.The authors conclude that(M,g)is smooth,closed n-dimensional(n≥ 3)Einstein manifolds whose scalar curvature is positive and not conformally diffeomorphic to the standard sphere,making conformal measures whose Q-curvature is constant unique in the sense of a constant difference.In this paper,the problem of predetermined Q-curvature in geometric analysis is transformed into the problem of properties of the solution of a fourth order equation on a manifold.On this basis,the author studies a more general fourth order equation and gets a better result.This paper verifies the calculation process of the author in detail.At the same time,in order to better explain the background of the problem and understand the author’s research ideas,this paper adds the introduction of Q-curvature and the source of the fourth order equation and the process of solving the second order equation by using the integration by parts technique,and compares and summarizes several key quantities from them with the important geometric quantities in the paper.