The Boundary Value Problem Of Poincare Einstein Metrics And Its Rigidity

In this paper,we give detailed proofs of Graham-Lee's results on the existence of the boundary value problem of the Einstein metric[1],and illustrate Qing's results on the uniqueness[2].About the boundary value problem,we know conformally compact Einstein manifolds play an important role in Ads/CFT correspondence physically,mathematically the study of conformally compact Einstein structure began with the work of Fefferman and Graham,the problem stems from the paper[3],the research object is the Einstein equation,and the boundary condition is that the boundary metric belongs to a conformal class:conformal infinity;We ill discuss the existence of the solutions of the boundary value problem on the sphere Bn+1 and its boundary manifold as(Sn,h),where h is a standard spherical metric,the main theorem used is the inverse function theorem between the Banach spaces,so the final result is the existence of a solution when the boundary metric and h.are sufficiently close under the corresponding norm;The paper will also construct and discuss the conditions to satisfy the inverse function theorem.The main results used to prove the existence of the solution are the asymptoticity of the normalized Einstein operator and the isomorphism theorem of the uniformly degenerate elliptic operator;Notice that since the Einstein equation is a non-linear equation,so Deturck trick is used in this paper and we also need to calculate its linearization.Due to the degeneracy of the equation at the boundary,we should discuss the asymptoticity,and asymptoticity can also help us construct the Banach space in the inverse function theorem;Regarding isomorphism,it is natural to satisfy the conditions of the inverse function theorem.It is worth noting that due to the requirements of the isomorphism theorem,the final existence theorem will have different results in terms of regularity with dimensionality.We will use the classical elliptical PDE theory in these parts,such as:the second existence theorem and regularity theorem of the weak solution in the L2 theory,the Schauder estimate,the extreme value principle on the manifold;It is only about the norm used in the paper and Schauder estimation.Since the current operator is a uniformly degenerate elliptic operator,we need to use the classic estimation to modify the estimation of this operator.About the uniqueness,we prove that the Einstein manifolds with conformal compactness and conformal infinite(Sn,h)have to be the classic hyperbolic space(Bn+1,h),where h is the standard hyperbolic metric and 3?n?6.The main theorem used is the positive mass theorem.The final result's proof is based on the construction for the eigenfunction and the conformal compactifications obtained using the eigenfunction;for function construction,we will use the expansion of metric in the literature[4]and a more general conclusion compared to the above isomorphism theorem.