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**Boundary regularity of conformally compact Einstein metrics.**
*(English)*
Zbl 1088.53031

Let \(\overline{M}\) be a smooth compact manifold with boundary, and denote by \(M\) its interior and by \(\partial M\) its boundary. A Riemannian metric \(g\) on \(M\) is conformally compact of class \(C^2\) if for some smooth defining function \(\rho\) for \(\partial M\) in \(\overline{M}\), \(\rho^2 g\) extends to a Riemannian metric \(\overline{g}\) of class \(C^2\) on \(\overline{M}\). The conformal class of the restriction of \(\overline{g}\) to \(\partial M\) is the conformal infinity of \(g\). It has been conjectured that conformally compact Einstein metrics with smooth conformal infinities have infinite-order asymptotic expansions in terms of \(\rho\) and \(\log(\rho)\). The purpose of the paper is to prove this conjecture.

Since only regularity at the boundary is an issue here, the authors can assume that \(\overline{M} = Y \times [0,1)\), \(M = Y \times (0,1)\) and \(\partial M = Y \times \{0\}\) with some \(n\)-dimensional connected compact smooth manifold \(Y\) without boundary. Let \(g\) be an Einstein metric on \(M\) with Einstein constant \(-n\), and assume that \(g\) is conformally compact of class \(C^2\) and that the representative \(\gamma\) of the conformal infinity of \(g\) is smooth. Let \(\widetilde{\gamma}\) be any smooth representative of the conformal class of \(\gamma\). The authors prove that for each \(\lambda \in (0,1)\) there exists \(R \in (0,1)\) and a \(C^{1,\lambda}\) collar diffeomorphism \(\Phi : Y \times [0,R] \to \overline{M}\) such that \(\Phi^*g\) can be written in the form \(\Phi^*g = \rho^{-2}(d\rho^2+G(\rho))\). Here, \(G(\rho)\), \(\rho \in (0,R]\), is a one-parameter family of smooth Riemannian metrics on \(Y\) and \(d\rho^2 + G(\rho)\) has a continuous extension to \(Y \times [0,R]\) with \(G(0) = \widetilde{\gamma}\) with the following properties: (1) If \(\dim M\) is even or equal to \(3\), then \(d\rho^2 + G(\rho)\) extends smoothly to \(Y \times [0,R]\), and hence \(\Phi^*g\) is conformally compact of class \(C^\infty\). (2) If \(\dim M\) is odd and greater than \(3\), then \(G\) can be written in the form \(G(\rho) = \varphi(\rho,\rho^n\log(\rho))\) with a two-parameter family \(\varphi(\rho,z)\) of Riemannian metrics on \(Y\) that is smooth in all of its arguments as a function on \(Y \times [0,R] \times [R^n\log(R),0]\). Furthermore, \(\Phi^*g\) is smoothly conformally compact if and only if \(\partial_z\varphi(0,0)\) vanishes identically on \(\partial M\).

The main idea of the proof is to use the harmonic map equation to put \(g\) into a gauge in which it satisfies an elliptic equation and then apply some polyhomogeneity results.

Since only regularity at the boundary is an issue here, the authors can assume that \(\overline{M} = Y \times [0,1)\), \(M = Y \times (0,1)\) and \(\partial M = Y \times \{0\}\) with some \(n\)-dimensional connected compact smooth manifold \(Y\) without boundary. Let \(g\) be an Einstein metric on \(M\) with Einstein constant \(-n\), and assume that \(g\) is conformally compact of class \(C^2\) and that the representative \(\gamma\) of the conformal infinity of \(g\) is smooth. Let \(\widetilde{\gamma}\) be any smooth representative of the conformal class of \(\gamma\). The authors prove that for each \(\lambda \in (0,1)\) there exists \(R \in (0,1)\) and a \(C^{1,\lambda}\) collar diffeomorphism \(\Phi : Y \times [0,R] \to \overline{M}\) such that \(\Phi^*g\) can be written in the form \(\Phi^*g = \rho^{-2}(d\rho^2+G(\rho))\). Here, \(G(\rho)\), \(\rho \in (0,R]\), is a one-parameter family of smooth Riemannian metrics on \(Y\) and \(d\rho^2 + G(\rho)\) has a continuous extension to \(Y \times [0,R]\) with \(G(0) = \widetilde{\gamma}\) with the following properties: (1) If \(\dim M\) is even or equal to \(3\), then \(d\rho^2 + G(\rho)\) extends smoothly to \(Y \times [0,R]\), and hence \(\Phi^*g\) is conformally compact of class \(C^\infty\). (2) If \(\dim M\) is odd and greater than \(3\), then \(G\) can be written in the form \(G(\rho) = \varphi(\rho,\rho^n\log(\rho))\) with a two-parameter family \(\varphi(\rho,z)\) of Riemannian metrics on \(Y\) that is smooth in all of its arguments as a function on \(Y \times [0,R] \times [R^n\log(R),0]\). Furthermore, \(\Phi^*g\) is smoothly conformally compact if and only if \(\partial_z\varphi(0,0)\) vanishes identically on \(\partial M\).

The main idea of the proof is to use the harmonic map equation to put \(g\) into a gauge in which it satisfies an elliptic equation and then apply some polyhomogeneity results.

Reviewer: Jürgen Berndt (Cork)

### MSC:

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |