On conformally compact Einstein manifolds. (English) Zbl 1508.53042
This paper surveys some recent results on compactness and uniqueness problems for some classes of conformally compact Einstein manifolds.
Let \(X\) be a smooth manifold of dimension \(d\geq3\) and with boundary \(\partial X\). A smooth conformally compact metric \(g^+\) on \(X\) is a Riemannian metric such that \(g=r^2g^+\) extends smoothly to the closure \(\bar X\) for some so-called defining function \(r\). A defining function is a smooth non-negative function on \(\bar X\) such that \(\partial X=\{r=0\}\) with the differential of \(r\) satisfying \(Dr\ne0\) on \(\partial X\). A conformally compact metric \(g^+\) on \(X\) is conformally compact Einstein (CCE) if its Ricci tensor satisfies \[ \mathrm{Ric}_{g^+}=-(d-1)g^+. \] Two basic examples of CCE manifolds are the hyperbolic ball with the Poincaré metric and the AdS-Schwarzschild space. A further class of basic examples mentioned in Section 2 of the paper is due to C. R. Graham and J. M. Lee [Adv. Math. 87, No. 2, 186–225 (1991; Zbl 0765.53034)].
The authors discuss (non-)existence and (non-)uniqueness results for CCE manifolds. The authors then discuss compactness results for sequences of CCE manifolds (Section 3) and uniqueness results for Graham-Lee metrics (Section 4) with a focus on high dimensions \(d\geq5\). The key results presented in Sections 3 and 4 appear to be taken from a paper currently in preparation by the authors. In Section 5 the authors discuss their previous work [Adv. Math. 340, 588–652 (2018; Zbl 1443.53027); Adv. Math. 373, Article ID 107325, 33 p. (2020; Zbl 1447.53043)] on compactness and uniqueness problems in dimension \(d=4\).
Let \(X\) be a smooth manifold of dimension \(d\geq3\) and with boundary \(\partial X\). A smooth conformally compact metric \(g^+\) on \(X\) is a Riemannian metric such that \(g=r^2g^+\) extends smoothly to the closure \(\bar X\) for some so-called defining function \(r\). A defining function is a smooth non-negative function on \(\bar X\) such that \(\partial X=\{r=0\}\) with the differential of \(r\) satisfying \(Dr\ne0\) on \(\partial X\). A conformally compact metric \(g^+\) on \(X\) is conformally compact Einstein (CCE) if its Ricci tensor satisfies \[ \mathrm{Ric}_{g^+}=-(d-1)g^+. \] Two basic examples of CCE manifolds are the hyperbolic ball with the Poincaré metric and the AdS-Schwarzschild space. A further class of basic examples mentioned in Section 2 of the paper is due to C. R. Graham and J. M. Lee [Adv. Math. 87, No. 2, 186–225 (1991; Zbl 0765.53034)].
The authors discuss (non-)existence and (non-)uniqueness results for CCE manifolds. The authors then discuss compactness results for sequences of CCE manifolds (Section 3) and uniqueness results for Graham-Lee metrics (Section 4) with a focus on high dimensions \(d\geq5\). The key results presented in Sections 3 and 4 appear to be taken from a paper currently in preparation by the authors. In Section 5 the authors discuss their previous work [Adv. Math. 340, 588–652 (2018; Zbl 1443.53027); Adv. Math. 373, Article ID 107325, 33 p. (2020; Zbl 1447.53043)] on compactness and uniqueness problems in dimension \(d=4\).
Reviewer: Andreas Vollmer (Hamburg)
MSC:
| 53C18 | Conformal structures on manifolds |
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
Keywords:
Einstein manifolds; compactness problems; uniqueness problems; Poincaré metric; AdS-Schwarzschild spaceReferences:
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