The mass of asymptotically hyperbolic manifolds.
Let \(\bar{X}\) be a compact \(n\)-dimensional manifold with boundary \(M\) and interior \(X\). A smooth function on \(\bar{X}\) with a first order zero on \(M\) and positive on \(X\) is called a defining function. Let \(g\) be a Riemannian metric on \(X\). \((X,g)\) is called conformally compact if, for any defining function \(r\), \(\bar{g}:=r^2 g\) extends as a \(C^3\) metric on \(\bar{X}\). The restriction of \(\bar{g}\) to \(M\) gives a metric and through it a conformal structure on \(M\). \((X,g)\) is called weakly asymptotically hyperbolic if \(| dr| _{\bar{g}}^2=1\) on \(M\). A weakly asymptotically hyperbolic manifold \((X,g)\) is called asymptotically hyperbolic if \(M\) is conformally isomorphic to the standard sphere \((S^{n-1},g_0)\) and near the infinity there is an expansion \[ g=\sinh^{-2}(r)[dr^2+g_0+\frac{r^n}{n}h +O(r^{n+1})], \] where \(r\) is a defining function and \(h\) is a symmetric 2-tensor on \(S^{n-1}\). The author proves that if \(X\) is an asymptotically hyperbolic spin manifold with scalar curvature \(R\geq -n(n-1)\), then it holds \[ \int_{S^{n-1}}\text{tr}_{g_0}(h) d\mu_{g_0}\geq | \int_{S^{n-1}} \text{tr}_{g_0}(h)x\,d\mu_{g_0}(x)| \] with equality precisely when \((X,g)\) is isometric to the hyperbolic space. Furthermore, it is proven that the quantity \[ A:=[\int_{S^2}\text{tr}_{g_0}(h) d\mu_{g_0}]^2 -| \int_{S^2}\text{tr}_{g_0} (h)x\,d\mu_{g_0}(x)| ^2 \] is an invariant for an asymptotically hyperbolic 3-manifold (which is automatically spin). If the scalar curvature of \(X\) is at least -6, then \(A\geq 0\), and the total mass of \(X\) is defined to be \(\frac{1}{16\pi}\sqrt{A}\).
scientific article; zbMATH DE number 1782680