The Gauss-Bonnet-Chern mass of conformally flat manifolds (Q2925298)

The authors consider the \(k\)th Gauß-Bonnet-Chern mass \(m_{k}\), introduced in [the authors, Adv. Math. 266, 84--119 (2014; Zbl 1301.53032)] and constructed from the \(k\)th Gauß-Bonnet curvature \(L_{k}\) on an asymptotically flat conformally flat manifold \((M,g)=(\mathbb{R}^{n}\backslash\Omega,e^{-2u}\delta)\) (\(n>2k\) and \(\Omega\subset\mathbb{R}^{n}\) open) of decay order \(\tau > \frac{n-2k}{k+1}\) on which \(L_{k}\) is integrable. The main results are two positive mass theorems and a Penrose-type inequality. The positive mass theorems state that if \(M=\mathbb{R}^{n}\) and either \(L_{j}\geq 0\) for all \(j\leq k\) or \(k\) is even and \((-1)^{j}L_{j}\geq 0\) for all \(j\leq k\), then \(m_{k}\geq 0\) and equality holds iff \(M\) is flat. Furthermore, the Penrose-type inequality reads NEWLINE\[NEWLINEm_{k}\geq \left(\frac{|\partial\Omega|}{\omega_{n-1}} \right)^{\frac{n-2k}{n-1}}NEWLINE\]NEWLINE whenever \(\Omega\) is convex, \(\partial\Omega\) is minimal in \((M,g)\) and \(u\) is constant on \(\partial\Omega\), where \(\omega_{n-1}={\mathrm{Vol}}(S^{n-1})\); moreover, if \(k\geq 2\), then the inequality NEWLINE\[NEWLINEm_{k}\geq \left(\frac{\int_{\partial\Omega}R}{(n-1)(n-2)\omega_{n-1}} \right)^{\frac{n-2k}{n-3}}NEWLINE\]NEWLINE also holds, where \(R\) is the scalar curvature of \(\partial\Omega\) considered as a submanifold of \(\mathbb{R}^{n}\). These results are meant to support a conjecture posed by the authors in [loc. cit.] on the positivity of the Gauß-Bonnet-Chern mass.NEWLINENEWLINEThe structure of the paper is as follows: In Section 1 the Gauß-Bonnet curvature and Gauß-Bonnet-Chern mass are defined and the main results are stated and elaborated upon. Section 2 recasts the Gauß-Bonnet curvatures of locally conformally flat metrics in terms of symmetric functions and the Schouten tensor, which then simplifies the computations carried out in the following sections. The positive mass theorems are then proved in Section 3 and Penrose-type inequalities in Section 4.NEWLINENEWLINEThroughout the paper the results are compared with analogous statements about the ADM mass (corresponding to the case \(k=1\)). An explicit computation of the \(k\)th Gauß-Bonnet-Chern mass of a manifold equipped with a general Schwarzschild metric is given in Section 4, as well as an explicit description of \(m_{k}\) more generally in the spherically symmetric case.