Curvature, cones and characteristic numbers (Q2841495)

Let \(M\) and \(\Sigma\) be smooth manifolds of dimensions \(n\) and \(n-2\) respectively and assume that \(\Sigma\) is smoothly embedded into \(M\). A Riemannian edge-cone metric of cone angle \(2\pi\beta\) is a Riemannian metric on \(M\) which is smooth on \(M \setminus \Sigma\) and, roughly speaking, smooth in directions parallel to \(\Sigma\) and modelled on a \(2\)-dimensional cone in the transverse directions. An edge-cone metric on \((M,\Sigma)\) is said to be Einstein if it is an Einstein metric on \(M \setminus \Sigma\). The authors study Einstein edge-cone metrics on smooth \(4\)-manifolds.NEWLINENEWLINEAssume that \(M\) is a smooth compact \(4\)-manifold and \(\Sigma\) is a smoothly embedded compact oriented surface. The main result of the paper states that if \((M,\Sigma)\) admits an Einstein edge-cone metric with cone angle \(2\pi\beta\) along \(\Sigma\), then \((M,\Sigma)\) must satisfy the two inequalities \((2\chi \pm 3\tau)(M) \geq (1-\beta)(2\chi(\Sigma) \pm (1+\beta)[\Sigma]^2)\). Here, \(\chi\) denotes the Euler characteristic, \(\tau\) the signature, and \([\Sigma]^2\) the self-intersection of the homology class of \(\Sigma\) in \(H_2(M) \cong H^2(M)\). For the proof the authors derive generalizations of the \(4\)-dimensional Gauss-Bonnet and signature formulae for arbitrary edge-cone metrics on compact \(4\)-manifolds.NEWLINENEWLINEThe result provides obstructions for the existence of Einstein edge-cone metrics in many cases. For example, if \(\Sigma\) is a connected oriented surface with either non-zero self-intersection or genus \(\geq 2\), then there is a real number \(\beta_0\) such that \((M,\Sigma)\) does not admit Einstein edge-cone metrics of cone angle \(2\pi\beta\) for any \(\beta \geq \beta_0\). As another application the authors consider the limit case. If \((M,\Sigma)\) admits a sequence \(g_j\) of Einstein edge-cone metrics with cone angles \(2\pi\beta_j \rightarrow 0\), then \((M,\Sigma)\) must satisfy the two inequalities \((2\chi \pm 3\tau)(M) \geq 2\chi(\Sigma) \pm [\Sigma]^2\).NEWLINENEWLINEIn the final part of the paper the authors discuss families of self-dual edge-cone metrics on \(4\)-manifolds and investigate their relation to certain gravitational instantons.