Desingularisation of Einstein metrics. I (Q1949230)

The author studies a new obstruction for a real Einstein 4-orbifold \((M_0,g_0)\) with \(A_1\)-singularity to be a limit of smooth Einstein 4-manifolds. The author proves that if \((M_0,g_0)\) with a nondegenerate asymptotically hyperbolic metric \(g_0\) has a singularity of the type \(\mathbb R^4\slash \mathbb Z_2\) at a point \(p_0\) and \(M\) is a manifold obtained from \(M_0\) by blowing up \(p_0\) to a sphere with self-intersection \(-2\) and \(\det(R^+_{g_0}(p_0))=0\), where \(R^+=W^++\frac{\mathrm{scal}}{12}I\), then there exists a family \(g_t\) of Einstein asymptotically hyperbolic metrics on \(M\) such that the volume of a blowed up sphere is \(t\) and if \(t\rightarrow 0\) then \(g_t\rightarrow g_0\) with the convergence of class \(C^{\infty}\) on all compact sets outside \(p_0\). If \(M_0\) has more singularities of the type \(\mathbb R^4\slash \mathbb Z_2\) the result remains true. The author also considers the Dirichlet problem of finding Einstein metrics with given conformal infinity on the boundary and proves ``that his obstruction defines a wall in the space of conformal metrics on the boundary and that all Einstein metrics must have their conformal infinity on one side of the wall''.