Scalar curvature deformation and a gluing construction for the Einstein constraint equations. (English) Zbl 1031.53064

The author proves for \(n\geq 3\) the existence on \({\mathbb R}^n\), of non-trivial Riemannian metrics with vanishing Ricci scalar that are asymptotically flat and spherically symmetric outside a compact set. Here ‘non-trivial’ means that the metrics are not spherically symmetric everywhere. Such Riemannian metrics can be used as time-symmetric Cauchy data for Einstein’s vacuum equation in \(n+1\) dimensions, yielding vacuum spacetimes which are Schwarzschild outside a spatially compact set. The proof proceeds in two steps. The first step is a gluing construction, yielding a metric which has all desired features except that the Ricci scalar is (small but) non-zero inside an annular region. The second step is a deformation construction which makes the Ricci scalar zero everywhere.


53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
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