## 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.

### MSC:

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