A solution operator for the linearized constant scalar curvature equation at the hyperbolic space

In the context of understanding the space of admissible initial data for the (Vacuum) Einstein Equations of General Relativity, namely Riemannian manifolds subject to the corresponding Einstein Constraint Equations (ECE),  gluing methods can be employed to construct a new solution to these highly underdetermined elliptic equations, containing two existing ones. Such methods rely on suitable iterative schemes by finding a sequence of solutions to the corresponding linearised problems. A recent simplified approach to address the linearized ECE at the flat space (reducing to vanishing scalar curvature in the so-called time symmetric regime) is established on a Bogovskii-type solution operator with good support propagation properties. We will introduce and discuss such an approach and how to obtain the corresponding solution operator for the linearized (negative constant) scalar curvature at the hyperbolic space.

This is based on joint work with P. Chuściel and A. Nützi.