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Description
We discuss a new approach to constructing a solution operator to the linearized scalar curvature equation at the hyperbolic metric, which has negative constant scalar curvature. Such an operator is of Bogovskii-type with good support propagation properties and gains two derivatives relative to standard norms; it extends the corresponding construction for the linearized scalar curvature operator at the Euclidean metric, which has vanishing scalar curvature. As an application, this provides a tool for gluing solutions of the Einstein Constraint Equations in the time-symmetric setting and hence for constructing new initial data for the Vacuum Einstein Equations of General Relativity: such gluing methods rely on suitable iterative schemes, in which one finds a sequence of solutions to the corresponding linearised problems. This is based on joint work with P. Chuściel and A. Nützi.