A Bogovskiǐ-type operator for Corvino-Schoen hyperbolic gluing

We construct a solution operator for the linearized constant scalar curvature equation at hyperbolic space in space dimension larger than or equal to two. The solution operator has good support propagation properties and gains two derivatives relative to standard norms. It can be used for Corvino-Schoen-type hyperbolic gluing, partly extending the recently introduced Mao-Oh-Tao gluing method to the hyperbolic setting.

💡 Research Summary

The paper addresses the problem of solving the linearized constant scalar curvature equation on hyperbolic space $\mathbb H^n$ for $n\ge 2$ and constructs a Bogovskiĭ‑type solution operator that enjoys two crucial properties: (i) it preserves the compact support of the source term, and (ii) it gains two derivatives compared with the standard Sobolev mapping properties of elliptic operators.

The authors begin by recalling that the constant scalar curvature equation $R(g)=-n(n-1)$ on a Riemannian manifold $(M,g)$ reduces, in the time‑symmetric vacuum setting, to a single scalar equation. On the model hyperbolic half‑space $H^n={x_1>0}$ the metric $b=\frac{1}{x_1^2}\sum_{i=1}^n dx_i\otimes dx_i$ solves the equation. Linearizing around $b$ yields the under‑determined elliptic operator
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