Mass in asymptotically locally hyperbolic 2+1 gravity is a minimised functional, not a single conserved number, and gluing theorems can construct initial data realising every mass aspect function.
Asymptotic gluing of asymptotically hyperbolic solutions to the Einstein constraint equations
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We show that asymptotically hyperbolic solutions of the Einstein constraint equations with constant mean curvature can be glued in such a way that their asymptotic regions are connected.
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Hyperbolic Mass in 2+1 Dimensions
Mass in asymptotically locally hyperbolic 2+1 gravity is a minimised functional, not a single conserved number, and gluing theorems can construct initial data realising every mass aspect function.