Volume Stability for Hyperbolic Manifolds and Applications to General Relativity

We prove a sharp volume-stability theorem for closed hyperbolic three-manifolds. Let $(M,h)$ be closed hyperbolic with $\operatorname{Ric}_h=-2h$, and let $g_i$ be smooth metrics on $M$ satisfying $R(g_i)\geq -6, \operatorname{Vol}_{g_i}(M)\longrightarrow \operatorname{Vol}_h(M)$. After passing to a subsequence, there exist $Z_i\subset M$, smooth domains $K_i\subset M$, and diffeomorphisms $\psi_i:K_i\longrightarrow M\setminus Z_i$ such that $\operatorname{Vol}_{g_i}(Z_i)\longrightarrow0, \operatorname{Vol}_h(M\setminus K_i)\longrightarrow0,$ and $\|\psi_i^*g_i-h\|_{C^0(K_i,h)}\longrightarrow0.$ Thus near-equality in the sharp hyperbolic volume bound forces tensorial $C^0$-convergence to the hyperbolic metric outside regions of vanishing volume. As an application, we establish stability of the Fischer--Moncrief reduced Hamiltonian at the Lorentz-cone ground state: after CMC normalization, near-minimizing compact vacuum data in the hyperbolic topological class converge, modulo sets of vanishing volume, to the hyperbolic Lorentz-cone geometry in tensorial $C^0$. This provides a rigorous volume-dominance formulation of the Fischer--Moncrief asymptotic picture.