2026/07/30 by Puskar Mondal, Shing-Tung Yau
Mathematics · Physics and Astronomy · #math.DG #gr-qc #math.AP
36 pages, comments welcome
arxiv created 2026/07/30 · arxiv updated 2026/07/31
We prove a sharp volume-stability theorem for closed hyperbolic three-manifolds. Let \((M,h)\) be closed hyperbolic with \(Rich=-2h\), and let \(gi\) be smooth metrics on \(M\) satisfying R(gi)≥ -6, Volgi(M)\longrightarrow Volh(M). After passing to a subsequence, there exist \(Zi⊂ M\), smooth domains \(Ki⊂ M\), and diffeomorphisms ψi:Ki\longrightarrow M∖ Zi such that Volgi(Zi)\longrightarrow0, Volh(M∖ Ki)\longrightarrow0, and ‖ψi^*gi-h‖C0(Ki,h)\longrightarrow0. Thus near-equality in the sharp hyperbolic volume bound forces tensorial \(C0\)-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 \(C0\). This provides a rigorous volume-dominance formulation of the Fischer--Moncrief asymptotic picture.