AI 中文总结
该研究证明闭双曲三维流形的精确体积稳定性定理，并将其应用于建立Fischer-Moncrief约化哈密顿量在洛伦兹锥基态下的稳定性，为相关渐近图像提供严格表述。
AI 中文摘要
我们证明了闭双曲三维流形的一个精确体积稳定性定理。设\u0028M,h\u0029为闭双曲流形，满足\uf061ic_h=-2h，且设g_i为M上的光滑度量，满足R(g_i)\u2265-6，Vol_{g_i}(M)\u2192Vol_h(M)。通过取子序列，存在M的子集Z_i、M的光滑区域K_i，以及微分同胚\uf076_i:K_i\u2192M\Z_i，使得Vol_{g_i}(Z_i)\u21920，Vol_h(M\K_i)\u21920，且||\uf076_i^*g_i-h||_{C^0(K_i,h)}\u21920。因此，精确双曲体积界的近等性迫使在体积趋于零的区域之外，张量C^0收敛到双曲度量。作为应用，我们在洛伦兹锥基态下建立了Fischer-Moncrief约化哈密顿量的稳定性：经CMC归一化后，双曲拓扑类中的近极小紧致真空数据，在体积趋于零的集合模下，以张量C^0收敛到双曲洛伦兹锥几何。这为Fischer-Moncrief渐近图像提供了严格的体积主导表述。
英文摘要
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 $ψ_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 $ \|ψ_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.
Comments36 pages, comments welcome
