AI 中文总结
该研究证明了带非紧边界及特殊渐近平坦半空间端的黎曼流形正质量定理，结合多种方法完成证明，还得到了对应的精确黎曼 Penrose 不等式。
AI 中文摘要
我们证明了带有非紧边界、一个特殊的渐近平坦半空间端以及有限多个无预设渐近行为的其他完备端的完备黎曼流形的正质量定理。当 $3\leq n\leq7$、数量曲率 $R_g\geq0$ 且边界平均曲率 $H_{\partial M}\geq0$ 时，质量 $\mathfrak m(M,g,\mathcal E)\geq0$。此外，等号成立当且仅当 $(M,g)$ 等距于欧几里得半空间。该证明结合了特殊端附近的密度变形、跨非紧边界的加倍、局部光滑化以及共形校正。当一个紧的最外层极小超曲面将 $\mathcal E$ 与所有其余端分隔开时，我们还得到了精确的黎曼 Penrose 不等式；等号恰好成立当外部区域为 Schwarzschild 半空间外部。
英文摘要
We prove a positive mass theorem for complete Riemannian manifolds with noncompact boundary, a distinguished asymptotically flat half-space end, and finitely many additional complete ends with no prescribed asymptotics. If $3\leq n\leq7$, $R_g\geq0$, and $H_{\partial M}\geq0$, then $ \mathfrak m(M,g,\mathcal E)\geq0$. Moreover, equality holds if and only if $(M,g)$ is isometric to the Euclidean half-space. The proof combines a density deformation near the distinguished end with doubling across the noncompact boundary, local smoothing, and a conformal correction. We also obtain the sharp Riemannian Penrose inequality when a compact outermost minimal hypersurface separates $\mathcal E$ from all the remaining ends; equality holds precisely when the exterior region is a Schwarzschild half-space exterior.
