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.
We establish index estimates for complete two-sided free boundary minimal surfaces in smooth mean-convex domains of $\mathbb{R}^3$ with noncompact boundary. We first prove $3{\rm Ind}_s(\Sigma)\geq 2g+b-1$, where $g$ and $b$ describe the conformal compactification. We then include all interior ends with their multiplic...
Let $M^3\to X^4$ be a complete, connected, two-sided stable minimal immersion. We prove that if the ambient sectional curvature is nonnegative and the ambient scalar curvature has a positive uniform lower bound, then $M$ is totally geodesic and its normal Ricci curvature vanishes. No weak bounded geometry assumption an...
Let $(M^3,g)$ be a complete Riemannian manifold diffeomorphic to $\R^3\setminus\{0\}$, with nonnegative scalar curvature. Assume that a distinguished end is asymptotically flat, with ADM mass $m_+$. For each $p\in(1,3)$, define $c_{O,p}$ as the infimum of the Schwarzschild-normalized $p$-capacity over outward-minimizin...
Let $S\subset\mathbb R^3$ be a properly embedded mean-convex planar surface with finitely many ends. Designate one end as asymptotically flat, assume that $H_S$ is integrable there, and denote its extrinsic mass by $m_+(S)$. Let $A_S$ be the infimum of the areas of compact surfaces separating the distinguished end from...
We establish the rigidity statement in the equality case of the hyperspherical-radius inequality of Brendle, Tsiamis, and Wang for compact spin fill-ins with scalar curvature bounded below. More precisely, let $(M^{n\geq 3},g)$ be a compact, connected Riemannian spin manifold having a connected boundary $\Sigma$ and sc...
We prove a positive mass theorem for one-ended asymptotically flat spin initial data with multiple time directions and a smooth compact boundary. Assuming the trace-norm dominant energy condition and pairwise commutativity of the second fundamental forms, a fixed auxiliary time direction on each boundary component dete...
Chang-Wen Fang· 0 citations
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