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1. Boundary behavior of compact manifolds with scalar curvature lower bounds and static quasi-local mass of tori

   https://doi.org/10.1090/proc/17223  

   Alaee, Aghil; Hung, Pei-Ken; Khuri, Marcus (March 2025, Proceedings of the American Mathematical Society)

   A classic result of Shi and Tam states that a 2-sphere of positive Gauss and mean curvature bounding a compact 3-manifold with nonnegative scalar curvature must have total mean curvature not greater than that of the isometric embedding into Euclidean 3-space, with equality only for domains in this reference manifold. We generalize this result to 2-tori of Gauss curvature greater than $-<\#comment/>1$, which bound a compact 3-manifold having scalar curvature not less than $-<\#comment/>6$and at least one other boundary component satisfying a ‘trapping condition’. The conclusion is that the total weighted mean curvature is not greater than that of an isometric embedding into the Kottler manifold, with equality only for domains in this space. Examples are given to show that the assumption of a secondary boundary component cannot be removed. The result gives a positive mass theorem for the static Brown-York mass of tori, in analogy to the Shi-Tam positivity of the standard Brown-York mass, and represents the first such quasi-local mass positivity result for nonspherical surfaces. Furthermore, we prove a Penrose-type inequality in this setting. 

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2. Weighted Poincaré inequality and the Poisson Equation

   https://doi.org/10.1090/tran/8291  

   Munteanu, Ovidiu; Sung, Chiung-Jue; Wang, Jiaping (March 2021, Transactions of the American Mathematical Society)

   We develop Green’s function estimates for manifolds satisfying a weighted Poincaré inequality together with a compatible lower bound on the Ricci curvature. This estimate is then applied to establish existence and sharp estimates of solutions to the Poisson equation on such manifolds. As an application, a Liouville property for finite energy holomorphic functions is proven on a class of complete Kähler manifolds. Consequently, such Kähler manifolds must be connected at infinity. 

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3. Nonnegative Ricci curvature, almost stability at infinity, and structure of fundamental groups

   https://doi.org/10.1016/j.aim.2025.110310  

   Pan, Jiayin (July 2025, Advances in Mathematics)

   We study the fundamental group of an open $$n$$-manifold $$M$$ of nonnegative Ricci curvature with additional stability conditions on $$\widetilde{M}$$, the Riemannian universal cover of $$M$$. We prove that if every asymptotic cone of $$\widetilde{M}$$ is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a prior fixed metric cone, then $$\pi_1(M)$$ is finitely generated and contains a normal abelian subgroup of finite index; if in addition $$\widetilde{M}$$ has Euclidean volume growth of constant at least $$L$$, then we can bound the index of that abelian subgroup by a constant $C(n,L)$. In particular, our result implies that if $$\widetilde{M}$$ has Euclidean volume growth of constant at least $$1-\epsilon(n)$$, then $$\pi_1(M)$$ is finitely generated and $C(n)$-abelian. 

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4. The Splitting Theorem and topology of noncompact spaces with nonnegative N-Bakry Émery Ricci curvature

   https://doi.org/10.1090/proc/15240  

   Lim, Alice (August 2021, Proceedings of the American Mathematical Society)

   In this paper, we generalize topological results known for noncompact manifolds with nonnegative Ricci curvature to spaces with nonnegative N N -Bakry Émery Ricci curvature. We study the Splitting Theorem and a property called the geodesic loops to infinity property in relation to spaces with nonnegative N N -Bakry Émery Ricci curvature. In addition, we show that if M n M^n is a complete, noncompact Riemannian manifold with nonnegative N N -Bakry Émery Ricci curvature where N > n N>n , then H n − 1 ( M , Z ) H_{n-1}(M,\mathbb {Z}) is 0 0 . 

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5. Capacity, quasi-local mass, and singular fill-ins

   https://doi.org/10.1515/crelle-2019-0040  

   Mantoulidis, Christos; Miao, Pengzi; Tam, Luen-Fai (December 2019, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown–York mass and the other is new. We argue by recasting the setup to the study of mean-convex fill-ins with nonnegative scalar curvature and, in the process, we consider fill-ins with singular metrics, which may have independent interest. Among other things, our work yields new variational characterizations of Riemannian Schwarzschild manifolds and new comparison results for surfaces in them. 

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