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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. Bartnik mass via vacuum extensions

   https://doi.org/10.1142/S0129167X19400068  

   Miao, Pengzi; Xie, Naqing (December 2019, International Journal of Mathematics)

   We construct asymptotically flat, scalar flat extensions of Bartnik data [Formula: see text], where [Formula: see text] is a metric of positive Gauss curvature on a two-sphere [Formula: see text], and [Formula: see text] is a function that is either positive or identically zero on [Formula: see text], such that the mass of the extension can be made arbitrarily close to the half area radius of [Formula: see text]. In the case of [Formula: see text], the result gives an analog of a theorem of Mantoulidis and Schoen [On the Bartnik mass of apparent horizons, Class. Quantum Grav. 32(20) (2015) 205002, 16 pp.], but with extensions that have vanishing scalar curvature. In the context of initial data sets in general relativity, the result produces asymptotically flat, time-symmetric, vacuum initial data with an apparent horizon [Formula: see text], for any metric [Formula: see text] with positive Gauss curvature, such that the mass of the initial data is arbitrarily close to the optimal value in the Riemannian Penrose inequality. The method we use is the Shi–Tam type metric construction from [Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62(1) (2002) 79–125] and a refined Shi–Tam monotonicity, found by the first named author in [On a localized Riemannian Penrose inequality, Commun. Math. Phys. 292(1) (2009) 271–284]. 

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3. Extremality and rigidity for scalar curvature in dimension four

   https://doi.org/10.1007/s00029-023-00892-5  

   Bettiol, Renato G.; Goodman, McFeely Jackson (February 2024, Selecta Mathematica)

   Following Gromov, a Riemannian manifold is called area-extremal if any modification that increases scalar curvature must decrease the area of some tangent 2-plane. We prove that large classes of compact 4-manifolds, with or without boundary, with nonnegative sectional curvature are area-extremal. We also show that all regions of positive sectional curvature on 4-manifolds are locally area-extremal. These results are obtained analyzing sections in the kernel of a twisted Dirac operator constructed from pairs of metrics, and using the Finsler–Thorpe trick for sectional curvature bounds in dimension 4. 

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4. Scalar curvature deformation and mass rigidity for ALH manifolds with boundary

   https://doi.org/10.1090/tran/8755  

   Huang, Lan-Hsuan; Jang, Hyun Chul (November 2022, Transactions of the American Mathematical Society)

   We study scalar curvature deformation for asymptotically locally hyperbolic (ALH) manifolds with nonempty compact boundary. We show that the scalar curvature map is locally surjective among either (1) the space of metrics that coincide exponentially toward the boundary, or (2) the space of metrics with arbitrarily prescribed nearby Bartnik boundary data. Using those results, we characterize the ALH manifolds that minimize the Wang-Chruściel-Herzlich mass integrals in great generality and establish the rigidity of the positive mass theorems. 

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5. Comparison Theorems for 3D Manifolds With Scalar Curvature Bound

   https://doi.org/10.1093/imrn/rnab307  

   Munteanu, Ovidiu; Wang, Jiaping (November 2021, International Mathematics Research Notices)

   Abstract Two sharp comparison results are derived for 3D complete noncompact manifolds with scalar curvature bounded from below. The 1st one concerns the Green’s function. When the scalar curvature is nonnegative, it states that the rate of decay of an energy quantity over the level set is strictly less than that of the Euclidean space unless the manifold itself is isometric to the Euclidean space. The result is in turn converted into a sharp area comparison for the level set of the Green’s function when in addition the Ricci curvature of the manifold is assumed to be asymptotically nonnegative at infinity. The 2nd result provides a sharp upper bound of the bottom spectrum in terms of the scalar curvature lower bound, in contrast to the classical result of Cheng, which involves a Ricci curvature lower bound. 

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