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1. 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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2. 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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3. Volume and macroscopic scalar curvature

   https://doi.org/10.1007/s00039-021-00588-y  

   Braun, Sabine; Sauer, Roman (December 2021, Geometric and Functional Analysis)

   Abstract We prove the macroscopic cousins of three conjectures: (1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, (2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, (3) a conjectural bound of $$\ell ^2$$ ℓ 2 -Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of 1-balls in the universal cover. 

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4. Curvature in the balance: the Weyl functional and scalar curvature of 4-manifolds

   https://doi.org/10.4310/PAMQ.2023.v19.n6.a5  

   LeBrun, Claude (January 2024, Pure and Applied Mathematics Quarterly)

   The infimum of the Weyl functional is shown to be surprisingly small on many compact 4-manifolds that admit positive- scalar-curvature metrics. Results are also proved that systematically compare the scalar and self-dual Weyl curvatures of certain almost-Kähler 4-manifolds. 

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5. 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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