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1. 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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2. Scalar curvature in discrete gravity

   https://doi.org/10.1140/epjc/s10052-022-10605-5  

   Chamseddine, Ali H.; Malaeb, Ola; Najem, Sara (July 2022, The European Physical Journal C)

   Abstract We focus on studying, numerically, the scalar curvature tensor in a two-dimensional discrete space. The continuous metric of a two-sphere is transformed into that of a lattice using two possible slicings. In the first, we use two integers, while in the second we consider the case where one of the coordinates is ignorable. The numerical results of both cases are then compared with the expected values in the continuous limit as the number of cells of the lattice becomes very large. 

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3. A Splitting Theorem for Scalar Curvature

   https://doi.org/10.1002/cpa.21803  

   Chodosh, Otis; Eichmair, Michael; Moraru, Vlad (December 2018, Communications on Pure and Applied Mathematics)
4. Scalar curvature rigidity of convex polytopes

   https://doi.org/10.1007/s00222-023-01229-x  

   Brendle, Simon (February 2024, Inventiones mathematicae)
5. Conformally Prescribed Scalar Curvature on Orbifolds

   https://doi.org/10.1007/s00220-022-04542-3  

   Ju, Tao; Viaclovsky, Jeff (November 2022, Communications in Mathematical Physics)

   Abstract We study the prescribed scalar curvature problem in a conformal class on orbifolds with isolated singularities. We prove a compactness theorem in dimension 4, and an existence theorem which holds in dimensions$$n \ge 4$$ $n\ge 4$. This problem is more subtle than the manifold case since the positive mass theorem does not hold for ALE metrics in general. We also determine the$$\textrm{U}(2)$$ $\text{U}\left(2\right)$-invariant Leray–Schauder degree for a family of negative-mass orbifolds found by LeBrun. 

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