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1. Scalar curvature rigidity of degenerate warped product spaces

   https://doi.org/10.1090/btran/206  

   Wang, Jinmin; Xie, Zhizhang (January 2025, Transactions of the American Mathematical Society, Series B)

   In this paper we prove the scalar curvature extremality and rigidity for a class of warped product spaces that are possibly degenerate at the two ends. The leaves of these warped product spaces can be any closed Riemannian manifolds with nonnegative curvature operators and nonvanishing Euler characteristics, flat tori, round spheres and their direct products. In particular, we obtain the scalar curvature extremality and rigidity for certain degenerate toric bands and also for round spheres with two antipodal points removed. This answers positively the corresponding questions of Gromov in all dimensions. 

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2. Spectral Torical Band Inequalities and Generalizations of the Schoen–Yau Black Hole Existence Theorem

   https://doi.org/10.1093/imrn/rnad129  

   Hirsch, Sven; Kazaras, Demetre; Khuri, Marcus; Zhang, Yiyue (June 2023, International Mathematics Research Notices)

   Abstract Generalized torical band inequalities give precise upper bounds for the width of compact manifolds with boundary in terms of positive pointwise lower bounds for scalar curvature, assuming certain topological conditions. We extend several incarnations of these results in which pointwise scalar curvature bounds are replaced with spectral scalar curvature bounds. More precisely, we prove upper bounds for the width in terms of the principal eigenvalue of the operator $$-\Delta +cR$$, where $$R$$ denotes scalar curvature and $c>0$ is a constant. Three separate strategies are employed to obtain distinct results holding in different dimensions and under varying hypotheses, namely we utilize spacetime harmonic functions, $$\mu $$-bubbles, and spinorial Callias operators. In dimension 3, where the strongest result is produced, we are also able to treat open and incomplete manifolds, and establish the appropriate rigidity statements. Additionally, a version of such spectral torus band inequalities is given where tori are replaced with cubes. Finally, as a corollary, we generalize the classical work of Schoen and Yau, on the existence of black holes due to concentration of matter, to higher dimensions and with alternate measurements of size. 

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3. Decay of scalar curvature on uniformly contractible manifolds with finite asymptotic dimension

   https://doi.org/10.1002/cpa.22128  

   Wang, Jinmin; Xie, Zhizhang; Yu, Guoliang (August 2023, Communications on Pure and Applied Mathematics)

   Abstract Gromov proved a quadratic decay inequality of scalar curvature for a class of complete manifolds. In this paper, we prove that for any uniformly contractible manifold with finite asymptotic dimension, its scalar curvature decays to zero at a rate depending only on the contractibility radius of the manifold and the diameter control of the asymptotic dimension. We construct examples of uniformly contractible manifolds with finite asymptotic dimension whose scalar curvature functions decay arbitrarily slowly. This shows that our result is the best possible. We prove our result by studying the index pairing between Dirac operators and compactly supported vector bundles with Lipschitz control. A key technical ingredient for the proof of our main result is a Lipschitz control for the topologicalK‐theory of finite dimensional simplicial complexes. 

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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. Initial Data Rigidity Results

   https://doi.org/10.1007/s00220-021-04033-x  

   Eichmair, Michael; Galloway, Gregory J.; Mendes, Abraão (August 2021, Communications in Mathematical Physics)

   null (Ed.)

   Abstract We prove several rigidity results related to the spacetime positive mass theorem. A key step is to show that certain marginally outer trapped surfaces are weakly outermost. As a special case, our results include a rigidity result for Riemannian manifolds with a lower bound on their scalar curvature. 

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