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1. Rigidity and compactness with constant mean curvature in warped product manifolds

   https://doi.org/10.1515/crelle-2025-0065  

   Maggi, Francesco; Santilli, Mario (October 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We prove the rigidity ofrectifiableboundaries with constantdistributionalmean curvature in the Brendle class of warped product manifolds (which includes important models in general relativity, like the de Sitter–Schwarzschild and Reissner–Nordstrom manifolds).As a corollary, we characterize limits of rectifiable boundaries whose mean curvatures converge, as distributions, to a constant.The latter result is new, and requires the full strength of distributional CMC-rigidity, even when one considers smooth boundaries whose mean curvature oscillations vanish in arbitrarily strong $C^k$C^{k}-norms.Our method also establishes that rectifiable boundaries of sets of finite perimeter in the hyperbolic space with constant distributional mean curvature are finite unions of possibly mutually tangent geodesic spheres. 

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2. Simple numerical solutions to the Einstein constraints on various three-manifolds

   https://doi.org/10.1007/s10714-022-03014-2  

   Zhang, Fan; Lindblom, Lee (October 2022, General Relativity and Gravitation)

   Abstract Numerical solutions to the Einstein constraint equations are constructed on a selection of compact orientable three-dimensional manifolds with non-trivial topologies. A simple constant mean curvature solution and a somewhat more complicated non-constant mean curvature solution are computed on example manifolds from three of the eight Thursten geometrization classes. The constant mean curvature solutions found here are also solutions to the Yamabe problem that transforms a geometry into one with constant scalar curvature. 

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3. The anisotropic min‐max theory: Existence of anisotropic minimal and CMC surfaces

   https://doi.org/10.1002/cpa.22189  

   De_Philippis, Guido; De_Rosa, Antonio (July 2024, Communications on Pure and Applied Mathematics)

   We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3–dimensional Riemannian manifolds. The constructed min‐max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard in dimension 3. 

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4. Colding–Minicozzi entropies in Cartan–Hadamard manifolds

   https://doi.org/10.1515/crelle-2025-0032  

   Bernstein, Jacob; Bhattacharya, Arunima (May 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We introduce a family of functionals defined on the set of submanifolds of Cartan–Hadamard manifolds which generalize the Colding–Minicozzi entropy of submanifolds of Euclidean space.We show these functionals are monotone under mean curvature flow under natural conditions.As a consequence, we obtain sharp lower bounds on these entropies for certain closed hypersurfaces and observe a novel rigidity phenomenon. 

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5. 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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