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1. Asymptotically hyperbolic Einstein constraint equations with apparent horizon boundary and the Penrose inequality for perturbations of Schwarzschild-AdS ^*

   https://doi.org/10.1088/1361-6382/acb24b  

   Khuri, Marcus; Kopiński, Jarosław (January 2023, Classical and Quantum Gravity)

   Abstract We prove the existence of asymptotically hyperbolic solutions to the vacuum Einstein constraint equations with a marginally outer trapped boundary of positive mean curvature, using the constant mean curvature conformal method. As an application of this result, we verify the Penrose inequality for certain perturbations of Schwarzschild Anti-de Sitter black hole initial data. 

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

   Maggi, Francesco; Santilli, Mario (March 2024, arxiv)

   We prove the rigidity of rectifiable boundaries with constant distributional mean curvature in the Brendle class of warped product manifolds (which includes important models in General Relativity, like the deSitter--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-norms. 

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4. 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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5. Improved Beckner's inequality for axially symmetric functions on $$\mathbb{S}^4$$

   https://doi.org/10.4171/rmi/1445  

   Gui, Changfeng; Hu, Yeyao; Xie, Weihong (February 2024, Revista Matemática Iberoamericana)

   We show that axially symmetric solutions on\mathbb{S}^4to a constantQ-curvature type equation (it may also be called fourth order mean field equation) must be constant, provided that the parameter\alphain front of the Paneitz operator belongs to the interval[\frac{473 + \sqrt{209329}}{1800}\approx 0.517, 1). This is in contrast to the case\alpha=1, where there exists a family of solutions, known as standard bubbles. The phenomenon resembles the Gaussian curvature equation on\mathbb{S}^2. As a consequence, we prove an improved Beckner's inequality on\mathbb{S}^4for axially symmetric functions with their centers of mass at the origin. Furthermore, we show uniqueness of axially symmetric solutions when\alpha=1/5by exploiting Pohozaev-type identities, and prove the existence of a non-constant axially symmetric solution for\alpha \in (1/5, 1/2)via a bifurcation method. 

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