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1. Total mean curvatures of Riemannian hypersurfaces

   https://doi.org/10.1515/ans-2022-0029  

   Ghomi, Mohammad ; Spruck, Joel ( January 2023 , Advanced Nonlinear Studies)

   Abstract We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces via Reilly’s identities. As applications, we derive several geometric inequalities for a convex hypersurface Γ \Gamma in a Cartan-Hadamard manifold M M . In particular, we show that the first mean curvature integral of a convex hypersurface γ \gamma nested inside Γ \Gamma cannot exceed that of Γ \Gamma , which leads to a sharp lower bound for the total first mean curvature of Γ \Gamma in terms of the volume it bounds in M M in dimension 3. This monotonicity property is extended to all mean curvature integrals when γ \gamma is parallel to Γ \Gamma , or M M has constant curvature. We also characterize hyperbolic balls as minimizers of the mean curvature integrals among balls with equal radii in Cartan-Hadamard manifolds. 

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2. Mean Curvature Flow in Null Hypersurfaces and the Detection of MOTS

   https://doi.org/10.1007/s00220-022-04326-9  

   Roesch, Henri ; Scheuer, Julian ( February 2022 , Communications in Mathematical Physics)

   We study the mean curvature flow in 3-dimensional null hypersurfaces. In a spacetime a hypersurface is called null, if its induced metric is degenerate. The speed of the mean curvature flow of spacelike surfaces in a null hypersurface is the projection of the codimension-two mean curvature vector onto the null hypersurface. We impose fairly mild conditions on the null hypersurface. Then for an outer un-trapped initial surface, a condition which resembles the mean-convexity of a surface in Euclidean space, we prove that the mean curvature flow exists for all times and converges smoothly to a marginally outer trapped surface (MOTS). As an application we obtain the existence of a global foliation of the past of an outermost MOTS, provided the null hypersurface admits an un-trapped foliation asymptotically.

    

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3. Hodge theory on ALG ∗ manifolds

   https://doi.org/10.1515/crelle-2023-0016  

   Chen, Gao ; Viaclovsky, Jeff ; Zhang, Ruobing ( May 2023 , Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We develop a Fredholm theory for the Hodge Laplacian in weighted spaces on ALG ∗ manifolds in dimension four.We then give several applications of this theory.First, we show the existence of harmonic functions with prescribed asymptotics at infinity.A corollary of this is a non-existence result for ALG ∗ manifolds with non-negative Ricci curvature having group Γ = { e } \Gamma=\{e\} at infinity.Next, we prove a Hodge decomposition for the first de Rham cohomology group of an ALG ∗ manifold.A corollary of this is vanishing of the first Betti number for any ALG ∗ manifold with non-negative Ricci curvature.Another application of our analysis is to determine the optimal order of ALG ∗ gravitational instantons. 

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4. Kasner-like regions near crushing singularities *

   https://doi.org/10.1088/1361-6382/abd3e1  

   Lott, John ( December 2020 , Classical and Quantum Gravity)

   Abstract We consider vacuum spacetimes with a crushing singularity. Under some scale-invariant curvature bounds, we relate the existence of Kasner-like regions to the asymptotics of spatial volume densities. 

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5. Role of curvature in acridone for 1O2 oxidation of a natural product homoallylic alcohol: A novel iso ‐hydroperoxide intermediate

   https://doi.org/10.1111/php.13843  

   Lapoot, Lloyd ; Jabeen, Shakeela ; Durantini, Andrés M. ; Greer, Alexander ( August 2023 , Photochemistry and Photobiology)

   A density functional theoretical (DFT) study is presented, implicating a¹O₂oxidation process to reach a dihydrobenzofuran from the reaction of the natural homoallylic alcohol, glycocitrine. Our results predict an interconversion between glycocitrine and aniso‐hydroperoxide intermediate [R(H)O⁺–O⁻] that provides a key path in the chemistry which then follows. Formations of allylic hydroperoxides are unlikely from a¹O₂‘ene’ reaction. Instead, the dihydrobenzofuran arises by¹O₂oxidation facilitated by a 16° curvature of the glycocitrine ring imposed by a pyramidalN‐methyl group. This curvature facilitates the formation of theiso‐hydroperoxide, which is analogous to theisospecies CH₂I⁺–I⁻and CHI₂⁺–I⁻formed by UV photolysis of CH₂I₂and CHI₃. Theiso‐hydroperoxide is also structurally reminiscent of carbonyl oxides (R₂C=O⁺–O⁻) formed in the reaction of carbenes and oxygen. Our DFT results point to intermolecular process, in which theiso‐hydroperoxide's fate relates to O‐transfer and H₂O dehydration reactions for new insight into the biosynthesis of dihydrobenzofuran natural products.

    

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