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1. Almost isotropic Kähler manifolds

   https://doi.org/10.1515/crelle-2019-0030  

   Schmidt, Benjamin; Shankar, Krishnan; Spatzier, Ralf (October 2020, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let M be a complete Riemannian manifold and suppose {p\in M} . For each unit vector {v\in T_{p}M} , the Jacobi operator , {\mathcal{J}_{v}:v^{\perp}\rightarrow v^{\perp}} is the symmetric endomorphism, {\mathcal{J}_{v}(w)=R(w,v)v} . Then p is an isotropic point if there exists a constant {\kappa_{p}\in{\mathbb{R}}} such that {\mathcal{J}_{v}=\kappa_{p}\operatorname{Id}_{v^{\perp}}} for each unit vector {v\in T_{p}M} . If all points are isotropic, then M is said to be isotropic; it is a classical result of Schur that isotropic manifolds of dimension at least 3 have constant sectional curvatures. In this paper we consider almost isotropic manifolds , i.e. manifolds having the property that for each {p\in M} , there exists a constant {\kappa_{p}\in\mathbb{R}} such that the Jacobi operators {\mathcal{J}_{v}} satisfy {\operatorname{rank}({\mathcal{J}_{v}-\kappa_{p}\operatorname{Id}_{v^{\perp}}}% )\leq 1} for each unit vector {v\in T_{p}M} . Our main theorem classifies the almost isotropic simply connected Kähler manifolds, proving that those of dimension {d=2n\geqslant 4} are either isometric to complex projective space or complex hyperbolic space or are totally geodesically foliated by leaves isometric to {{\mathbb{C}}^{n-1}} . 

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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. An upper bound for the first nonzero Steklov eigenvalue

   https://doi.org/10.1051/cocv/2024088  

   Li, Xiaolong; Wang, Kui; Wu, Haotian (January 2025, ESAIM: Control, Optimisation and Calculus of Variations)

   Let (Mⁿ,g) be a complete simply connectedn-dimensional Riemannian manifold with curvature bounds Sect_g≤ κ for κ ≤ 0 and Ric_g≥ (n− 1)KgforK≤ 0. We prove that for any bounded domain Ω ⊂Mⁿwith diameterdand Lipschitz boundary, if Ω* is a geodesic ball in the simply connected space form with constant sectional curvature κ enclosing the same volume as Ω, then σ₁(Ω) ≤Cσ₁(Ω*), where σ₁(Ω) and σ₁(Ω*) denote the first nonzero Steklov eigenvalues of Ω and Ω* respectively, andC=C(n, κ,K,d) is an explicit constant. When κ =K, we haveC= 1 and recover the Brock–Weinstock inequality, asserting that geodesic balls uniquely maximize the first nonzero Steklov eigenvalue among domains of the same volume, in Euclidean space and the hyperbolic space. 

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4. Geodesic planes in geometrically finite acylindrical -manifolds

   https://doi.org/10.1017/etds.2021.19  

   BENOIST, YVES; OH, HEE (February 2022, Ergodic Theory and Dynamical Systems)

   Abstract Let M be a geometrically finite acylindrical hyperbolic $$3$$ -manifold and let $M^*$ denote the interior of the convex core of M . We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many closed geodesic planes in $M^*$ . These results were obtained by McMullen, Mohammadi and Oh [Geodesic planes in hyperbolic 3-manifolds. Invent. Math. 209 (2017), 425–461; Geodesic planes in the convex core of an acylindrical 3-manifold. Duke Math. J. , to appear, Preprint , 2018, arXiv:1802.03853] when M is convex cocompact. As a corollary, we obtain that when M covers an arithmetic hyperbolic $$3$$ -manifold $$M_0$$ , the topological behavior of a geodesic plane in $M^*$ is governed by that of the corresponding plane in $$M_0$$ . We construct a counterexample of this phenomenon when $$M_0$$ is non-arithmetic. 

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5. SQUARE-INTEGRABILITY OF THE MIRZAKHANI FUNCTION AND STATISTICS OF SIMPLE CLOSED GEODESICS ON HYPERBOLIC SURFACES

   https://doi.org/10.1017/fms.2019.49  

   ARANA-HERRERA, FRANCISCO; ATHREYA, JAYADEV S. (January 2020, Forum of Mathematics, Sigma)

   null (Ed.)

   Given integers $$g,n\geqslant 0$$ satisfying $2-2g-n<0$ , let $${\mathcal{M}}_{g,n}$$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $$g$$ with $$n$$ cusps. We study the global behavior of the Mirzakhani function $$B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$$ which assigns to $$X\in {\mathcal{M}}_{g,n}$$ the Thurston measure of the set of measured geodesic laminations on $$X$$ of hyperbolic length $${\leqslant}1$$ . We improve bounds of Mirzakhani describing the behavior of this function near the cusp of $${\mathcal{M}}_{g,n}$$ and deduce that $$B$$ is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of $$B$$ to statistics of counting problems for simple closed hyperbolic geodesics. 

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