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1. Half-Space Intersection Properties for Minimal Hypersurfaces

   https://doi.org/10.1007/s12220-025-02006-3  

   Naff, Keaton; Zhu, Jonathan J (June 2025, The Journal of Geometric Analysis)

   We prove ``half-space" intersection properties in three settings: the hemisphere, half-geodesic balls in space forms, and certain subsets of Gaussian space. For instance, any two embedded minimal hypersurfaces in the sphere must intersect in every closed hemisphere. Two approaches are developed: one using classifications of stable minimal hypersurfaces, and the second using conformal change and comparison geometry for $$\alpha$$-Bakry-\'{E}mery-Ricci curvature. Our methods yield the analogous intersection properties for free boundary minimal hypersurfaces in space form balls, even when the interior or boundary curvature may be negative. Finally, Colding and Minicozzi recently showed that any two embedded shrinkers of dimension $$n$$ must intersect in a large enough Euclidean ball of radius $R(n)$. We show that $$R(n) \leq 2 \sqrt{n}$$. 

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2. Non-vanishing of class group L -functions for number fields with a small regulator

   https://doi.org/10.1112/S0010437X20007472  

   Khayutin, Ilya (November 2020, Compositio Mathematica)

   null (Ed.)

   Let $$E/\mathbb {Q}$$ be a number field of degree $$n$$ . We show that if $$\operatorname {Reg}(E)\ll _n |\!\operatorname{Disc}(E)|^{1/4}$$ then the fraction of class group characters for which the Hecke $$L$$ -function does not vanish at the central point is $$\gg _{n,\varepsilon } |\!\operatorname{Disc}(E)|^{-1/4-\varepsilon }$$ . The proof is an interplay between almost equidistribution of Eisenstein periods over the toral packet in $$\mathbf {PGL}_n(\mathbb {Z})\backslash \mathbf {PGL}_n(\mathbb {R})$$ associated to the maximal order of $$E$$ , and the escape of mass of the torus orbit associated to the trivial ideal class. 

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3. An inhomogeneous Dirichlet theorem via shrinking targets

   https://doi.org/10.1112/S0010437X1900719X  

   Kleinbock, Dmitry; Wadleigh, Nick (July 2019, Compositio Mathematica)

   We give an integrability criterion on a real-valued non-increasing function $$\unicode[STIX]{x1D713}$$ guaranteeing that for almost all (or almost no) pairs $$(A,\mathbf{b})$$ , where $$A$$ is a real $$m\times n$$ matrix and $$\mathbf{b}\in \mathbb{R}^{m}$$ , the system $$\begin{eqnarray}\Vert A\mathbf{q}+\mathbf{b}-\mathbf{p}\Vert ^{m}<\unicode[STIX]{x1D713}(T),\quad \Vert \mathbf{q}\Vert ^{n} 

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4. The anisotropic Bernstein problem

   https://doi.org/10.1007/s00222-023-01222-4  

   Mooney, Connor; Yang, Yang (January 2024, Inventiones mathematicae)

   Abstract We construct nonlinear entire anisotropic minimal graphs over$$\mathbb{R}^{4}$$ $R^4$, completing the solution to the anisotropic Bernstein problem. The examples we construct have a variety of growth rates, and our approach both generalizes to higher dimensions and recovers and elucidates known examples of nonlinear entire minimal graphs over$$\mathbb{R}^{n},\, n \geq 8$$ $R^n,n\ge 8$. 

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5. Absence of bubbling phenomena for non-convex anisotropic nearly umbilical and quasi-Einstein hypersurfaces

   https://doi.org/10.1515/crelle-2021-0038  

   De Rosa, Antonio; Gioffrè, Stefano (August 2021, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We prove that, for every closed (not necessarily convex) hypersurface Σ in ℝ n + 1 {\mathbb{R}^{n+1}} and every p > n {p>n} , the L p {L^{p}} -norm of the trace-free part of the anisotropic second fundamental form controls from above the W 2 , p {W^{2,p}} -closeness of Σ to the Wulff shape. In the isotropic setting, we provide a simpler proof. This result is sharp since in the subcritical regime p ≤ n {p\leq n} , the lack of convexity assumptions may lead in general to bubbling phenomena.Moreover, we obtain a stability theorem for quasi-Einstein (not necessarily convex) hypersurfaces and we improve the quantitative estimates in the convex setting. 

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