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1. Integral Ricci curvature and the mass gap of Dirichlet Laplacians on domains

   https://doi.org/10.1002/mana.202100523  

   Ramos Olivé, Xavier; Rose, Christian; Wang, Lili; Wei, Guofang (June 2023, Mathematische Nachrichten)

   Abstract We obtain a fundamental gap estimate for classes of bounded domains with quantitative control on the boundary in a complete manifold with integral bounds on the negative part of the Ricci curvature. This extends the result of Oden, Sung, and Wang [Trans. Amer. Math. Soc. 351 (1999), no. 9, 3533–3548] to ‐Ricci curvature assumptions, . To achieve our result, it is shown that the domains under consideration are John domains, what enables us to obtain an estimate on the first nonzero Neumann eigenvalue, which is of independent interest. 

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2. Hyperbolicity and Uniformity of Varieties of Log General type

   https://doi.org/10.1093/imrn/rnaa186  

   Ascher, Kenneth; DeVleming, Kristin; Turchet, Amos (August 2020, International Mathematics Research Notices)

   Abstract Projective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. A major hurdle is that the naive generalization is false—the log cotangent bundle is never ample. Instead, we define a notion called almost ample that roughly asks that it is as positive as possible. We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. In addition, if one assumes globally generated then we obtain that such varieties contain finitely many integral points. In another direction, we show that the Lang–Vojta conjecture implies the number of stably integral points on curves of log general type, and surfaces of log general type with almost ample log cotangent sheaf are uniformly bounded. 

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3. Jackiw-Teitelboim and Kantowski-Sachs quantum cosmology

   https://doi.org/10.1088/1475-7516/2022/03/056  

   Fanaras, Georgios; Vilenkin, Alexander (March 2022, Journal of Cosmology and Astroparticle Physics)

   Abstract We study quantum cosmology of the 2DJackiw-Teitelboim (JT) gravity with Λ > 0 and calculate the Hartle-Hawking (HH) wave function for this model in the minisuperspace framework. Our approach is guided by the observation that the JT dynamics can be mapped exactly onto that of the Kantowski-Sachs (KS) model describing a homogeneous universe with spatial sections ofS¹×S²topology. This allows us to establish a JT-KS correspondence between the wave functions of the models. We obtain the semiclassical Hartle-Hawking wave function by evaluating the path integral with appropriate boundary conditions and employing the methods of Picard-Lefschetz theory. The JT-KS connection formulas allow us to translate this result to JT gravity, define the HH wave function and obtain a probability distribution for the dilaton field. 

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4. Kleinian sphere packings, reflection groups, and arithmeticity

   https://doi.org/10.1090/mcom/3858  

   Bogachev, Nikolay; Kolpakov, Alexander; Kontorovich, Alex (January 2024, Mathematics of Computation)

   In this paper we study crystallographic sphere packings and Kleinian sphere packings, introduced first by Kontorovich and Nakamura in 2017 and then studied further by Kapovich and Kontorovich in 2021. In particular, we solve the problem of existence of crystallographic sphere packings in certain higher dimensions posed by Kontorovich and Nakamura. In addition, we present a geometric doubling procedure allowing to obtain sphere packings from some Coxeter polyhedra without isolated roots, and study “properly integral” packings (that is, ones which are integral but not superintegral). Our techniques rely extensively on computations with Lorentzian quadratic forms, their orthogonal groups, and associated higher–dimensional hyperbolic polyhedra. 

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5. Convergence of the self‐dual U (1)‐Yang–Mills–Higgs energies to the (n−2)$(n-2)$‐area functional

   https://doi.org/10.1002/cpa.22150  

   Parise, Davide; Pigati, Alessandro; Stern, Daniel (January 2024, Communications on Pure and Applied Mathematics)

   Abstract Given a hermitian line bundle on a closed Riemannian manifold , the self‐dual Yang–Mills–Higgs energies are a natural family of functionalsdefined for couples consisting of a section  and a hermitian connection ∇ with curvature . While the critical points of these functionals have been well‐studied in dimension two by the gauge theory community, it was shown in [52] that critical points in higher dimension converge as (in an appropriate sense) to minimal submanifolds of codimension two, with strong parallels to the correspondence between the Allen–Cahn equations and minimal hypersurfaces. In this paper, we complement this idea by showing the Γ‐convergence of to (2π times) the codimension two area: more precisely, given a family of couples with , we prove that a suitable gauge invariant Jacobian converges to an integral ‐cycle Γ, in the homology class dual to the Euler class , with mass . We also obtain a recovery sequence, for any integral cycle in this homology class. Finally, we apply these techniques to compare min‐max values for the ‐area from the Almgren–Pitts theory with those obtained from the Yang–Mills–Higgs framework, showing that the former values always provide a lower bound for the latter. As an ingredient, we also establish a Huisken‐type monotonicity result along the gradient flow of . 

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