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Journal für die reine und angewandte Mathematik

Edited by: Tobias Colding , Daniel Huybrechts , Jun-Muk Hwang , James Newton , Olivier Schiffmann , Jakob Stix and Gábor Székelyhidi

Languages: English, German, French

First published: January 1, 1826

Publication Frequency: 12 issues per year

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About this journal

The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of "Crelle's Journal". In the 190 years of its existence, "Crelle's Journal" has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals. It belongs to the very top mathematics periodicals, as listed in ISI's Journal Citation Report.

Open Access
Revisiting generic mean curvature flow in ℝ³

October 8, 2025

Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis, Felix Schulze

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Bamler–Kleiner recently proved a multiplicity-one theorem for mean curvature flow in R 3 \mathbb{R}^{3} and combined it with the authors’ work on generic mean curvature flows to fully resolve Huisken’s genericity conjecture. In this paper, we show that a short density-drop theorem plus the Bamler–Kleiner multiplicity-one theorem for tangent flows at the first nongeneric singular time suffice to resolve Huisken’s conjecture – without relying on the strict genus-drop theorem for one-sided ancient flows previously established by the authors.
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Zariski-dense non-tempered subgroups in higher rank of nearly optimal growth

July 8, 2025

Mikołaj Frączyk, Hee Oh

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We construct the first example of a Zariski-dense, discrete, non-lattice subgroup Γ 0 \Gamma_{0} of a higher rank simple Lie group 𝐺, which is non-tempered in the sense that the quasi-regular representation L 2 ⁢ ( Γ 0 \ G ) L^{2}(\Gamma_{0}\backslash G) is non-tempered. More precisely, let n ≥ 3 n\geq 3 and let Γ be the fundamental group of a closed hyperbolic 𝑛-manifold that contains a properly embedded totally geodesic hyperplane. We show that there exists a non-empty open subset 𝒪 of Hom ⁡ ( Γ , SO ⁡ ( n , 2 ) ) \operatorname{Hom}(\Gamma,\operatorname{SO}(n,2)) such that, for any σ ∈ O \sigma\in\mathcal{O} , the subgroup σ ⁢ ( Γ ) \sigma(\Gamma) is a Zariski-dense and non-tempered Anosov subgroup of SO ⁡ ( n , 2 ) \operatorname{SO}(n,2) . In addition, the growth indicator of σ ⁢ ( Γ ) \sigma(\Gamma) is nearly optimal: it almost realizes the supremum of growth indicators among all non-lattice discrete subgroups, a bound imposed by property (T) of SO ⁡ ( n , 2 ) \operatorname{SO}(n,2) .
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On the field of meromorphic functions on a Stein surface

October 29, 2025

Olivier Benoist

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We prove that fields of meromorphic functions on Stein surfaces have cohomological dimension 2, and solve the period-index problem and Serre’s conjecture II for these fields. We obtain analogous results for fields of real meromorphic functions on Stein surfaces equipped with an antiholomorphic involution. We deduce an optimal quantitative solution to Hilbert’s 17th problem on analytic surfaces.
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Transverse surfaces and pseudo-Anosov flows

November 11, 2025

Michael P. Landry, Yair N. Minsky, Samuel J. Taylor

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Let 𝜑 be a transitive pseudo-Anosov flow on an oriented, compact 3-manifold 𝑀, possibly with toral boundary. We characterize the surfaces in 𝑀 that are (almost) transverse to 𝜑. When 𝜑 has no perfect fits (e.g. 𝜑 is the suspension flow of a pseudo-Anosov homeomorphism), we prove that any Thurston norm-minimizing surface 𝑆 that pairs nonnegatively with the closed orbits of 𝜑 is almost transverse to 𝜑, up to isotopy. This answers a question of Cooper–Long–Reid. Our main tool is a correspondence between surfaces that are almost transverse to 𝜑 and those that are relatively carried by any associated veering triangulation. The correspondence also allows us to investigate the uniqueness of almost transverse position, to extend Mosher’s Transverse Surface Theorem to the case with boundary, and more generally to characterize when relative homology classes represent Birkhoff surfaces.
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Second adjointness and cuspidal supports at the categorical level

November 14, 2025

Yuta Takaya

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We prove the second adjointness in the setting of the categorical local Langlands correspondence. Moreover, we study the relation between Eisenstein series and cuspidal supports and present a conjectural characterization of irreducible smooth representations with supercuspidal 𝐿-parameters regarding geometric constant terms. The main technical ingredient is an induction principle for geometric Eisenstein series which allows us to reduce to the situations already treated in the literature.
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Inductive methods for counting number fields

November 7, 2025

Brandon Alberts, Robert J. Lemke Oliver, Jiuya Wang, Melanie Matchett Wood

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We give a new method for counting extensions of a number field asymptotically by discriminant, which we employ to prove many new cases of Malle’s Conjecture and counterexamples to Malle’s Conjecture. We consider families of extensions whose Galois closure is a fixed permutation group 𝐺. Our method relies on having asymptotic counts for 𝑇-extensions for some normal subgroup 𝑇 of 𝐺, uniform bounds for the number of such 𝑇-extensions, and possibly weak bounds on the asymptotic number of G / T G/T -extensions. However, we do not require that most 𝑇-extensions of a G / T G/T -extension are 𝐺-extensions. Our new results use 𝑇 either abelian or S 3 m S_{3}^{m} , though our framework is general.
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Linear Shafarevich conjecture in positive characteristic, hyperbolicity and applications

November 18, 2025

Ya Deng, Katsutoshi Yamanoi

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Given a complex quasi-projective normal variety 𝑋 and a linear representation ϱ : π 1 ⁢ ( X ) → GL N ⁡ ( K ) \varrho\colon\pi_{1}(X)\to\operatorname{GL}_{N}(K) with 𝐾 any field of positive characteristic, we mainly establish the following results. The construction of the Shafarevich morphism sh ϱ : X → Sh ϱ ⁢ ( X ) \mathrm{sh}_{\varrho}\colon X\to\mathrm{Sh}_{\varrho}(X) associated with 𝜚. In cases where 𝑋 is projective, 𝜚 is faithful and the Γ-dimension of 𝑋 is at most two (e.g. dim X = 2 \dim X=2 ), we prove that the Shafarevich conjecture holds for 𝑋: the universal covering of 𝑋 is holomorphically convex. In cases where 𝜚 is big, we prove that the Green–Griffiths–Lang conjecture holds for 𝑋: 𝑋 is of log general type if and only it is pseudo-Picard or pseudo-Brody hyperbolic. When 𝜚 is big and the Zariski closure of ϱ ⁢ ( π 1 ⁢ ( X ) ) \varrho(\pi_{1}(X)) is a semisimple algebraic group, we prove that 𝑋 is pseudo-Picard hyperbolic, and strongly of log general type. If 𝑋 is special or ℎ-special, then ϱ ⁢ ( π 1 ⁢ ( X ) ) \varrho(\pi_{1}(X)) is virtually abelian. We also prove Claudon–Höring–Kollár’s conjecture for complex projective manifolds with linear fundamental groups of any characteristic.
Open Access
Semistable Lefschetz pencils

November 20, 2025

Hélène Esnault, Moritz Kerz

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We study the geometry and cohomology of Lefschetz pencils for semistable schemes over a discrete valuation ring. We relate the global cohomological properties of the Lefschetz pencil and the monodromy-weight conjecture; in particular, we show that if one assumes the monodromy-weight conjecture in smaller dimensions, then one can obtain a rather complete understanding of the relative cohomology of the pencil. This reduces the monodromy-weight conjecture to an arithmetic variant of a conjecture of Kashiwara for the projective line.
Open Access
On the structure of singularities of weak mean curvature flows with mean curvature bounds

November 27, 2025

Maxwell Stolarski

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This paper studies singularities of mean curvature flows with integral mean curvature bounds H ∈ L ∞ ⁢ L loc p H\in L^{\infty}L^{p}_{\mathrm{loc}} for some p ∈ ( n , ∞ ] p\in(n,\infty] . For such flows, any backward tangent flow is given by the flow of a stationary cone 𝐂. When p = ∞ p=\infty and 𝐂 is a regular cone, we prove that the backward tangent flow is unique. These results hold for general integral Brakke flows of arbitrary codimension in an open subset U ⊆ R N U\subseteq\mathbb{R}^{N} with H ∈ L ∞ ⁢ L loc p H\in L^{\infty}L^{p}_{\mathrm{loc}} . For smooth, codimension-one mean curvature flows with H ∈ L ∞ ⁢ L loc ∞ H\in L^{\infty}L^{\infty}_{\mathrm{loc}} , we also show that, at points where a backward tangent flow is given by an area-minimizing Simons cone, there is an accompanying limit flow given by a smooth Hardt–Simon minimal surface.
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On the minimal number of closed geodesics on positively curved Finsler spheres

November 8, 2025

Huagui Duan, Dong Xie

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In this paper, we prove that, for every Finsler metric on S n S^{n} ( n ≥ 4 n\geq 4 ) with reversibility 𝜆 and flag curvature 𝐾 satisfying ( 2 ⁢ n − 3 n − 1 ) 2 ⁢ ( λ λ + 1 ) 2 < K ≤ 1 (\frac{2n-3}{n-1})^{2}(\frac{\lambda}{\lambda+1})^{2}<K\leq 1 and λ < n − 1 n − 2 \lambda<\frac{n-1}{n-2} , there exist at least 𝑛 prime closed geodesics on ( S n , F ) (S^{n},F) , which solves a conjecture of Katok and Anosov for such positively curved spheres when 𝑛 is even. Furthermore, if the number of closed geodesics on such positively curved Finsler S n S^{n} is finite, then there exist at least 2 ⁢ [ n 2 ] − 1 2[\frac{n}{2}]-1 non-hyperbolic closed geodesics.
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On the geometry of spaces of filtrations on local rings

November 14, 2025

Lu Qi

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We study the geometry of spaces of filtrations on a Noetherian local domain. We introduce a metric d 1 d_{1} on the space of saturated filtrations, inspired by the Darvas metric in complex geometry, such that it is a geodesic metric space. In the toric case, using Newton–Okounkov bodies, we identify the space of saturated monomial filtrations with a subspace of L loc 1 L^{1}_{\mathrm{loc}} . We also consider several other topologies on such spaces and study the semi-continuity of the log canonical threshold function in the spirit of Demailly–Kollár. Moreover, there is a natural lattice structure on the space of saturated filtrations, which is a generalization of the classical result that the ideals of a ring form a lattice.
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Rigidity results for initial data sets satisfying the dominant energy condition

November 8, 2025

Christian Bär, Simon Brendle, Tsz-Kiu Aaron Chow, Bernhard Hanke

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Our work proves rigidity theorems for initial data sets associated with compact smooth spin manifolds with boundary and with compact convex polytopes, subject to the dominant energy condition. For manifolds with smooth boundary, this is based on the solution of a boundary value problem for Dirac operators. For convex polytopes, we use approximations by manifolds with smooth boundary.
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A new energy method for shortening and straightening complete curves

November 5, 2025

Tatsuya Miura, Fabian Rupp

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We introduce a novel energy method that reinterprets “curve shortening” as “tangent aligning”. This conceptual shift enables the variational study of infinite-length curves evolving by the curve shortening flow, as well as higher-order flows such as the elastic flow, which involves not only the curve shortening but also the curve straightening effect. For the curve shortening flow, we prove convergence to a straight line under mild assumptions on the ends of the initial curve. For the elastic flow, we establish a global well-posedness theory, and investigate the precise long-time behavior of solutions. In fact, our method applies to a more general class of geometric evolution equations including the surface diffusion flow, Chen’s flow, and the free elastic flow.
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Exceptional Tannaka groups only arise from cubic threefolds

November 11, 2025

Thomas Krämer, Christian Lehn, Marco Maculan

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We show that, under mild assumptions, the Fano surfaces of lines on smooth cubic threefolds are the only smooth subvarieties of abelian varieties whose Tannaka group for the convolution of perverse sheaves is an exceptional simple group. This in particular leads to a considerable strengthening of our previous work on the Shafarevich conjecture. A key idea is to control the Hodge decomposition on cohomology by a cocharacter of the Tannaka group of Hodge modules, and to play this off against an improvement of the Hodge number estimates for irregular varieties by Lazarsfeld–Popa and Lombardi.
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Gradient estimates for the conductivity problem with imperfect bonding interfaces

November 5, 2025

Hongjie Dong, Zhuolun Yang, Hanye Zhu

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We study the field concentration phenomenon between two closely spaced perfect conductors with imperfect bonding interfaces of low conductivity type. The boundary condition on these interfaces is given by a Robin-type boundary condition. We discover a new dichotomy for the field concentration depending on the bonding parameter 𝛾. Specifically, we show that the gradient of solution is uniformly bounded independent of 𝜀 (the distance between two inclusions) when 𝛾 is sufficiently small. However, the gradient may blow up when 𝛾 is large. Moreover, we identify the threshold of 𝛾 and the optimal blow-up rates under certain symmetry assumptions. The proof relies on a crucial anisotropic gradient estimate in the thin neck between two inclusions. We develop a general framework for establishing such estimate, which is applicable to a wide range of elliptic equations and boundary conditions.
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Complete minimal surfaces of finite topology in the doubled Schwarzschild 3-manifold

November 25, 2025

Jaigyoung Choe, Jaehoon Lee, Eungbeom Yeon

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We construct complete embedded minimal surfaces of arbitrary genus in the doubled Schwarzschild 3-manifold. A classical desingularization method is used for the construction.
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On explicit bounds of Fano threefolds

August 28, 2025

Caucher Birkar, Jihao Liu

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In this paper, we study the explicit geometry of threefolds, in particular, Fano varieties. We find an explicitly computable positive integer 𝑁 such that all but a bounded family of Fano threefolds have 𝑁-complements. This result has many applications on finding explicit bounds of algebraic invariants for threefolds. We provide explicit lower bounds for the first gap of the ℝ-complementary thresholds for threefolds, the first gap of the global lc thresholds, the smallest minimal log discrepancy of exceptional threefolds, and the volume of log threefolds with reduced boundary and ample log canonical divisor. We also provide an explicit upper bound of the anti-canonical volume of exceptional threefolds. While the bounds in this paper may not and are not expected to be optimal, they are the first explicit bounds of these invariants in dimension three.
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Uniform K-stability of 𝐺-varieties of complexity 1

November 20, 2025

Yan Li, Zhenye Li

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Let k \mathrm{k} be an algebraically closed field of characteristic 0 and 𝐺 a connected, reductive, linear algebraic group of simply connected type over k \mathrm{k} . Let 𝑋 be a projective 𝐺-variety of complexity 1. We classify 𝐺-equivariant normal test configurations of 𝑋 with integral central fibre via the combinatorial data. We also give a formula of anti-canonical divisors on 𝑋. Based on this formula, when 𝑋 is ℚ-Fano, we give an expression of the Futaki invariant, and derive a criterion of uniform K-stability in terms of the combinatorial data.

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Journal Impact Factor	1.3	2024, Journal Citation Reports (Clarivate, 2025)
5-year Journal Impact Factor	1.6	2024, Journal Citation Reports (Clarivate, 2025)
Journal Citation Indicator	1.10	2024, Journal Citation Reports (Clarivate, 2025)
CiteScore	2.4	2024, Scopus (Elsevier B.V., 2025)
SCImago Journal Rank	2.189	2024, SJR (Scimago Lab, 2025; Data Source: Scopus)
Source Normalized Impact per Paper	1.750	2024, CWTS Journal Indicators (CWTS B.V., 2025; Data Source: Scopus)
Mathematical Citation Quotient	1.87

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Please submit your paper via our online submission system accessible here, following the Information for Authors. Each paper should include a short but informative abstract, as well as the Mathematics Subject Classification 2010 representing the primary and secondary subjects of the article.

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Crelle's Journal follows the traditional style of presentation and layout that has been part of this journal since its inception. This applies to formulas, references and other. Our copyediting will help you with adapting your accepted manuscripts to the Crelle style
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References should be listed alphabetically and conform to the following example:
[3] H. Hasse, Symmetrische Matrizen im Körper der reellen Zahlen, J. reine angew. Math. 153 (1923), 12–43.
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Editors

Daniel Huybrechts; Mathematisches Institut, Universität Bonn; Endenicher Allee 60, 53115 Bonn; Germany; email: huybrech@math.uni-bonn.de

Tobias H. Colding; Department of Mathematics, Massachusetts Institute of Technology; 77 Massachusetts Avenue, Cambridge, MA 02139-4307; USA; email: journals@math.mit.edu

Jun-Muk Hwang; Institute for Basic Science, Center for Complex Geometry, Daejeon, Korea (South); email: jmhwang@ibs.re.kr

James Newton; Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG: email: newton@maths.ox.ac.uk

Olivier Schiffmann; Institut de Mathématiques d’Orsay, Faculté des Sciences d’Orsay, Bat. 307, 91405 Orsay Cedex, France; email: olivier.schiffmann@cnrs.fr

Jakob Stix; Institut für Mathematik, Johann Wolfgang Goethe-Universität, Robert-Mayer-Str. 6-8, 60325 Frankfurt am Main, Germany; email: stix.crelle@math.uni-frankfurt.de

Gábor Székelyhidi; Mathematics Department, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208; USA; email: gaborsz@northwestern.edu

Managing Editor

Daniel Huybrechts; Mathematisches Institut, Universität Bonn; Endenicher Allee 60, 53115 Bonn; Germany; email: crelle@math.uni-bonn.de

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eISSN:

1435-5345

ISSN:

0075-4102

Languages:

English, German, French

Publisher:

De Gruyter

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First published:

January 1, 1826

Publication Frequency:

12 issues per year

eISSN: 1435-5345

ISSN: 0075-4102

Audience(s) for this journal

Researchers interested in pure and applied mathematics

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Journal für die reine und angewandte Mathematik

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