journal: Journal für die reine und angewandte Mathematik

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Zeitschrift

Journal für die reine und angewandte Mathematik

Herausgegeben von: Tobias Colding , Daniel Huybrechts , Jun-Muk Hwang , James Newton , Olivier Schiffmann , Jakob Stix und Gábor Székelyhidi

Sprachen: Englisch, Deutsch, Französisch

Erstveröffentlichung: 1. Januar 1826

Erscheinungsweise: 12 Hefte pro Jahr

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Über diese Zeitschrift

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

8. Oktober 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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Rigidity and compactness with constant mean curvature in warped product manifolds

2. Oktober 2025

Francesco Maggi, Mario Santilli

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We prove the rigidity of rectifiable boundaries with constant distributional mean 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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A new monotone quantity in mean curvature flow implying sharp homotopic criteria

30. September 2025

Chung-Jun Tsai, Mao-Pei Tsui, Mu-Tao Wang

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A new monotone quantity in graphical mean curvature flows of higher codimensions is identified in this work. The submanifold deformed by the mean curvature flow is the graph of a map between Riemannian manifolds, and the quantity is monotone increasing under the area-decreasing condition of the map. The flow provides a natural homotopy of the corresponding map and leads to sharp criteria regarding the homotopic class of maps between complex projective spaces, and maps from spheres to complex projective spaces, among others.
Open Access
On explicit bounds of Fano threefolds

28. August 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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Zariski-dense non-tempered subgroups in higher rank of nearly optimal growth

8. Juli 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) .
Open Access
On the field of meromorphic functions on a Stein surface

1. Januar 2000

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

1. Januar 2000

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.
Open Access
Inductive methods for counting number fields

1. Januar 2000

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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Classification of torsion of elliptic curves over quartic fields

1. Januar 2000

Maarten Derickx, Filip Najman

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Let 𝐸 be an elliptic curve over a quartic field 𝐾. By the Mordell–Weil theorem, E ⁢ ( K ) E(K) is a finitely generated group. We determine all the possibilities for the torsion group E ⁢ ( K ) tors E(K)_{\mathrm{tors}} , where 𝐾 ranges over all quartic fields 𝐾 and 𝐸 ranges over all elliptic curves over 𝐾. We show that there are no sporadic torsion groups, or in other words, that all torsion groups either do not appear or they appear for infinitely many non-isomorphic elliptic curves 𝐸. Proving this requires showing that numerous modular curves X 1 ⁢ ( m , n ) X_{1}(m,n) have no non-cuspidal degree 4 points. We deal with almost all the curves using one of 3 methods: a method for the rank 0 cases requiring no computation; the Hecke sieve, a local method requiring computer-assisted computations; and the global method, an argument for the positive rank cases also requiring no computation. We deal with the handful of remaining cases using ad hoc methods.
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Volumes of subvarieties of complex ball quotients and sparsity of rational points

1. Januar 2000

Soheil Memariansorkhabi

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Let X = Γ ∖ B n X=\Gamma\setminus\mathbb{B}^{n} be an 𝑛-dimensional complex ball quotient by a torsion-free non-uniform lattice Γ whose parabolic subgroups are unipotent. We prove that the volumes of subvarieties of 𝑋 are controlled by the systole of 𝑋, which is the length of a shortest closed geodesic of 𝑋. There are a number of arithmetic and geometric consequences: the systole of 𝑋 controls the growth rate of rational points on 𝑋, uniformly in the field of definition. Also, we obtain effective global generation and very ampleness results for multiples of the canonical bundle K X ̄ K_{\overline{X}} , where X ̄ \overline{X} is the toroidal compactification of 𝑋. These follow from the bound we find for the Seshadri constant of K X ̄ K_{\overline{X}} in terms of the systole.
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Some obstructions to positive scalar curvature on a noncompact manifold

1. Januar 2000

John Lott

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We give obstructions for a noncompact manifold to admit a complete Riemannian metric with (nonuniformly) positive scalar curvature. We treat both the finite volume and infinite volume cases.
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On the minimal number of closed geodesics on positively curved Finsler spheres

1. Januar 2000

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.
Open Access
Equations involving the modular 𝑗-function and its derivatives

1. Januar 2000

Vahagn Aslanyan, Sebastian Eterović, Vincenzo Mantova

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We show that, for any polynomial F ⁢ ( X , Y 0 , Y 1 , Y 2 ) ∈ C ⁢ [ X , Y 0 , Y 1 , Y 2 ] F(X,Y_{0},Y_{1},Y_{2})\in\mathbb{C}[X,Y_{0},Y_{1},Y_{2}] , the equation F ⁢ ( z , j ⁢ ( z ) , j ′ ⁢ ( z ) , j ′′ ⁢ ( z ) ) = 0 F(z,j(z),j^{\prime}(z),j^{\prime\prime}(z))=0 has a Zariski dense set of solutions in the hypersurface F ⁢ ( X , Y 0 , Y 1 , Y 2 ) = 0 F(X,Y_{0},Y_{1},Y_{2})=0 , unless 𝐹 is in C ⁢ [ X ] \mathbb{C}[X] or it is divisible by Y 0 Y_{0} , Y 0 − 1728 Y_{0}-1728 , or Y 1 Y_{1} . Our methods establish criteria for finding solutions to more general equations involving periodic functions. Furthermore, they produce a qualitative description of the distribution of these solutions.
Open Access
Rigidity results for initial data sets satisfying the dominant energy condition

1. Januar 2000

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.
Open Access
A new energy method for shortening and straightening complete curves

1. Januar 2000

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.
Open Access
Algebraic Lagrangian cobordisms, flux and the Lagrangian Ceresa cycle

1. Januar 2000

Alexia Corradini

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We introduce an equivalence relation for Lagrangians in a symplectic manifold known as algebraic Lagrangian cobordism , which is meant to mirror algebraic equivalence of cycles. From this, we prove a symplectic, mirror-symmetric analogue of the statement “the Ceresa cycle is non-torsion in the Griffiths group of the Jacobian of a generic genus 3 curve”. Namely, we show that, for a family of tropical curves, the Lagrangian Ceresa cycle , which is the Lagrangian lift of their tropical Ceresa cycle to the corresponding Lagrangian torus fibration, is non-torsion in its oriented algebraic Lagrangian cobordism group. We proceed by developing the notions of tropical (resp. symplectic) flux, which are morphisms from the tropical Griffiths (resp. algebraic Lagrangian cobordism) groups.
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Exceptional Tannaka groups only arise from cubic threefolds

1. Januar 2000

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.
Open Access
Gradient estimates for the conductivity problem with imperfect bonding interfaces

1. Januar 2000

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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Bounding volume of compact domains via mean curvature and bottom spectrum

1. Januar 2000

Ovidiu Munteanu, Jiaping Wang

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The classical Minkowski inequality implies that the volume of a bounded convex domain in the Euclidean space is controlled from above by the integral of the mean curvature of its boundary. In this note, an analogous inequality is established without assuming convexity, valid for all bounded smooth domains in a complete manifold whose bottom spectrum is suitably large relative to its Ricci curvature lower bound. An immediate consequence is the nonexistence of embedded closed minimal hypersurfaces in such manifolds. The same nonexistence issue is also addressed for steady and expanding Ricci solitons. The proofs are very much inspired by a sharp monotonicity formula, derived for positive harmonic functions on manifolds with positive spectrum.

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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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[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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1435-5345

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Erstveröffentlichung:

1. Januar 1826

Erscheinungsweise:

12 Hefte pro Jahr

eISSN: 1435-5345

ISSN: 0075-4102

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