- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 20:35:18
15
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=15&iss=3&update_since=2024-03-28
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
15
2013
3
Amenability of linear-activity automaton groups
Gideon
Amir
Bar-Ilan University, RAMAT GAN, ISRAEL
Omer
Angel
University of British Columbia, VANCOUVER, CANADA
Bálint
Virág
University of Toronto, TORONTO, ONTARIO, CANADA
amenability, automaton groups, self-similar, random walk, entropy
We prove that every linear-activity automaton group is amenable. The proof is based on showing that a random walk on a specially constructed degree 1 automaton group – the mother group – has asymptotic entropy 0. Our result answers an open question by Nekrashevych in the Kourovka notebook, and gives a partial answer to a question of Sidki.
Probability theory and stochastic processes
Group theory and generalizations
General
705
730
10.4171/JEMS/373
http://www.ems-ph.org/doi/10.4171/JEMS/373
Line bundles with partially vanishing cohomology
Burt
Totaro
University of Cambridge, CAMBRIDGE, UNITED KINGDOM
Vanishing theorems, ample line bundles, q-ample line bundles, Castelnuovo–Mumford regularity, Koszul algebras, q-convexity
Define a line bundle $L$ on a projective variety to be $q$-ample, for a natural number $q$, if tensoring with high powers of $L$ kills coherent sheaf cohomology above dimension $q$. Thus 0-ampleness is the usual notion of ampleness. We show that $q$-ampleness of a line bundle on a projective variety in characteristic zero is equivalent to the vanishing of an explicit finite list of cohomology groups. It follows that $q$-ampleness is a Zariski open condition, which is not clear from the definition.
Algebraic geometry
Several complex variables and analytic spaces
General
731
754
10.4171/JEMS/374
http://www.ems-ph.org/doi/10.4171/JEMS/374
Separable solutions of quasilinear Lane–Emden equations
Alessio
Porretta
Università di Roma, ROMA, ITALY
Laurent
Véron
Université François Rabelais, TOURS, FRANCE
Quasilinear elliptic equations, $p$-Laplacian, cones, Leray–Schauder degree
For $0
Partial differential equations
Operator theory
Global analysis, analysis on manifolds
General
755
774
10.4171/JEMS/375
http://www.ems-ph.org/doi/10.4171/JEMS/375
Invariants for the modular cyclic group of prime order via classical invariant theory
David
Wehlau
Royal Military College of Canada, KINGSTON, ONTARIO, CANADA
Modular invariant theory, cyclic group, classical invariant theory, Roberts' isomorphism
Let $\mathbb F$ be any field of characteristic $p$. It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_1,V_2,\dots,V_p$ of $C_p$ defined over $\mathbb F$. Thus if $V$ is any finite dimensional $C_p$-representation there are non-negative integers $0\leq n_1,n_2,\dots, n_k \leq p-1$ such that $V \cong \oplus_{i=1}^k V_{n_i+1}$. It is also well-known there is a unique (up to equivalence) $d+1$ dimensional irreducible complex representation of SL$_2(\mathbb C)$ given by its action on the space $R_d$ of $d$ forms. Here we prove a conjecture, made by R.~J.~Shank, which reduces the computation of the ring of $C_p$-invariants $\mathbb F[ \oplus_{i=1}^k V_{n_i+1}]^{C_p}$ to the computation of the classical ring of invariants (or covariants) $\mathbb C[R_1 \oplus (\oplus_{i=1}^k R_{n_i})]^{\mathrm {SL}_2(\mathbb C)}$. This shows that the problem of computing modular $C_p$ invariants is equivalent to the problem of computing classical SL$_2(\mathbb C)$ invariants. This allows us to compute for the first time the ring of invariants for many representations of $C_p$. In particular, we easily obtain from this generators for the rings of vector invariants $\mathbb F[m\,V_2]^{C_p}$, $\mathbb F[m\,V_3]^{C_p}$ and $\mathbb F[m\,V_4]^{C_p}$for all $m \in \mathbb N$. This is the first computation of the latter two families of rings of invariants.
Commutative rings and algebras
Group theory and generalizations
General
775
803
10.4171/JEMS/376
http://www.ems-ph.org/doi/10.4171/JEMS/376
A sharp Strichartz estimate for the wave equation with data in the energy space
Neal
Bez
University of Birmingham, EDGBASTON, UNITED KINGDOM
Keith
Rogers
CSIC-UAM-UC3M-UCM, MADRID, SPAIN
Strichartz estimates, wave equation, sharp constants
We prove a sharp bilinear estimate for the wave equation from which we obtain the sharp constant in the Strichartz estimate which controls the $L^{4}_{t,x}(\mathbb R^{5+1})$ norm of the solution in terms of the energy. We also characterise the maximisers.
Partial differential equations
General
805
823
10.4171/JEMS/377
http://www.ems-ph.org/doi/10.4171/JEMS/377
Local null controllability of a fluid-solid interaction problem in dimension 3
Muriel
Boulakia
Université Pierre et Marie Curie, PARIS, FRANCE
Sergio
Guerrero
Université Pierre et Marie Curie, PARIS, FRANCE
Fluid-solid interaction, controllability, Carleman estimate
We are interested by the three-dimensional coupling between an incompressible fluid and a rigid body. The fluid is modeled by the Navier-Stokes equations, while the solid satisfies the Newton's laws. In the main result of the paper we prove that, with the help of a distributed control, we can drive the fluid and structure velocities to zero and the solid to a reference position provided that the initial velocities are small enough and the initial position of the structure is close to the reference position. This is done without any condition on the geometry of the rigid body.
Partial differential equations
Mechanics of deformable solids
Systems theory; control
General
825
856
10.4171/JEMS/378
http://www.ems-ph.org/doi/10.4171/JEMS/378
A stability theorem for elliptic Harnack inequalities
Richard
Bass
University of Connecticut, STORRS, UNITED STATES
Harnack inequality, random walks on graphs, Poincaré inequality, cutoff inequality, metric measure space
We prove a stability theorem for the elliptic Harnack inequality: if two weighted graphs are equivalent, then the elliptic Harnack inequality holds for harmonic functions with respect to one of the graphs if and only if it holds for harmonic functions with respect to the other graph. As part of the proof, we give a characterization of the elliptic Harnack inequality.
Potential theory
Probability theory and stochastic processes
General
857
876
10.4171/JEMS/379
http://www.ems-ph.org/doi/10.4171/JEMS/379
Limiting Sobolev inequalities for vector fields and canceling linear differential operators
Jean
Van Schaftingen
Université Catholique de Louvain, LOUVAIN-LA-NEUVE, BELGIUM
Sobolev embedding, overdetermined elliptic operator, compatibility conditions, homogeneous differential operator, canceling operator, cocanceling operator, exterior derivative, symmetric derivative, homogeneous Triebel−Lizorkin space, homogeneous Besov space, Lorentz space, homogeneous fractional Sobolev−Slobodeckiĭ space, Korn−Sobolev inequality, Hodge inequality, Saint-Venant
The estimate \[ \|{D^{k-1}u}\|_{L^{n/(n-1)}} \le \|{A(D)u}\|_{L^1} \] is shown to hold if and only if \(A(D)\) is elliptic and canceling. Here \(A(D)\) is a homogeneous linear differential operator \(A(D)\) of order \(k\) on \(\mathbb R^n\) from a vector space \(V\) to a vector space \(E\). The operator \(A(D)\) is defined to be canceling if \[ \bigcap_{\xi \in \mathbb R^n \setminus \{0\}} A(\xi)[V]=\{0\}. \] This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous linear differential operator \(L(D)\) of order \(k\) on \(\mathbb R^n\) from a vector space \(E\) to a vector space \(F\) is introduced. It is proved that \(L(D)\) is cocanceling if and only if for every \(f \in L^1(\mathbb R^n; E)\) such that \(L(D)f=0\), one has \(f \in \dot{W}^{-1, n/(n-1)}(\mathbb R^n; E)\). The results extend to fractional and Lorentz spaces and can be strengthened using some tools of J. Bourgain and H. Brezis.
Functional analysis
Real functions
Fourier analysis
General
877
921
10.4171/JEMS/380
http://www.ems-ph.org/doi/10.4171/JEMS/380
Linearized plasticity is the evolutionary $\Gamma$-limit of finite plasticity
Alexander
Mielke
Angewandte Analysis und Stochastik, BERLIN, GERMANY
Ulisse
Stefanelli
Università di Pavia, PAVIA, ITALY
Finite-strain elastoplasticity, linearized elastoplasticity, gamma-convergence, rate-independent processes
We provide a rigorous justification of the classical linearization approach in plasticity. By taking the small-deformations limit, we prove via $\Gamma$-convergence for rate-independent processes that energetic solutions of the quasi-static finite-strain elastoplasticity system converge to the unique strong solution of linearized elastoplasticity.
Mechanics of deformable solids
Calculus of variations and optimal control; optimization
General
923
948
10.4171/JEMS/381
http://www.ems-ph.org/doi/10.4171/JEMS/381
The automorphism group of $\overline{M}_{0,n}$
Andrea
Bruno
Università degli studi Roma Tre, ROMA, ITALY
Massimiliano
Mella
Università di Ferrara, FERRARA, ITALY
Moduli space of curves, pointed rational curves, fibrations, automorphism
The paper studies fiber type morphisms between moduli spaces of pointed rational curves. Via Kapranov’s description we are able to prove that the only such morphisms are forgetful maps. This allows us to show that the automorphism group of $\overline{M}_{0,n}$ is the permutation group on n elements as soon as $n ≥ 5$.
Algebraic geometry
General
949
968
10.4171/JEMS/382
http://www.ems-ph.org/doi/10.4171/JEMS/382
Legendrian and transverse twist knots
John
Etnyre
Georgia Institute of Technology, ATLANTA, UNITED STATES
Lenhard
Ng
Duke University, DURHAM, UNITED STATES
Vera
Vértesi
Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES
Legendrian knot, transverse knot, twist knots
In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the $m(5_2)$ knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least $n$ different Legendrian representatives with maximal Thurston--Bennequin number of the twist knot $K_{-2n}$ with crossing number $2n+1$. In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that $K_{-2n}$ has exactly $\lceil\frac{n^2}2\rceil$ Legendrian representatives with maximal Thurston–Bennequin number, and $\lceil\frac{n}{2}\rceil$ transverse representatives with maximal self-linking number. Our techniques include convex surface theory, Legendrian ruling invariants, and Heegaard–Floer homology.
Manifolds and cell complexes
Differential geometry
General
969
995
10.4171/JEMS/383
http://www.ems-ph.org/doi/10.4171/JEMS/383
The boundary value problem for Dirac-harmonic maps
Qun
Chen
Wuhan University, WUHAN, HUBEI, CHINA
Jürgen
Jost
Mathematik in den Naturwissenschaften, LEIPZIG, GERMANY
Guofang
Wang
Universität Freiburg, FREIBURG, GERMANY
Miaomiao
Zhu
ETH Zürich, ZÜRICH, SWITZERLAND
Dirac-harmonic map, regularity, boundary value
Dirac-harmonic maps are a mathematical version (with commuting variables only) of the solutions of the field equations of the non-linear supersymmetric sigma model of quantum field theory. We explain this structure, including the appropriate boundary conditions, in a geometric framework. The main results of our paper are concerned with the analytic regularity theory of such Dirac-harmonic maps. We study Dirac-harmonic maps from a Riemannian surface to an arbitrary compact Riemannian manifold. We show that a weakly Dirac-harmonic map is smooth in the interior of the domain. We also prove regularity results for Dirac-harmonic maps at the boundary when they solve an appropriate boundary value problem which is the mathematical interpretation of the D-branes of superstring theory.
Global analysis, analysis on manifolds
Differential geometry
Partial differential equations
General
997
1031
10.4171/JEMS/384
http://www.ems-ph.org/doi/10.4171/JEMS/384
About the Calabi problem: a finite-dimensional approach
H.-D.
Cao
Lehigh University, BETHLEHEM, UNITED STATES
J.
Keller
Université de Provence, MARSEILLE CEDEX 13, FRANCE
Calabi problem, Balanced metrics, canonical flow, Kähler geometry, moment map, Bergman kernel, asymptotics, quantization
Let us consider a projective manifold $M^n$ and a smooth volume form $\Omega$ on $M$. We define the gradient flow associated to the problem of $\Omega$-balanced metrics in the quantum formalism, the $\Omega$-balancing flow. At the limit of the quantization, we prove that (see Theorem 1) the $\Omega$-balancing flow converges towards a natural flow in Kähler geometry, the $\Omega$-Kähler flow. We also prove the long time existence of the $\Omega$-Kähler flow and its convergence towards Yau's solution to the Calabi conjecture of prescribing the volume form in a given Kähler class (see Theorem 2). We derive some natural geometric consequences of our study.
Differential geometry
Several complex variables and analytic spaces
General
1033
1065
10.4171/JEMS/385
http://www.ems-ph.org/doi/10.4171/JEMS/385
Singular sets of holonomy maps for algebraic foliations
Gabriel
Calsamiglia
Universidade Federal Fluminense - UFF, NITERÓI - RJ, BRAZIL
Bertrand
Deroin
Université Paris-Sud, ORSAY CEDEX, FRANCE
Sidney
Frankel
Rensselaer Polytechnic Institute, TROY, UNITED STATES
Adolfo
Guillot
U.N.A.M., CUERNAVACA, MORELOS, MEXICO
Holomorphic foliation, holonomy map, analytic extension
In this article we investigate the natural domain of definition of a holonomy map associated to a singular holomorphic foliation of the complex projective plane. We prove that germs of holonomy between algebraic curves can have large sets of singularities for the analytic continuation. In the Riccati context we provide examples with natural boundary and maximal sets of singularities. In the generic case we provide examples having at least a Cantor set of singularities and even a nonempty open set of singularities. The examples provided are based on the presence of sufficiently rich contracting dynamics in the holonomy pseudogroup of the foliation. This gives answers to some questions and conjectures of Loray and Ilyashenko, which follow-up on an approach to the associated ODE’s developed notably by Painlevé.
Ordinary differential equations
Several complex variables and analytic spaces
General
1067
1099
10.4171/JEMS/386
http://www.ems-ph.org/doi/10.4171/JEMS/386
Best constants for the isoperimetric inequality in quantitative form
Marco
Cicalese
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Gian Paolo
Leonardi
Università di Modena e Reggio Emilia, MODENA, ITALY
Best constants, isoperimetric inequality, quasiminimizers of the perimeter
We prove some results in the context of isoperimetric inequalities with quantitative terms. In the $2$-dimensional case, our main contribution is a method for determining the optimal coefficients $c_{1},\dots,c_{m}$ in the inequality $\delta P(E) \geq \sum_{k=1}^{m}c_{k}\alpha (E)^{k} + o(\alpha (E)^{m})$, valid for each Borel set $E$ with positive and finite area, with $\delta P(E)$ and $\alpha (E)$ being, respectively, the \textit{isoperimetric deficit} and the \textit{Fraenkel asymmetry} of $E$. In $n$ dimensions, besides proving existence and regularity properties of minimizers for a wide class of \textit{quantitative isoperimetric quotients} including the lower semicontinuous extension of $\frac{\delta P(E)}{\alpha (E)^{2}}$, we describe the general technique upon which our $2$-dimensional result is based. This technique, called Iterative Selection Principle, extends the one introduced in [12].
Convex and discrete geometry
Measure and integration
Calculus of variations and optimal control; optimization
General
1101
1129
10.4171/JEMS/387
http://www.ems-ph.org/doi/10.4171/JEMS/387