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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=11&update_since=2024-03-29
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
11
2009
1
Non-generic blow-up solutions for the critical focusing NLS in 1-D
Joachim
Krieger
Bâtiment des Mathématiques, LAUSANNE, SWITZERLAND
Wilhelm
Schlag
University of Chicago, CHICAGO, UNITED STATES
Nonlinear Schrödinger equations, L2-critical NLS, pseudo-conformal blow-up
We consider the L2-critical focussing nonlinear Schrödinger equation in 1+1-D. We demonstrate the existence of a large set of initial data close to the ground state soliton resulting in the pseudo-conformal type blow up behavior. More specifically, we prove a version of a conjecture of Perelman, establishing the existence of a co-dimension one stable blow up manifold in the measurable category.
Partial differential equations
General
1
125
10.4171/JEMS/143
http://www.ems-ph.org/doi/10.4171/JEMS/143
On the Ladyzhenskaya�Smagorinsky turbulence model of the Navier�Stokes equations in smooth domains. The regularity problem
Hugo
Beirão da Veiga
Università di Pisa, PISA, ITALY
We establish regularity results up to the boundary for solutions to generalized Stokes and Navier-Stokes systems of equations in the stationary and in the evolutive cases. Generalized here means the presence of a shear dependent viscosity. We treat here the case $p\geq 2$. Actually, we are interested in proving regularity results in $L^q(\Omega)$ spaces for all the second order derivatives of the velocity and all the first order derivatives of the pressure. The main aim of the present paper is to extend our previous scheme, introduced in references \cite{bvlali} and \cite{bvcubo} for the flat-boundary case, to the case of curvilinear boundaries.
Partial differential equations
Fluid mechanics
General
127
167
10.4171/JEMS/144
http://www.ems-ph.org/doi/10.4171/JEMS/144
Single-point blow-up for a semilinear parabolic system
Philippe
Souplet
Université de Paris XIII/CNRS, VILLETANEUSE, FRANCE
Semilinear parabolic system, reaction-diffusion, power nonlinearities, blow-up set, single point blow-up
We consider positive solutions of a semilinear parabolic system coupled by power nonlinearities, in a ball or in the whole space. Relatively little is known on the blow-up set for parabolic systems and, up to now, no result was available for this basic system except for the very special case of equal powers. Here we prove single-point blow-up for a large class of radial decreasing solutions. This in particular solves a problem left open in a paper of A.~Friedman and Y.~Giga (1987). We also obtain lower pointwise estimates for the final blow-up profiles.
Partial differential equations
General
169
188
10.4171/JEMS/145
http://www.ems-ph.org/doi/10.4171/JEMS/145
On the topology of positively curved Bazaikin spaces
Luis
Florit
Estrada Dona Castorina 110, RIO DE JANEIRO RJ, BRAZIL
Wolfgang
Ziller
University of Pennsylvania, PHILADELPHIA, UNITED STATES
Positive curvature, Bazaikin spaces
In this note we study the topology of the positively curved Bazaikin spaces. We show that the only Bazaikin space that is homotopically equivalent to a homogeneous space is the Berger space. Moreover, we compute their Pontryagin classes and linking form to conclude that there is no pair of positively curved Bazaikin spaces which are homeomorphic, at least if the order of the sixth cohomology group with integer coefficients is less than 108.
Differential geometry
General
189
205
10.4171/JEMS/146
http://www.ems-ph.org/doi/10.4171/JEMS/146
Liouville theorems for self-similar solutions of heat flows
Jiayu
Li
Chinese Academy of Sciences, BEIJING, CHINA
Meng
Wang
Zhejiang University, HANGZHOU, CHINA
Harmonic sphere, self-similar solution, quasi-harmonic sphere, heat flow
Let $N$ be a compact Riemannian manifold. A quasi-harmonic sphere is a harmonic map from $({\bf R}^m, e^{-|x|^2/2(m-2)}ds_0^2)$ to $N$ ($m\geq 3$) with finite energy ([LnW]). Here $ds_0^2$ is the Euclidean metric in ${\bf R}^m$. It arises from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target $N$. We also derive gradient estimates and Liouville theorems for positive quasi-harmonic functions.
Partial differential equations
Global analysis, analysis on manifolds
General
207
221
10.4171/JEMS/147
http://www.ems-ph.org/doi/10.4171/JEMS/147
2
On the stabilization problem for nonholonomic distributions
Ludovic
Rifford
Université de Nice, NICE CEDEX 2, FRANCE
Emmanuel
Trélat
Université Pierre et Marie Curie (Paris 6), PARIS CEDEX 05, FRANCE
Nonholonomic distributions, stabilization, SRS feedback, minimizing singular path, Martinet case, nonsmooth analysis
Let $M$ be a smooth connected and complete manifold of dimension $n$, and $\Delta$ be a smooth nonholonomic distribution of rank $m\leq n$ on $M$. We prove that, if there exists a smooth Riemannian metric on $\Delta$ for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of $\Delta$ on $M$. Moreover, in dimension three, the assumption of the absence of singular minimizing horizontal paths can be dropped in the Martinet case. The proofs are based on the study, using specific results of nonsmooth analysis, of an optimal control problem of Bolza type, for which we prove that the corresponding value function is semiconcave and is a viscosity solution of a Hamilton-Jacobi equation, and establish fine properties of optimal trajectories.
Systems theory; control
Ordinary differential equations
General
223
255
10.4171/JEMS/148
http://www.ems-ph.org/doi/10.4171/JEMS/148
Convergence of singular integrals with general measures
Pertti
Mattila
University of Helsinki, HELSINKI, FINLAND
Joan
Verdera
Universitat Autónoma de Barcelona, BARCELONA, SPAIN
Singular integrals, principal values, martingales
We show that $L^2$-bounded singular integrals in metric spaces with respect to general measures and kernels converge weakly. This implies a kind of average convergence almost everywhere. For measures with zero density we prove the almost everywhere existence of principal values.
Fourier analysis
General
257
271
10.4171/JEMS/149
http://www.ems-ph.org/doi/10.4171/JEMS/149
Three-space problems for the approximation property
A.
Szankowski
The Hebrew University of Jerusalem, JERUSALEM, ISRAEL
Quotients of Banach spaces, approximation property
It is shown that there is a subspace $Z_q$ of $\ell_q$ for $1
Functional analysis
General
273
282
10.4171/JEMS/150
http://www.ems-ph.org/doi/10.4171/JEMS/150
Strong spectral gaps for compact quotients of products of PSL(2,ℝ)
Dubi
Kelmer
University of Chicago, CHICAGO, UNITED STATES
Peter
Sarnak
Princeton University, PRINCETON, UNITED STATES
Spectral gap, Selberg trace formula, product of hyperbolic planes
The existence of a strong spectral gap for quotients $\Gamma\bs G$ of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the Ramanujan-Selberg Conjectures. If $G$ has no compact factors then for general lattices a spectral gap can still be established, however, there is no uniformity and no effective bounds are known. This note is concerned with the spectral gap for an irreducible co-compact lattice $\Gamma$ in $G=\PSL(2,\bbR)^d$ for $d\geq 2$ which is the simplest and most basic case where the congruence subgroup property is not known. The method used here gives effective bounds for the spectral gap in this setting.
Topological groups, Lie groups
General
283
313
10.4171/JEMS/151
http://www.ems-ph.org/doi/10.4171/JEMS/151
Polarizations of Prym varieties for Weyl groups via abelianization
Christian
Pauly
Université de Montpellier II, MONTPELLIER CEDEX 5, FRANCE
Christoph
Scheven
Friedrich-Alexander-Universität Erlangen, ERLANGEN, GERMANY
Prym variety, principal G-bundle, abelianization, moduli stack
Let $\pi: Z \ra X$ be a Galois covering of smooth projective curves with Galois group the Weyl group of a simple and simply connected Lie group $G$. For any dominant weight $\lambda$ consider the curve $Y = Z/\Stab(\lambda)$. The Kanev correspondence defines an abelian subvariety $P_\lambda$ of the Jacobian of $Y$. We compute the type of the polarization of the restriction of the canonical principal polarization of $\Jac(Y)$ to $P_\lambda$ in some cases. In particular, in the case of the group $E_8$ we obtain families of Prym-Tyurin varieties. The main idea is the use of an abelianization map of the Donagi-Prym variety to the moduli stack of principal $G$-bundles on the curve $X$.
Algebraic geometry
General
315
349
10.4171/JEMS/152
http://www.ems-ph.org/doi/10.4171/JEMS/152
Convergence of a two-grid algorithm for the control of the wave equation
Liviu
Ignat
Universidad Autónoma de Madrid, MADRID, SPAIN
Enrique
Zuazua
Universidad Autónoma de Madrid, MADRID, SPAIN
Waves, finite difference approximation, propagation, observation, control, two-grid
We analyze the problem of boundary observability of the finite-difference space semi-discretizations of the 2-d wave equation in the square. We prove the uniform (with respect to the mesh-size) boundary observability for the solutions obtained by the two-grid preconditioner introduced by Glowinski \cite{0763.76042}. Our method uses previously known uniform observability inequalities for low frequency solutions and a dyadic spectral time decomposition. As a consequence we prove the convergence of the two-grid algorithm for computing the boundary controls for the wave equation. The method can be applied in any space dimension, for more general domains and other discretization schemes.
Partial differential equations
Numerical analysis
Systems theory; control
General
351
391
10.4171/JEMS/153
http://www.ems-ph.org/doi/10.4171/JEMS/153
Cycles on algebraic models of smooth manifolds
Wojciech
Kucharz
Jagiellonian University, KRAKOW, POLAND
Real algebraic sets, algebraic cohomology classes, algebraic models
Every compact smooth manifold $M$ is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of $M$. We study modulo $2$ homology classes represented by algebraic subsets of $X$, as $X$ runs through the class of all algebraic models of $M$. Our main result concerns the case where $M$ is a spin manifold.
Global analysis, analysis on manifolds
Algebraic geometry
General
393
405
10.4171/JEMS/154
http://www.ems-ph.org/doi/10.4171/JEMS/154
Cambrian fans
Nathan
Reading
University of Michigan, ANN ARBOR, UNITED STATES
David
Speyer
Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES
For a finite Coxeter group~$W$ and a Coxeter element~$c$ of $W,$ the $c$-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of~$W\!$. Its maximal cones are naturally indexed by the $c$-sortable elements of~$W\!$. The main result of this paper is that the known bijection $\cl_c$ between $c$-sortable elements and $c$-clusters induces a combinatorial isomorphism of fans. In particular, the $c$-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for~$W\!$. The rays of the $c$-Cambrian fan are generated by certain vectors in the $W$-orbit of the fundamental weights, while the rays of the $c$-cluster fan are generated by certain roots. For particular (``bipartite'') choices of~$c$, we show that the $c$-Cambrian fan is linearly isomorphic to the $c$-cluster fan. We characterize, in terms of the combinatorics of clusters, the partial order induced, via the map $\cl_c$, on $c$-clusters by the $c$-Cambrian lattice. We give a simple bijection from $c$-clusters to $c$-noncrossing partitions that respects the refined (Narayana) enumeration. We relate the Cambrian fan to well known objects in the theory of cluster algebras, providing a geometric context for $\mathbf{g}$-vectors and quasi-Cartan companions.
Group theory and generalizations
Associative rings and algebras
General
407
447
10.4171/JEMS/155
http://www.ems-ph.org/doi/10.4171/JEMS/155
3
Face enumeration—from spheres to manifolds
Ed
Swartz
Cornell University, ITHACA, UNITED STATES
We prove a number of new restrictions on the enumerative properties of homology manifolds and semi-Eulerian complexes and posets. These include a determination of the affine span of the fine h-vector of balanced semi-Eulerian complexes and the toric h-vector of semi-Eulerian posets. The lower bounds on simplicial homology manifolds, when combined with higher dimensional analogues of Walkup’s 3-dimensional constructions [47], allow us to give a complete characterization of the f-vectors of arbitrary simplicial triangulations of S1 × S3 , ℂP2, K3 surfaces, and (S2 × S2) # (S2 × S2). We also establish a principle which leads to a conjecture for homology manifolds which is almost logically equivalent to the g-conjecture for homology spheres. Lastly, we show that with sufficiently many vertices, every triangulable homology manifold without boundary of dimension three or greater can be triangulated in a 2-neighborly fashion.
Manifolds and cell complexes
Combinatorics
General
449
485
10.4171/JEMS/156
http://www.ems-ph.org/doi/10.4171/JEMS/156
Toric structures on near-symplectic 4-manifolds
David
Gay
University of Cape Town, RONDEBOSCH, SOUTH AFRICA
Margaret
Symington
Mercer University, MACON, UNITED STATES
Symplectic, near-symplectic, toric, torus action, four-manifold, Hamiltonian, Lagrangian fibration
A near-symplectic structure on a 4-manifold is a closed 2-form that is symplectic away from the 1-dimensional submanifold along which it vanishes and that satisfies a certain transversality condition along this vanishing locus. We investigate near-symplectic 4-manifolds equipped with singular Lagrangian torus fibrations which are locally induced by effective Hamiltonian torus actions. We show how such a structure is completely characterized by a singular integral affine structure on the base of the fibration whenever the vanishing locus is nonempty. The base equipped with this geometric structure generalizes the moment map image of a toric 4-manifold in the spirit of earlier work by the second author on almost toric symplectic 4-manifolds. We use the geometric structure on the base to investigate the problem of making given smooth torus actions on 4-manifolds symplectic or Hamiltonian with respect to near-symplectic structures and to give interesting constructions of structures which are locally given by torus actions but have nontrivial global monodromy.
Manifolds and cell complexes
General
487
520
10.4171/JEMS/157
http://www.ems-ph.org/doi/10.4171/JEMS/157
Which 3-manifold groups are Kähler groups?
Alexandru
Dimca
Université de Nice Sophia Antipolis, NICE, FRANCE
Alexander
Suciu
Northeastern University, BOSTON, UNITED STATES
Kähler manifold, 3-manifold, fundamental group, cohomology ring, resonance variety, isotropic subspace
The question in the title, first raised by Goldman and Donaldson, was partially answered by Reznikov. We give a complete answer, as follows: if G can be realized as both the fundamental group of a closed 3-manifold and of a compact Kähler manifold, then G must be finite—and thus belongs to the well-known list of finite subgroups of O(4), acting freely on S3.
Group theory and generalizations
Several complex variables and analytic spaces
General
521
528
10.4171/JEMS/158
http://www.ems-ph.org/doi/10.4171/JEMS/158
Existence of rational points on smooth projective varieties
Bjorn
Poonen
Building 2, Room 244, CAMBRIDGE, UNITED STATES
Brauer–Manin obstruction, Hasse principle, Châtelet surface, conic bundle, rational points
Fix a number field k. We prove that if there is an algorithm for deciding whether a smooth projective geometrically integral k-variety has a k-point, then there is an algorithm for deciding whether an arbitrary k-variety has a k-point and also an algorithm for computing X(k) for any k-variety X for which X(k) is finite. The proof involves the construction of a one-parameter algebraic family of Châtelet surfaces such that exactly one of the surfaces fails to have a k-point.
Algebraic geometry
Number theory
General
529
543
10.4171/JEMS/159
http://www.ems-ph.org/doi/10.4171/JEMS/159
Positive solutions for nonlinear Schrödinger equations with deepening potential well
Zhengping
Wang
Chinese Academy of Sciences, WUHAN, HUBEI, CHINA
Huan-Song
Zhou
Chinese Academy of Sciences, WUHAN, HUBEI, CHINA
Nonlinear Schrödinger equation, mountain pass theorem, potential well, asymptotically linear
Consider the following nonlinear Schrödinger equation: (*) -Δu + (1 + λg(x))u = f(u) and u> 0 in ℝN, u ∈ H1.(ℝN), N ≥ 3, where λ ≥ 0 is a parameter, g ∈ L∞(ℝN) vanishes on a bounded domain in ℝN, and the function f is such that lim(s→0) f(s)/s = 0 and 1 ≤ α + 1 = lim(s→∞) f(s)/s < ∞. We are interested in whether problem (*) has a solution for any given α, λ ≥ 0. It is shown in [14] and [31] that problem (*) has solutions for some α and λ. In this paper, we establish the existence of solution of (*) for all α and λ by using a variant of the Mountain Pass Theorem. Based on these results, we give a diagram in the (λ,α)-plane showing how the solvability of problem (*) depends on the parameters α and λ.
Partial differential equations
General
545
573
10.4171/JEMS/160
http://www.ems-ph.org/doi/10.4171/JEMS/160
Stability of closed characteristics on compact convex hypersurfaces in ℝ6
Wei
Wang
Peking University, BEIJING, CHINA
Compact convex hypersurfaces, closed characteristics, Hamiltonian systems, Morse theory, mean index identity, stability
Let Σ ⊂ ℝ6 be a compact convex hypersurface. We prove that if Σ carries only finitely many geometrically distinct closed characteristics, then at least two of them must have irrational mean indices. Moreover, if Σ carries exactly three geometrically distinct closed characteristics, then at least two of them must be elliptic.
Global analysis, analysis on manifolds
Dynamical systems and ergodic theory
General
575
596
10.4171/JEMS/161
http://www.ems-ph.org/doi/10.4171/JEMS/161
Relative integral functors for singular fibrations and singular partners
Daniel
Hernández Ruipérez
Universidad de Salamanca, SALAMANCA, SPAIN
Ana Cristina
López Martín
Universidad de Salamanca, SALAMANCA, SPAIN
Fernando
Sancho de Salas
Universidad de Salamanca, SALAMANCA, SPAIN
Geometric integral functors, Fourier–Mukai, Cohen–Macaulay, fully faithful, elliptic fibration, equivalence of categories
We study relative integral functors for singular schemes and characterise those which preserve boundedness and those which have integral right adjoints. We prove that a relative integral functor is an equivalence if and only if its restriction to every fibre is an equivalence. This allows us to construct a non-trivial auto-equivalence of the derived category of an arbitrary genus one fibration with no conditions on either the base or the total space and getting rid of the usual assumption of irreducibility of the fibres. We also extend to Cohen–Macaulay schemes the criterion of Bondal and Orlov for an integral functor to be fully faithful in characteristic zero and give a different criterion which is valid in arbitrary characteristic. Finally, we prove that for projective schemes both the Cohen–Macaulay and the Gorenstein conditions are invariant under Fourier–Mukai functors.
Category theory; homological algebra
Commutative rings and algebras
Algebraic geometry
General
597
625
10.4171/JEMS/162
http://www.ems-ph.org/doi/10.4171/JEMS/162
On Kahane's ultraflat polynomials
Enrico
Bombieri
Institute for Advanced Study, PRINCETON, UNITED STATES
Jean
Bourgain
Institute for Advanced Study, PRINCETON, UNITED STATES
Trigonometric polynomials, probabilistic methods, exponential sums
This paper is devoted to the construction of polynomials of almost constant modulus on the unit circle, with coefficients of constant absolute value. In particular, one obtains a much improved estimate for the error term. A major part of this paper deals also with the long-standing problem of the effective construction of ultraflat polynomials.
Field theory and polynomials
Number theory
General
627
703
10.4171/JEMS/163
http://www.ems-ph.org/doi/10.4171/JEMS/163
4
Critical points via Γ-convergence: general theory and applications
Robert
Jerrard
University of Toronto, TORONTO, ONTARIO, CANADA
Peter
Sternberg
Indiana University, BLOOMINGTON, UNITED STATES
Gamma-convergence, critical points, Allen–Cahn, Ginzburg–Landau
It is well-known that Γ-convergence of functionals provides a tool for studying global and local minimizers. Here we present a general result establishing the existence of critical points of a Γ-converging sequence of functionals provided the associated Γ-limit possesses a nondegenerate critical point, subject to certain mild additional hypotheses. We then go on to prove a theorem that describes suitable nondegenerate critical points for functionals, involving the arclength of a limiting singular set, that arise as Γ-limits in a number of problems. Finally, we apply the general theory to prove some new results, and give new proofs of some known results, establishing the existence of critical points of the 2d Modica–Mortola (Allen–Cahn) energy and 3d Ginzburg–Landau energy with and without magnetic field, and various generalizations, all in a unified framework.
Calculus of variations and optimal control; optimization
General
705
753
10.4171/JEMS/164
http://www.ems-ph.org/doi/10.4171/JEMS/164
Limits of Calabi–Yau metrics when the Kähler class degenerates
Valentino
Tosatti
Harvard University, CAMBRIDGE, UNITED STATES
Calabi–Yau manifolds, Ricci-flat metrics, degenerate complex Monge–Ampère equations
We study the behavior of families of Ricci-flat Kähler metrics on a projective Calabi– Yau manifold when the Kähler classes degenerate to the boundary of the ample cone. We prove that if the limit class is big and nef the Ricci-flat metrics converge smoothly on compact sets outside a subvariety to a limit incomplete Ricci-flat metric. The limit can also be understood from algebraic geometry.
Several complex variables and analytic spaces
Algebraic geometry
General
755
776
10.4171/JEMS/165
http://www.ems-ph.org/doi/10.4171/JEMS/165
Hypoellipticity, fundamental solution and Liouville type theorem for matrix-valued differential operators in Carnot groups
Annalisa
Baldi
Università di Bologna, BOLOGNA, ITALY
Bruno
Franchi
Università di Bologna, BOLOGNA, ITALY
Maria Carla
Tesi
Università di Bologna, BOLOGNA, ITALY
Hypoelliptic operators, maximal hypoelliptic operators, subelliptic operators, Carnot groups, fundamental solution
Let ℒ be a non-negative self-adjoint N × N matrix-valued operator of order a ≤ Q on a Carnot group G. Here Q is the homogeneous dimension of G. The aim of this paper is to investigate the relationship between hypoellipticity and maximal hypoellipticity (i.e. sharp L2 estimates in appropriate Sobolev spaces), Lp-maximal hypoellipticity (i.e. sharp Lp estimates in appropriate Sobolev spaces for 1 < p < ∞), and what we call maximal subellipticity of ℒ (which is basically a sharp higher order energy estimate).
Partial differential equations
Commutative rings and algebras
Algebraic geometry
General
777
798
10.4171/JEMS/166
http://www.ems-ph.org/doi/10.4171/JEMS/166
K3 surfaces with a symplectic automorphism of order 11
Igor
Dolgachev
University of Michigan, ANN ARBOR, UNITED STATES
JongHae
Keum
Korea Institute for Advanced Study (KIAS), SEOUL, SOUTH KOREA
K3 surfaces, positive characteristic, automorphism groups, wild action, Mathieu groups
We classify possible finite groups of symplectic automorphisms of K3 surfaces of order divisible by 11. The characteristic of the ground field must be equal to 11. The complete list of such groups consists of five groups: the cyclic group C11 of order 11, C11 ⋊ C5 , PSL2(F11) and the Mathieu groups M11, M22. We also show that a surface X admitting an automorphism g of order 11 admits a g-invariant elliptic fibration with the Jacobian fibration isomorphic to one of explicitly given elliptic K3 surfaces.
Algebraic geometry
General
799
818
10.4171/JEMS/167
http://www.ems-ph.org/doi/10.4171/JEMS/167
Saddle-shaped solutions of bistable diffusion equations in all of ℝ2m
Xavier
Cabré
Universitat Politècnica de Catalunya, BARCELONA, SPAIN
Joana
Terra
Departament de Matematica Aplicada I, BARCELONA, SPAIN
Allen–Cahn equation, saddle-shaped solutions, Simons cone, instability, Morse index, conjecture of De Giorgi on 1D symmetry
We study the existence and instability properties of saddle-shaped solutions of the semilinear elliptic equation −∆ u = f(u) in the whole ℝ2m, where f is of bistable type. It is known that in dimension 2m = 2 there exists a saddle-shaped solution. This is a solution which changes sign in ℝ2 and vanishes only on {|x1 | = |x2 |}. It is also known that this solution is unstable. In this article we prove the existence of saddle-shaped solutions in every even dimension, as well as their instability in the case of dimension 2m = 4. More precisely, our main result establishes that if 2m = 4, every solution vanishing on the Simons cone {(x1, x2) ∈ ℝ2 × ℝ2 : |x1| = |x2|} is unstable outside every compact set and, as a consequence, has infinite Morse index. These results are relevant in connection with a conjecture of De Giorgi extensively studied in recent years and for which the existence of a counter-example in high dimensions is still an open problem.
Partial differential equations
Differential geometry
General
819
943
10.4171/JEMS/168
http://www.ems-ph.org/doi/10.4171/JEMS/168
Skeletons, bodies and generalized E(R)-algebras
Rüdiger
Göbel
Universität Duisburg-Essen, ESSEN, GERMANY
Daniel
Herden
Universität Duisburg-Essen, ESSEN, GERMANY
Saharon
Shelah
The Hebrew University of Jerusalem, JERUSALEM, ISRAEL
Endomorphism rings, indecomposable modules, E-rings
In this paper we want to solve a fifty year old problem on R-algebras over cotorsionfree commutative rings R with 1. For simplicity (but only for the abstract) we will assume that R is any countable principal ideal domain, but not a field. For example R can be the ring ℤ or the polynomial ring ℚ[x]. An R-algebra A is called a generalized E(R)-algebra if its algebra EndR A of R-module endomorphisms of the underlying R-module RA is isomorphic to A (as an R-algebra). Properties, including the existence of such algebras are derived in various papers ([5, 6, 9, 10, 20, 22, 24, 25]). The study was stimulated by Fuchs [13], and specially by Schultz [26]. But due to [26] the investigation concentrated on ordinary commutative E(R)-algebras. A substantial part of problem 45 (p. 232) in the monograph [13] (repeated in later publications, e.g. [27]), which will be answered positively for all rings R above in this paper, remained open: Can we find non-commutative generalized E(R)-algebras? In Theorem 1.5 we will show that for all countable, principal ideal domains R which are not fields and for any infinite cardinal κ there is a non-commutative R-algebra A of cardinality |A| = κℵ0 with EndR A ≅ A, so A is a non-commutative generalized E(R)-algebra, and—not too surprisingly—there is a proper class of examples. The new strategy should be interesting and useful for other problems as well: We will first translate the heart of the algebraic question on the existence of certain monoids via model theory into geometric structures leading to a special class of (decorated) trees and solve this problem introducing products of trees etc. This can be compared with the well-known, but different process of translating group problems to small cancelations in groups via the van Kampen lemma. By small cancelation of trees we are able to find a suitable monoid and thus a non-commutative algebra A with an important non-canonical embedding A → EndR A, our ∗-scalar multiplication. In a second part of this paper we must enlarge A to get rid of all undesired endomorphisms. This can be done more easily (thus first) with the help of additional set theory (Jensen’s diamond predictions), which will support the reader to understand more quickly the last steps of the proof. In a final chapter we will also give an argument (for removing unwanted endomorphisms) which is based on ordinary set theory of ZFC only (using our favored Black Box predictions; see [20]). Thus we get the result as stated above. Furthermore, this last chapter includes a construction for rigid systems of non-commutative generalized E(R)-algebras.
Group theory and generalizations
Associative rings and algebras
General
845
901
10.4171/JEMS/169
http://www.ems-ph.org/doi/10.4171/JEMS/169
Hypersurfaces in ℍn+1 and conformally invariant equations: the generalized Christoffel and Nirenberg problems
José
Espinar
Universidad de Granada, GRANADA, SPAIN
José
Gálvez
Universidad de Granada, GRANADA, SPAIN
Gabriel
Soler López
Universidad Politécnica de Cartagena, CARTAGENA, SPAIN
Christoffel problem, Nirenberg problem, Kazdan-Warner conditions, Schouten tensor, hyperbolic Gauss map, Weingarten hypersurfaces
Our first objective in this paper is to give a natural formulation of the Christoffel problem for hypersurfaces in ℍn+1, by means of the hyperbolic Gauss map and the notion of hyperbolic curvature radii for hypersurfaces. Our second objective is to provide an explicit equivalence of this Christoffel problem with the famous problem of prescribing scalar curvature on Sn for conformal metrics, posed by Nirenberg and Kazdan–Warner. This construction lets us translate into the hyperbolic setting the known results for the scalar curvature problem, and also provides a hypersurface theory interpretation of such an intrinsic problem from conformal geometry. Our third objective is to place the above result in a more general framework. Specifically, we will show how the problem of prescribing the hyperbolic Gauss map and a given function of the hyperbolic curvature radii in ℍn+1 is strongly related to some important problems on conformally invariant PDEs in terms of the Schouten tensor. This provides a bridge between the theory of conformal metrics on Sn and the theory of hypersurfaces with prescribed hyperbolic Gauss map in ℍn+1. The fourth objective is to use the above correspondence to prove that for a wide family of Weingarten functionals W(κ1, . . . , κn), the only compact immersed hypersurfaces in ℍn+1 on which W is constant are round spheres.
Differential geometry
General
903
939
10.4171/JEMS/170
http://www.ems-ph.org/doi/10.4171/JEMS/170
5
Holomorphic functions and subelliptic heat kernels over Lie groups
Bruce
Driver
University of California, San Diego, LA JOLLA, UNITED STATES
Leonard
Gross
Cornell University, ITHACA, UNITED STATES
Laurent
Saloff-Coste
Cornell University, ITHACA, UNITED STATES
Subelliptic, heat kernel, complex groups, universal enveloping algebra, Taylor map Schrödinger equations, L2-critical NLS, pseudo-conformal blow-up
A Hermitian form q on the dual space, g∗, of the Lie algebra, g, of a Lie group, G, determines a sub-Laplacian, ∆, on G. It will be shown that Hörmander’s condition for hypoellipticity of the sub-Laplacian holds if and only if the associated Hermitian form, induced by q on the dual of the universal enveloping algebra, U', is non-degenerate. The subelliptic heat semigroup, et∆/4, is given by convolution by a C∞ probability density ρt. When G is complex and u : G → C is a holomorphic function, the collection of derivatives of u at the identity in G gives rise to an element, û(e) ∈ U'. We will show that if G is complex, connected, and simply connected then the “Taylor” map, u ↦ û(e), defines a unitary map from the space of holomorphic functions in L2(G, ρt) onto a natural Hilbert space lying in U'.
Several complex variables and analytic spaces
General
941
978
10.4171/JEMS/171
http://www.ems-ph.org/doi/10.4171/JEMS/171
Construction of Kähler surfaces with constant scalar curvature
Yann
Rollin
Imperial College London, LONDON, UNITED KINGDOM
Michael
Singer
University of Edinburgh, EDINBURGH, UNITED KINGDOM
Kähler surfaces, constant scalar curvature, Stability, Gluing
We present new constructions of Kähler metrics with constant scalar curvature on complex surfaces, in particular on certain del Pezzo surfaces. Some higher-dimensional examples are provided as well.
Differential geometry
General
979
997
10.4171/JEMS/172
http://www.ems-ph.org/doi/10.4171/JEMS/172
Singular Dynamics for Semiconcave Functions
Piermarco
Cannarsa
Università di Roma, ROMA, ITALY
Yifeng
Yu
University of Texas at Austin, AUSTIN, UNITED STATES
Semiconcave functions, singularities, Hamilton-Jacobi equations, Monge-Ampère equations, weak KAM theory
Semiconcave functions are a well-known class of nonsmooth functions that possess deep connections with optimization theory and nonlinear pde’s. Their singular sets exhibit interesting structures that we investigate in this paper. First, by an energy method, we analyze the curves along which the singularities of semiconcave solutions to Hamilton–Jacobi equations propagate—the socalled generalized characteristics. This part of the paper improves the main result in [P. Albano, P. Cannarsa, Propagation of singularities for solutions of nonlinear first order partial differential equations, Arch. Ration. Mech. Anal. 162 (2002), 1–23] and simplifies the construction therein. As applications, we recover some known results for gradient flows and conservation laws. Then we derive a simple dynamics for the propagation of singularities of general semiconcave functions. This part of the work is also used to study the singularities of generalized solutions to Monge–Ampère equations. We conclude with a global propagation result for the singularities of solutions e in weak KAM theory.
Real functions
Partial differential equations
Calculus of variations and optimal control; optimization
General
999
1024
10.4171/JEMS/173
http://www.ems-ph.org/doi/10.4171/JEMS/173
Resonant normal form for even periodic FPU chains
Andreas
Henrici
University of Zürich, ZÜRICH, SWITZERLAND
Thomas
Kappeler
University of Zürich, ZÜRICH, SWITZERLAND
We investigate periodic FPU chains with an even number of particles. We show that near the equilibrium point, any such chain admits a resonant Birkhoff normal form of order four which is completely integrable—an important fact which helps explain the numerical experiments of Fermi, Pasta, and Ulam. We analyze the moment map of the integrable approximation of an even FPU chain. Unlike the case of odd FPU chains these integrable systems (generically) exhibit hyperbolic dynamics. As an application we prove that any FPU chain with Dirichlet boundary conditions admits a Birkhoff normal form up to order four and show that a KAM theorem applies.
Partial differential equations
General
1025
1056
10.4171/JEMS/174
http://www.ems-ph.org/doi/10.4171/JEMS/174
Expansion and random walks in SLd(ℤ/pnℤ): II
Jean
Bourgain
Institute for Advanced Study, PRINCETON, UNITED STATES
Alex
Gamburd
University of California at Santa Cruz, SANTA CRUZ, UNITED STATES
We prove that Cayley graphs of SLd(ℤ/pnℤ) are expanders with respect to the projection of any fixed elements in SL2(ℤ) generating a Zariski-dense subgroup.
Group theory and generalizations
Combinatorics
Computer science
General
1057
1103
10.4171/JEMS/175
http://www.ems-ph.org/doi/10.4171/JEMS/175
The sharp Sobolev inequality in quantitative form
Andrea
Cianchi
Universita di Firenze, FIRENZE, ITALY
Nicola
Fusco
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Francesco
Maggi
The University of Texas at Austin, AUSTIN, UNITED STATES
Aldo
Pratelli
Universität Erlangen-Nürnberg, ERLANGEN, GERMANY
A quantitative version of the sharp Sobolev inequality in W1,p (ℝn), 1 < p < n, is established with a remainder term involving the distance from the family of extremals.
Partial differential equations
General
1105
1139
10.4171/JEMS/176
http://www.ems-ph.org/doi/10.4171/JEMS/176
6
Local rigidity in quaternionic hyperbolic space
Inkang
Kim
KIAS, SEOUL, SOUTH KOREA
Pierre
Pansu
Université Paris-Sud 11, ORSAY CEDEX, FRANCE
Quaternionic hyperbolic space, rank one symmetric space, quasifuchsian representation, bending, rigidity, group cohomology
We study deformations of quaternionic hyperbolic lattices in larger quaternionic hyperbolic spaces and prove local rigidity results. On the other hand, surface groups are shown to be more flexible in quaternionic hyperbolic plane than in complex hyperbolic plane.
Geometry
Manifolds and cell complexes
General
1141
1164
10.4171/JEMS/177
http://www.ems-ph.org/doi/10.4171/JEMS/177
On the structure of Hardy–Sobolev–Maz'ya inequalities
Stathis
Filippas
University of Crete, HERAKLION-CRETE, GREECE
Achilles
Tertikas
University of Crete, HERAKLION-CRETE, GREECE
Jesper
Tidblom
, WIEN, AUSTRIA
We establish new improvements of the optimal Hardy inequality in the half-space. We first add all possible linear combinations of Hardy type terms, thus revealing the structure of this type of inequalities and obtaining best constants. We then add the critical Sobolev term and obtain necessary and sufficient conditions for the validity of Hardy–Sobolev–Maz’ya type inequalities.
Partial differential equations
General
1165
1185
10.4171/JEMS/178
http://www.ems-ph.org/doi/10.4171/JEMS/178
Confirmation of Matheron's conjecture on the covariogram of a planar convex body
Gennadiy
Averkov
Otto-von-Guericke-Universität, MAGDEBURG, GERMANY
Gabriele
Bianchi
Università degli Studi di Firenze, FIRENZE, ITALY
Autocorrelation, covariogram, cut-and-project scheme, geometric tomography, image analysis, phase retrieval, quasicrystal, set covariance
The covariogram gK of a convex body K in Ed is the function which associates to each x ∈ Ed the volume of the intersection of K with K + x. In 1986 G. Matheron conjectured that for d = 2 the covariogram gK determines K within the class of all planar convex bodies, up to translations and reflections in a point. This problem is equivalent to some problems in stochastic geometry and probability as well as to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its X-ray diffraction image. In this paper we confirm Matheron’s conjecture completely. This problem is equivalent to some problems in stochastic geometry and probability as well as to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its X-ray diffraction image. In this paper we confirm Matheron's conjecture completely.
Probability theory and stochastic processes
Fourier analysis
Convex and discrete geometry
General
1187
1202
10.4171/JEMS/179
http://www.ems-ph.org/doi/10.4171/JEMS/179
The cubic nonlinear Schrödinger equation in two dimensions with radial data
Rowan
Killip
University of California Los Angeles, LOS ANGELES, UNITED STATES
Terence
Tao
University of California Los Angeles, LOS ANGELES, UNITED STATES
Monica
Visan
UCLA, LOS ANGELES, UNITED STATES
Nonlinear Schrödinger equation, scattering, mass-critical, virial identity
We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrödinger equation iut + u = ±|u|2 u for large spherically symmetric Lx2(ℝ2) initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state. As a consequence, we deduce that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time. We also establish some partial results towards the analogous claims in other dimensions and without the assumption of spherical symmetry.
Partial differential equations
General
1203
1258
10.4171/JEMS/180
http://www.ems-ph.org/doi/10.4171/JEMS/180
Metrical theory for α-Rosen fractions
Karma
Dajani
Universiteit Utrecht, UTRECHT, NETHERLANDS
Cor
Kraaikamp
Delft University of Technology, CD DELFT, NETHERLANDS
Wolfgang
Steiner
Université Paris 7, Denis Diderot, PARIS CEDEX 05, FRANCE
Rosen fractions, natural extension, Diophantine approximation
The Rosen fractions form an infinite family which generalizes the nearest-integer continued fractions. In this paper we introduce a new class of continued fractions related to the Rosen fractions, the α-Rosen fractions. The metrical properties of these α-Rosen fractions are studied. We find planar natural extensions for the associated interval maps, and show that their domains of definition are closely related to the domains of the ‘classical’ Rosen fractions. This unifies and generalizes results of diophantine approximation from the literature.
Number theory
General
1259
1283
10.4171/JEMS/181
http://www.ems-ph.org/doi/10.4171/JEMS/181
Separable p-harmonic functions in a cone and related quasilinear equations on manifolds
Alessio
Porretta
Università di Roma, ROMA, ITALY
Laurent
Véron
Université François Rabelais, TOURS, FRANCE
p-harmonic functions, conical singularities, Ricci curvature, ergodic constant
In considering a class of quasilinear elliptic equations on a Riemannian manifold with nonnegative Ricci curvature, we give a new proof of Tolksdorf's result on the construction of separable p-harmonic functions in a cone.
Partial differential equations
Global analysis, analysis on manifolds
General
1285
1305
10.4171/JEMS/182
http://www.ems-ph.org/doi/10.4171/JEMS/182
Heegard Floer invariants of Legendrian knots in contact three-manifolds
Paolo
Lisca
Università di Pisa, PISA, ITALY
Peter
Ozsváth
Princeton University, PRINCETON, UNITED STATES
András
Stipsicz
Hungarian Academy of Sciences, BUDAPEST, HUNGARY
Zoltán
Szabó
Princeton University, PRINCETON, UNITED STATES
Legendrian knots, transverse knots, Heegaard Floer homology
We define invariants of null-homologous Legendrian and transverse knots in contact 3manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non-loose knots in overtwisted 3-spheres. Moreover, we apply the invariants to find transversely non-simple knot types in many overtwisted contact 3-manifolds.
Manifolds and cell complexes
General
1307
1363
10.4171/JEMS/183
http://www.ems-ph.org/doi/10.4171/JEMS/183
Pólya's conjecture in the presence of a constant magnetic field
Rupert
Frank
Caltech, PASADENA, UNITED STATES
Michael
Loss
Georgia Institute of Technology, ATLANTA, UNITED STATES
Timo
Weidl
Universität Stuttgart, STUTTGART, GERMANY
Eigenvalue bounds, semi-classical estimates, Pólya's conjecture, Laplace operator, magnetic Schrödinger operators.
We consider the Dirichlet Laplacian with a constant magnetic field in a two-dimensional domain of finite measure. We determine the sharp constants in semi-classical eigenvalue estimates and show, in particular, that Pólya's conjecture is not true in the presence of a magnetic field.
Partial differential equations
General
1365
1383
10.4171/JEMS/184
http://www.ems-ph.org/doi/10.4171/JEMS/184
Geometry of the theta divisor of a compactified jacobian
Lucia
Caporaso
Università degli studi Roma Tre, ROMA, ITALY
Nodal curve, line bundle, compactified Picard scheme, theta divisor, Abel map, hyperelliptic stable curve.
The object of this paper is the theta divisor of the compactified jacobian of a nodal curve. We determine its irreducible components and give it a geometric interpretation. A characterization of hyperelliptic irreducible stable curves is appended as an application.
Algebraic geometry
General
1385
1427
10.4171/JEMS/185
http://www.ems-ph.org/doi/10.4171/JEMS/185
Formal power series rings over a π-domain
Byung Gyun
Kang
Pohang University of Science and Technology, POHANG CITY, KYUNGBUK, SOUTH KOREA
Dong Yeol
Oh
National Institute for Mathematical Sciences, DAEJEON, SOUTH KOREA
Krull domain, π-domain, unique factorization domain, formal power series ring, invertible ideal, class group, Picard group
Let R be an integral domain, Χ be a set of indeterminates over R, and R[[Χ]]3 be the full ring of formal power series in Χ over R. We show that the Picard group of R[[Χ]]3 is isomorphic to the Picard group of R. An integral domain is called a π-domain if every principal ideal is a product of prime ideals. An integral domain is a π-domain if and only if it is a Krull domain that is locally a unique factorization domain. We show that R[[Χ]]3 is a π-domain if R[[Χ1 , . . . , Χn]] is a π-domain for every n ≥ 1. In particular, R[[Χ]]3 is a π-domain if R is a Noetherian regular domain. We extend these results to rings with zero-divisors. A commutative ring R with identity is called a π-ring if every principal ideal is a product of prime ideals. We show that R[[Χ]]3 is a π-ring if R is a Noetherian regular ring.
Commutative rings and algebras
General
1429
1443
10.4171/JEMS/186
http://www.ems-ph.org/doi/10.4171/JEMS/186