- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 01:54:59
41
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=RMI&vol=24&update_since=2024-03-29
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
24
2008
1
A moduli approach to quadratic $\mathbb{Q}$-curves realizing projective mod $p$ Galois representations
Julio
Fernández
Universitat Politècnica de Catalunya, VILANOVA I LA GELTRÚ, SPAIN
mod $p$ Galois representations, elliptic curves, $p$-torsion points, quadratic $\mathbb{Q}$-curves, twisted modular curves, moduli problem
For a fixed odd prime $p$ and a representation $\varrho$ of the absolute Galois group of $\mathbb{Q}$ into the projective group ${\rm PGL}_2(\mathbb{F}_p)$, we provide the twisted modular curves whose rational points supply the quadratic $\mathbb{Q}$-curves of degree $N$ prime to $p$ that realize $\varrho$ through the Galois action on their $p$-torsion modules. The modular curve to twist is either the fiber product of $X_0(N)$ and $X(p)$ or a certain quotient of Atkin-Lehner type, depending on the value of $N$ mod $p$. For our purposes, a special care must be taken in fixing rational models for these modular curves and in studying their automorphisms. By performing some genus computations, we obtain as a by-product some finiteness results on the number of quadratic $\mathbb{Q}$-curves of a given degree $N$ realizing $\varrho$.
Number theory
Algebraic geometry
General
1
30
10.4171/RMI/527
http://www.ems-ph.org/doi/10.4171/RMI/527
Some asymptotic properties of the hybrids of empirical and partial-sum processes
Sergio
Alvarez-Andrade
Université de Technologie de Compiègne, COMPIÈGNE, FRANCE
Local times, compensated Poisson process, hybrids of empirical and partialsum processes, Brownian motion
The motivation of this paper is to study some properties of the local times (when it exists) of the hybrids of empirical and partial-sum processes defined by $$ \bar{A}_n(t)=\sum_{1\leq i \leq n} H(X_i)1_{\{X_i\leq t\}} \epsilon_i, \quad - \infty #x003C; t #x003C; \infty , $$ namely by using knowing results on empirical process and Brownian local times.
Probability theory and stochastic processes
Statistics
General
31
41
10.4171/RMI/528
http://www.ems-ph.org/doi/10.4171/RMI/528
Sums of Toeplitz products with harmonic symbols
Boo Rim
Choe
Korea University, SEOUL, SOUTH KOREA
Hyungwoon
Koo
Korea University, SEOUL, SOUTH KOREA
Young Joo
Lee
Chonnam National University, GWANGJU, SOUTH KOREA
Toeplitz operators, Bergman space, finite rank operators
On the Bergman space of the unit disk, we consider a class of operators which contain sums of finitely many Toeplitz products with harmonic symbols. We give characterizations of when an operator in that class has finite rank or is compact. Our results provide a unified way of treating several known results.
Operator theory
Several complex variables and analytic spaces
General
43
70
10.4171/RMI/529
http://www.ems-ph.org/doi/10.4171/RMI/529
Transformations between surfaces in $\mathbb{R}^4$ with flat normal and/or tangent bundles
Angel
Montesinos-Amilibia
Universitat de València, BURJASSOT (VALENCIA), SPAIN
flat, semiumbilical surfaces in $\mathbb{R}^4$, Bianchi permutability, Bäcklund transformation, evolute
We exhibit several transformations of surfaces $M$ in $\mathbb{R}^4$: a transformation of flat surfaces that gives surfaces with flat normal bundle (semiumbilical surfaces); and its inverse that from a semiumbilical surface obtains a flat surface; then a one-parameter family of transformations $f$ on flat semiumbilical immersed surfaces (FSIS), such that $df(T_pM)$ is totally orthogonal to $T_pM,$ and that give FSIS. This family satisfies a Bianchi type of permutability property.
Differential geometry
General
71
90
10.4171/RMI/530
http://www.ems-ph.org/doi/10.4171/RMI/530
A finiteness theorem for the space of $L^{p}$ harmonic sections
Stefano
Pigola
Università dell'Insubria, COMO, ITALY
Marco
Rigoli
Università di Milano, MILANO, ITALY
Alberto
Setti
Università dell'Insubria, COMO, ITALY
Riemannian vector bundles, harmonic sections, Morse index
In this paper we give a unified and improved treatment to finite dimensionality results for subspaces of $L^{p}$ harmonic sections of Riemannian or Hermitian vector bundles over complete manifolds. The geometric conditions on the manifold are subsumed by the assumption that the Morse index of a related Schr#x00F6;dinger operator is finite. Applications of the finiteness theorem to concrete geometric situations are also presented.
Differential geometry
Partial differential equations
General
91
116
10.4171/RMI/531
http://www.ems-ph.org/doi/10.4171/RMI/531
The algebro-geometric Toda hierarchy initial value problem for complex-valued initial data
Fritz
Gesztesy
Baylor University, WACO, UNITED STATES
Helge
Holden
University of Trondheim, TRONDHEIM, NORWAY
Gerald
Teschl
Universität Wien, WIEN, AUSTRIA
Toda hierarchy, complex-valued solutions, initial value problem
We discuss the algebro-geometric initial value problem for the Toda hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Toda hierarchy, which is of independent interest as it solves the inverse algebro-geometric spectral problem for generally non-self-adjoint Jacobi operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin equations), this yields the construction of global algebro-geometric solutions of the time-dependent Toda hierarchy. The inherent non-self-adjointness of the underlying Lax (i.e., Jacobi) operator associated with complex-valued coefficients for the Toda hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Toda hierarchy but applies generally to $1+1$-dimensional completely integrable discrete soliton equations.
Dynamical systems and ergodic theory
Partial differential equations
Operator theory
General
117
182
10.4171/RMI/532
http://www.ems-ph.org/doi/10.4171/RMI/532
Flux terms and Robin boundary conditions as limit of reactions and potentials concentrating at the boundary
José
Arrieta
Universidad Complutense de Madrid, MADRID, SPAIN
Angela
Jiménez-Casas
Universidad Pontificia Comillas de Madrid, MADRID, SPAIN
Aníbal
Rodríguez-Bernal
Universidad Complutense de Madrid, MADRID, SPAIN
Elliptic equations, concentrating terms, boundary reaction, boundary potential
We analyze the limit of the solutions of an elliptic problem when some reaction and potential terms are concentrated in a neighborhood of a portion $\Gamma$ of the boundary and this neighborhood shrinks to $\Gamma$ as a parameter goes to zero. We prove that this family of solutions converges in certain Sobolev spaces and also in the sup norm, to the solution of an elliptic problem where the reaction term and the concentrating potential are transformed into a flux condition and a potential on $\Gamma$.
Partial differential equations
General
183
211
10.4171/RMI/533
http://www.ems-ph.org/doi/10.4171/RMI/533
Rees algebras on smooth schemes: integral closure and higher differential operator
Orlando
Villamayor U.
Universidad Autónoma de Madrid, MADRID, SPAIN
Integral closure, Rees algebras
Let $V$ be a smooth scheme over a field $k$, and let $\{I_n, n\geq 0\}$ be a filtration of sheaves of ideals in $\mathcal{O}_V$, such that $I_0=\mathcal{O}_V$, and $I_s\cdot I_t\subset I_{s+t}$. In such case $\bigoplus I_n$ is called a Rees algebra. A Rees algebra is said to be a differential algebra if, for any two integers $N > n$ and any differential operator $D$ of order $n$, $D(I_N)\subset I_{N-n}$. Any Rees algebra extends to a smallest differential algebra. There are two extensions of Rees algebras of interest in singularity theory: one defined by taking integral closures, and another by extending the algebra to a differential algebra. We study here some relations between these two extensions, with particular emphasis on the behavior of higher order differentials over arbitrary fields.
Algebraic geometry
General
213
242
10.4171/RMI/534
http://www.ems-ph.org/doi/10.4171/RMI/534
Heat kernel transform for nilmanifolds associated to the Heisenberg group
Bernhard
Krötz
Universität Paderborn, PADERBORN, GERMANY
Sundaram
Thangavelu
Indian Institute of Science, BANGALORE, INDIA
Yuan
Xu
University of Oregon, EUGENE, UNITED STATES
Heisenberg group, nilmanifolds, Bergman spaces, heat kernel, Hermite and Laguerre functions
We study the heat kernel transform on a nilmanifold $M$ of the Heisenberg group. We show that the image of $L^2(M)$ under this transform is a direct sum of weighted Bergman spaces which are related to twisted Bergman and Hermite-Bergman spaces.
Topological groups, Lie groups
Partial differential equations
Global analysis, analysis on manifolds
General
243
266
10.4171/RMI/535
http://www.ems-ph.org/doi/10.4171/RMI/535
Comparison of the classical BMO with the BMO spaces associated with operators and applications
Donggao
Deng
Zhongshan University, GUANGZHOU, GUANGDONG, CHINA
Xuan Thinh
Duong
Macquarie University, SYDNEY, NSW, AUSTRALIA
Adam
Sikora
Macquarie University, SYDNEY, NSW, AUSTRALIA
Lixin
Yan
Zhongshan University, GUANGZHOU, GUANGDONG, CHINA
BMO space, Hardy space, Dirichlet and Neumann Laplacians, semigroup, Gaussian bounds, fractional powers, purely imaginary powers, spectral multiplier
Let $L$ be a generator of a semigroup satisfying the Gaussian upper bounds. A new ${\rm BMO}_L$ space associated with $L$ was recently introduced in [Duong, X. T. and Yan, L.: {New function spaces of BMO type, the John-Nirenberg inequality, interpolation and applications}. \textit{Comm. Pure Appl. Math.} {\bf 58} (2005), 1375-1420] and [Duong, X. T. and Yan, L.: {Duality of Hardy and BMO spaces associated with operators with heat kernels bounds}. \textit{J. Amer. Math. Soc.} {\bf 18} (2005), 943-973]. We discuss applications of the new ${\rm BMO}_L$ spaces in the theory of singular integration. For example we obtain ${\rm BMO}_L$ estimates and interpolation results for fractional powers, purely imaginary powers and spectral multipliers of self adjoint operators. We also demonstrate that the space ${\rm BMO}_L$ might coincide with or might be essentially different from the classical BMO space.
Fourier analysis
Operator theory
General
267
296
10.4171/RMI/536
http://www.ems-ph.org/doi/10.4171/RMI/536
Bound state solutions for a class of nonlinear Schrödinger equations
Denis
Bonheure
Université libre de Bruxelles, BRUXELLES, BELGIUM
Jean
Van Schaftingen
Université Catholique de Louvain, LOUVAIN-LA-NEUVE, BELGIUM
Nonlinear Schrödinger equation, semi-classical states, concentration, vanishing potentials, unbounded competition functions
We deal with the existence of positive bound state solutions for a class of stationary nonlinear Schrödinger equations of the form $$ -\varepsilon^2\Delta u + V(x) u = K(x) u^p,\qquad x\in\mathbb{R}^N, $ where $V, K$ are positive continuous functions and $p > 1$ is subcritical, in a framework which may exclude the existence of ground states. Namely, the potential $V$ is allowed to vanish at infinity and the competing function $K$ does not have to be bounded. In the \emph{semi-classical limit}, i.e. for $\varepsilon\sim 0$, we prove the existence of bound state solutions localized around local minimum points of the auxiliary function $\mathcal{A} = V^\theta K^{-\frac{2}{p-1}}$, where $\theta=(p+1)/(p-1)-N/2$. A special attention is devoted to the qualitative properties of these solutions as $\varepsilon$ goes to zero.
Partial differential equations
General
297
351
10.4171/RMI/537
http://www.ems-ph.org/doi/10.4171/RMI/537
Sampling Sets for the Nevanlinna class
Xavier
Massaneda
Universitat de Barcelona, BARCELONA, SPAIN
Pascal
Thomas
Université Paul Sabatier, TOULOUSE CEDEX 9, FRANCE
Sampling sets, determination sets, Nevanlinna class, Smirnov class
We propose a definition of sampling set for the Nevanlinna and Smirnov classes in the disk and show its equivalence with the notion of determination set for the same classes. We also show the relationship with determination sets for related classes of functions and deduce a characterization of Smirnov sampling sets. For Nevanlinna sampling we give general conditions (necessary or sufficient), from which we obtain precise geometric descriptions in several regular cases.
Functions of a complex variable
General
353
385
10.4171/RMI/538
http://www.ems-ph.org/doi/10.4171/RMI/538
Erratum: A Parabolic Quasilinear Problem for Linear Growth Functionals
Fuensanta
Andreu
Universitat de Valencia, BURJASSOT (VALENCIA), SPAIN
Vicent
Caselles
Universitat Pompeu-Fabra, BARCELONA, SPAIN
José
Mazón
Universitat de Valencia, BURJASSOT (VALENCIA), SPAIN
Linear growth functionals, nonlinear parabolic equations, accretive operators, nonlinear semigroups
We give the correct proof of Lemma 3.6 of the paper {\it A Parabolic Quasilinear Problem for Linear Growth Functionals} (Rev. Mat. Iberoamericana {\bf 18} (2002), no. 1, 135-185).
Partial differential equations
Operator theory
General
387
390
10.4171/RMI/539
http://www.ems-ph.org/doi/10.4171/RMI/539
2
On the number of ovals of a symmetry of a compact Riemann surface
Emilio
Bujalance
UNED, MADRID, SPAIN
Francisco Javier
Cirre
UNED, MADRID, SPAIN
José Manuel
Gamboa
Universidad Complutense de Madrid, MADRID, SPAIN
Grzegorz
Gromadzki
University of Gdańsk, GDAŃSK, POLAND
Riemann surface, symmetries, ovals
Let $X$ be a symmetric compact Riemann surface whose full group of conformal automorphisms is cyclic. We derive a formula for counting the number of ovals of the symmetries of $X$ in terms of few data of the monodromy of the covering $X\rightarrow X/G$, where $G=\mbox{\rm Aut\/}^\pm X$ is the full group of conformal and anticonformal automorphisms of $X$.
Functions of a complex variable
Algebraic geometry
General
391
405
10.4171/RMI/540
http://www.ems-ph.org/doi/10.4171/RMI/540
Entropy methods for reaction-diffusion equations: slowly growing a-priori bounds
Laurent
Desvillettes
, CACHAN CEDEX, FRANCE
Klemens
Fellner
Universität Wien, WIEN, AUSTRIA
Reaction-diffusion, entropy method, exponential decay, slowly growing a-priori estimates
In the continuation of [Desvillettes, L., Fellner, K.: Exponential Decay toward Equilibrium via Entropy Methods for Reaction-Diffusion Equations. J. Math. Anal. Appl. 319 (2006), no. 1, 157-176], we study reversible reaction-diffusion equations via entropy methods (based on the free energy functional) for a 1D system of four species. We improve the existing theory by getting 1) almost exponential convergence in $L^1$ to the steady state via a precise entropy-entropy dissipation estimate, 2) an explicit global $L^{\infty}$ bound via interpolation of a polynomially growing $H^1$ bound with the almost exponential $L^1$ convergence, and 3), finally, explicit exponential convergence to the steady state in all Sobolev norms.
Partial differential equations
General
407
431
10.4171/RMI/541
http://www.ems-ph.org/doi/10.4171/RMI/541
Soluble products of connected subgroups
M. Pilar
Gállego
Universidad de Zaragoza, ZARAGOZA, SPAIN
Peter
Hauck
Universität Tübingen, TÜBINGEN, GERMANY
M. Dolores
Pérez-Ramos
Universitat de València, BURJASSOT (VALENCIA), SPAIN
Finite groups, soluble groups, 2-generated subgroups, product of subgroups, metanilpotent groups, Fitting series
The main result in the paper states the following: For a finite group $G=AB$, which is the product of the soluble subgroups $A$ and $B$, if $\langle a,b \rangle$ is a metanilpotent group for all $a\in A$ and $b\in B$, then the factor groups $\langle a,b \rangle F(G)/F(G)$ are nilpotent, $F(G)$ denoting the Fitting subgroup of $G$. A particular generalization of this result and some consequences are also obtained. For instance, such a group $G$ is proved to be soluble of nilpotent length at most $l+1$, assuming that the factors $A$ and $B$ have nilpotent length at most $l$. Also for any finite soluble group $G$ and $k\geq 1$, an element $g\in G$ is contained in the preimage of the hypercenter of $G/F_{k-1}(G)$, where $F_{k-1}(G)$ denotes the ($k-1$)th term of the Fitting series of $G$, if and only if the subgroups $\langle g,h\rangle$ have nilpotent length at most $k$ for all $h\in G$.
Group theory and generalizations
General
433
461
10.4171/RMI/542
http://www.ems-ph.org/doi/10.4171/RMI/542
Global infinite energy solutions of the critical semilinear wave equation
Pierre
Germain
New York University, NEW YORK, UNITED STATES
Critical wave equation, global solutions, infinite energy, Besov spaces
We consider the critical semilinear wave equation \begin{equation*} (NLW)_{2^*-1} \;\;\; \left\{ \begin{aligned} \square u + |u|^{2^*-2} u & = 0 \\ u_{|t=0} & = u_0 \\ \partial_t u_{|t=0} & = u_1 \, \,, \end{aligned} \right. \end{equation*} set in $\mathbb{R}^d$, $d \geq 3$, with $2^* = \frac{2d}{d-2} \,\cdotp$ Shatah and Struwe [Shatah, J. and Struwe, M.: Geometric wave equations. Courant Lecture Notes in Mathematics 2. New York University, Courant Institute of Mathematical Sciences. American Mathematical Society, RI, 1998] proved that, for finite energy initial data (ie if $(u_0,u_1) \in \dot{H}^1 \times L^2$), there exists a global solution such that $(u,\partial_t u)\in \mathcal{C}(\mathbb{R},\dot{H}^1 \times L^2)$. Planchon [Planchon, F.: Self-similar solutions and semi-linear wave equations in Besov spaces. J. Math. Pures Appl. (9) 79 (2000), no. 8, 809-820] showed that there also exists a global solution for certain infinite energy initial data, namely, if the norm of $(u_0,u_1)$ in $\dot{B}^1_{2,\infty} \times \dot{B}^0_{2,\infty}$ is small enough. In this article, we build up global solutions of $(NLW)_{2^*-1}$ for arbitrarily big initial data of infinite energy, by using two methods which enable to interpolate between finite and infinite energy initial data: the method of Calderón, and the method of Bourgain. These two methods give complementary results.
Partial differential equations
Functional analysis
General
463
497
10.4171/RMI/543
http://www.ems-ph.org/doi/10.4171/RMI/543
Interpolation and Sampling for Generalized Bergman Spaces on finite Riemann surfaces
Alexander
Schuster
San Francisco State University, SAN FRANCISCO, UNITED STATES
Dror
Varolin
Stony Brook University, STONY BROOK, UNITED STATES
Finite Riemann surfaces, Green’s function, Evans kernel, Beurling density, Ohsawa’s theorem
We find sufficient conditions for a discrete sequence to be interpolating or sampling for certain generalized Bergman spaces on open Riemann surfaces. As in previous work of Bendtsson, Ortega-Cerdá, Seip, Wallsten and others, our conditions for interpolation and sampling are as follows: If a certain upper density of the sequence has value less that 1, then the sequence is interpolating, while if a certain lower density has value greater than 1, then the sequence is sampling. Unlike previous works, we introduce a family of densities all of which provide sufficient conditions. Thus we obtain new results even in classical cases, some of which might be useful in industrial applications. The main point of the article is to demonstrate the interaction between the potential theory of the Riemann surface and its interpolation and sampling properties.
Functions of a complex variable
General
499
530
10.4171/RMI/544
http://www.ems-ph.org/doi/10.4171/RMI/544
$L^2$ boundedness for commutator of rough singular integral with variable kernel
Yanping
Chen
University of Sciences and Technology, BEIJING, CHINA
Yong
Ding
Beijing Normal University, BEIJING, CHINA
Commutator, singular integral, variable kernel, BMO, spherical harmonic function
n this paper the authors prove the $L^2(\mathbb{R}^n)$ boundedness of the commutator of the singular integral operator with rough variable kernels, which is a substantial improvement and extension of some known results.
Fourier analysis
General
531
547
10.4171/RMI/545
http://www.ems-ph.org/doi/10.4171/RMI/545
Geometric optics with critical vanishing viscosity for one-dimensional semilinear initial value problems
Stéphane
Junca
Université de Nice Sophia Antipolis, NICE, FRANCE
Nonlinear geometric optics, small viscosity, profile, phase, non stationary phase, maximum principle, energy estimates, interpolation, weakly coupled parabolic systems
We study the propagation of high frequency oscillations for one dimensional semi-linear hyperbolic systems with small parabolic perturbations. We obtain a new degenerate parabolic system for the profile, and valid an asymptotic development in the spirit of Joly, Métivier and Rauch.
Partial differential equations
General
549
566
10.4171/RMI/546
http://www.ems-ph.org/doi/10.4171/RMI/546
Non-uniqueness in a free boundary problem
Björn
Bennewitz
University of Jyväskylä, JYVÄSKYLÄ, FINLAND
p Laplacian, overdetermined, elliptic, Hausdorff measure
We show that a result of Lewis and Vogel on uniqueness in a free boundary problem for the $p$-Laplace operator is sharp in two dimensions.
Partial differential equations
General
567
595
10.4171/RMI/547
http://www.ems-ph.org/doi/10.4171/RMI/547
Quasilinear equations with natural growth
David
Arcoya
Universidad de Granada, GRANADA, SPAIN
Pedro
Martínez-Aparicio
Universidad de Granada, GRANADA, SPAIN
Quasilinear elliptic equations, critical growth, singular nonlinearity
We study the existence of positive solution $w\in H_0^1(\Omega)$ of the quasilinear equation $-\Delta w+ g(w)|\nabla w|^2=a(x)$, $x\in \Omega$, where $\Omega$ is a bounded domain in $\mathbb R^N$, $0\leq a\in L^\infty (\Omega )$ and $g$ is a nonnegative continuous function on $(0,+\infty)$ which may have a singularity at zero.
Partial differential equations
General
597
616
10.4171/RMI/548
http://www.ems-ph.org/doi/10.4171/RMI/548
On the verbal width of finitely generated pro-$p$ groups
Andrei
Jaikin-Zapirain
Universidad Autónoma de Madrid, MADRID, SPAIN
pro-p group, verbal subgroup, verbal width, p-adic analytic group
Let $p$ be a prime. It is proved that a non-trivial word $w$ from a free group $F$ has finite width in every finitely generated pro-$p$ group if and only if $w\not \in (F^\prime)^{p} F^{\prime\prime}$. Also it is shown that any word $w$ has finite width in a compact $p$-adic group.
Group theory and generalizations
Topological groups, Lie groups
General
617
630
10.4171/RMI/549
http://www.ems-ph.org/doi/10.4171/RMI/549
Notes on the roots of Steiner polynomials
Martin
Henk
Technische Universität Berlin, BERLIN, GERMANY
María
Hernández Cifre
Universidad de Murcia, MURCIA, SPAIN
Steiner polynomial, Teissier’s problem, tangential bodies, circumradius, inradius
We study the location and the size of the roots of Steiner polynomials of convex bodies in the Minkowski relative geometry. Based on a problem of Teissier on the intersection numbers of Cartier divisors of compact algebraic varieties it was conjectured that these roots have certain geometric properties related to the in- and circumradius of the convex body. We show that the roots of 1-tangential bodies fulfill the conjecture, but we also present convex bodies violating each of the conjectured properties.
Convex and discrete geometry
Functions of a complex variable
General
631
644
10.4171/RMI/550
http://www.ems-ph.org/doi/10.4171/RMI/550
Measure density and extendability of Sobolev functions
Piotr
Hajłasz
University of Pittsburgh, PITTSBURGH, UNITED STATES
Pekka
Koskela
University of Jyväskylä, JYVÄSKYLÄ, FINLAND
Heli
Tuominen
University of Jyväskylä, JYVÄSKYLÄ, FINLAND
Sobolev extension, doubling measure
We study necessary and sufficient conditions for a domain to be a Sobolev extension domain in the setting of metric measure spaces. In particular, we prove that extension domains must satisfy a measure density condition.
Functional analysis
General
645
669
10.4171/RMI/551
http://www.ems-ph.org/doi/10.4171/RMI/551
On the NLS dynamics for infinite energy vortex configurations on the plane
Fabrice
Bethuel
Université Pierre et Marie Curie, PARIS, FRANCE
Robert
Jerrard
University of Toronto, TORONTO, ONTARIO, CANADA
Didier
Smets
UPMC, Université Paris 06,, PARIS CEDEX 05, FRANCE
Vortex dynamics, NLS equation, superfluids
We derive the asymptotical dynamical law for Ginzburg-Landau vortices in the plane under the Schrödinger dynamics, as the Ginz\-burg-Landau parameter goes to zero. The limiting law is the well-known point-vortex system. This result extends to the whole plane previous results of [Colliander, J.E. and Jerrard, R.L.: Vortex dynamics for the Ginzburg-Landau-Schrödinger equation. Internat. Math. Res. Notices 1998, no. 7, 333-358; Lin, F.-H. and Xin, J.\,X.: On the incompressible fluid limit and the vortex motion law of the nonlinear Schr\"{o}dinger equation. Comm. Math. Phys. 200 (1999), 249-274] established for bounded domains, and holds for arbitrary degree at infinity. When this degree is non-zero, the total Ginzburg-Landau energy is infinite.
Partial differential equations
Statistical mechanics, structure of matter
General
671
702
10.4171/RMI/552
http://www.ems-ph.org/doi/10.4171/RMI/552
Stable Higgs $G$-sheaves
Tomás
Gómez
Universidad Complutense de Madrid, MADRID, SPAIN
Ignacio
Sols
Universidad Complutense de Madrid, MADRID, SPAIN
Moduli spaces, principal bundles, Higgs bundles
For a connected reductive group $G$, we generalize the notion of (semi)stable Higgs $G$-bundles on curves to smooth projective schemes of higher dimension, allowing also Higgs $G$-sheaves, and construct the corresponding moduli space.
Algebraic geometry
General
703
719
10.4171/RMI/553
http://www.ems-ph.org/doi/10.4171/RMI/553
3
The Walsh model for $M_2^*$ Carleson
Ciprian
Demeter
Indiana University, BLOOMINGTON, UNITED STATES
Michael
Lacey
Georgia Institute of Technology, ATLANTA, UNITED STATES
Terence
Tao
University of California Los Angeles, LOS ANGELES, UNITED STATES
Christoph
Thiele
Universität Bonn, BONN, GERMANY
Carleson’s operator, multiplier norm
We study the Walsh model of a certain maximal truncation of Carleson's operator related to the Return Times Theorem.
Fourier analysis
General
721
744
10.4171/RMI/554
http://www.ems-ph.org/doi/10.4171/RMI/554
Infinite groups with many permutable subgroups
Adolfo
Ballester-Bolinches
Universitat de València, BURJASSOT (VALENCIA), SPAIN
L.
Kurdachenko
National Dnepropetrovsk University, DNEPROPETROVSK, UKRAINE
J.
Otal
Universidad de Zaragoza, ZARAGOZA, SPAIN
T.
Pedraza
Universidad Politécnica de Valencia, VALENCIA, SPAIN
radical groups, hyper--$\mathfrak{X}$--groups, $AP$--groups, $PT$--groups
A subgroup $H$ of a group $G$ is said to be \textit{permutable in $G$}, if $HK = KH$ for every subgroup $K$ of $G$. A result due to Stonehewer asserts that every permutable subgroup is ascendant although the converse is false. In this paper we study some infinite groups whose ascendant subgroups are permutable ($AP$--groups). We show that the structure of radical hyperfinite $AP$--groups behave as that of finite soluble groups in which the relation \textit{to be a permutable subgroup} is transitive ($PT$--groups).
Group theory and generalizations
General
745
764
10.4171/RMI/555
http://www.ems-ph.org/doi/10.4171/RMI/555
The linear fractional model on the ball
Frédéric
Bayart
Université Blaise Pascal, AUBIÈRE CEDEX, FRANCE
Linear fractional maps, iteration
Given a holomorphic self-map $\varphi$ of the ball of $\mathbb{C}^N$, we study whether there exists a map $\sigma$ and a linear fractional transformation $A$ such that $\sigma\circ\varphi=A\circ\sigma$. This is an important result when $N=1$ with a great number of applications. We extend this result to the multi-dimensional setting for a large class of maps. Applications to commuting holomorphic self-maps are given.
Several complex variables and analytic spaces
General
765
824
10.4171/RMI/556
http://www.ems-ph.org/doi/10.4171/RMI/556
Large-scale Sobolev inequalities on metric measure spaces and applications
Romain
Tessera
Université Paris-Sud, CNRS, ORSAY CEDEX, FRANCE
Large-scale analysis on metric spaces, coarse equivalence, symmetric random walks on groups, Sobolev inequalities, isoperimetry
For functions on a metric measure space, we introduce a notion of "gradient at a given scale''. This allows us to define Sobolev inequalities at a given scale. We prove that satisfying a Sobolev inequality at a large enough scale is invariant under large-scale equivalence, a metric-measure version of coarse equivalence. We prove that for a Riemmanian manifold satisfying a local Poincaré inequality, our notion of Sobolev inequalities at large scale is equivalent to its classical version. These notions provide a natural and efficient point of view to study the relations between the large time on-diagonal behavior of random walks and the isoperimetry of the space. Specializing our main result to locally compact groups, we obtain that the $L^p$-isoperimetric profile, for every $1\leq p\leq \infty$ is invariant under quasi-isometry between amenable unimodular compactly generated locally compact groups. A qualitative application of this new approach is a very general characterization of the existence of a spectral gap on a quasi-transitive measure space $X$, providing a natural point of view to understand this phenomenon.
Group theory and generalizations
Topological groups, Lie groups
General
825
864
10.4171/RMI/557
http://www.ems-ph.org/doi/10.4171/RMI/557
The real genus of the alternating groups
José Javier
Etayo Gordejuela
Universidad Complutense de Madrid, MADRID, SPAIN
Ernesto
Martínez
UNED, MADRID, SPAIN
Alternating groups, real genus, M*-groups, bordered Klein surfaces
A Klein surface with boundary of algebraic genus $\mathfrak{p}\geq 2$, has at most $12(\mathfrak{p}-1)$ automorphisms. The groups attaining this upper bound are called $M^{\ast}$-groups, and the corresponding surfaces are said to have maximal symmetry. The $M^{\ast}$-groups are characterized by a partial presentation by generators and relators. The alternating groups $A_{n}$ were proved to be $M^{\ast}$-groups when $n\geq 168$ by M. Conder. In this work we prove that $A_{n}$ is an $M^{\ast }$-group if and only if $n\geq 13$ or $n=5,10$. In addition, we describe topologically the surfaces with maximal symmetry having $A_{n}$ as automorphism group, in terms of the partial presentation of the group. As an application we determine explicitly all such surfaces for $n\leq 14$. Each finite group $G$ acts as an automorphism group of several Klein surfaces. The minimal genus of these surfaces is called the real genus of the group, $\rho(G)$. If $G$ is an $M^{\ast}$-group then $\rho(G)=\frac{o(G)}{12}+1$. We end our work by calculating the real genus of the alternating groups which are not $M^{\ast}$-groups.
Group theory and generalizations
Functions of a complex variable
General
865
894
10.4171/RMI/558
http://www.ems-ph.org/doi/10.4171/RMI/558
Projections of hypersurfaces in the hyperbolic space to hyperhorospheres and hyperplanes
Shyuichi
Izumiya
Hokkaido University, SAPPORO, JAPAN
Farid
Tari
Universidade de São Paulo, SÃO CARLOS, SP, BRAZIL
Bifurcation sets, contours, Legendrian duality, projections, profiles, hyperbolic space, singularities, de Sitter space, lightcone
We study in this paper orthogonal projections in a hyperbolic space to hyperhorospheres and hyperplanes. We deal in more details with the case of embedded surfaces $M$ in $H^3_+(-1)$. We study the generic singularities of the projections of $M$ to horospheres and planes. We give geometric characterizations of these singularities and prove duality results concerning the bifurcation sets of the families of projections. We also prove Koenderink type theorems that give the curvature of the surface in terms of the curvatures of the profile and the normal section of the surface.
Differential geometry
Global analysis, analysis on manifolds
General
895
920
10.4171/RMI/559
http://www.ems-ph.org/doi/10.4171/RMI/559
Reflections of regular maps and Riemann surfaces
Adnan
Melekoğlu
Adnan Menderes University, AYDIN, TURKEY
David
Singerman
University of Southampton, SOUTHAMPTON, UNITED KINGDOM
Regular map, Riemann surface, Platonic surface, M-surface, (M−1)-surface
A compact Riemann surface of genus $g$ is called an M-surface if it admits an anti-conformal involution that fixes $g+1$ simple closed curves, the maximum number by Harnack's Theorem. Underlying every map on an orientable surface there is a Riemann surface and so the conclusions of Harnack's theorem still apply. Here we show that for each genus $g ϯ 1$ there is a unique M-surface of genus $g$ that underlies a regular map, and we prove a similar result for Riemann surfaces admitting anti-conformal involutions that fix $g$ curves.
Combinatorics
Functions of a complex variable
General
921
939
10.4171/RMI/560
http://www.ems-ph.org/doi/10.4171/RMI/560
Tropical resultants for curves and stable intersection
Luis Felipe
Tabera Alonso
Universidad de Cantabria, SANTANDER, SPAIN
Tropical geometry, resultants, plane curves
We introduce the notion of resultant of two planar curves in the tropical geometry framework. We prove that the tropicalization of the algebraic resultant can be used to compute the stable intersection of two tropical plane curves. It is shown that, for two generic preimages of the curves to an algebraic framework, their intersection projects exactly onto the stable intersection of the curves. It is also given sufficient conditions for such a generality in terms of the residual coefficients of the algebraic coefficients of defining equations of the curves.
Algebraic geometry
Convex and discrete geometry
General
941
961
10.4171/RMI/561
http://www.ems-ph.org/doi/10.4171/RMI/561
Homology exponents for $H$-spaces
Alain
Clément
, VEVEY, SWITZERLAND
Jérôme
Scherer
EPFL, LAUSANNE, SWITZERLAND
Homology exponent, H-space, loop space, Steenrod algebra
We say that a space $X$ admits a \emph{homology exponent} if there exists an exponent for the torsion subgroup of $H^*(X;\mathbb Z)$. Our main result states that if an $H$-space of finite type admits a homology exponent, then either it is, up to $2$-completion, a product of spaces of the form $B\mathbb Z/2^r$, $S^1$, $\mathbb C P^\infty$, and $K(\mathbb Z,3)$, or it has infinitely many non-trivial homotopy groups and $k$-invariants. Relying on recent advances in the theory of $H$-spaces, we then show that simply connected $H$-spaces whose mod $2$ cohomology is finitely generated as an algebra over the Steenrod algebra do not have homology exponents, except products of mod $2$ finite $H$-spaces with copies of $\mathbb C P^\infty$ and $K(\mathbb Z,3)$.
Manifolds and cell complexes
Algebraic topology
General
963
980
10.4171/RMI/562
http://www.ems-ph.org/doi/10.4171/RMI/562
A note on boundaries of open polynomial images of $\mathbb R^2$
Carlos
Ueno
IES Jandía, MORRO JABLE, SPAIN
polynomial images, semialgebraic sets, parametric semilines
We construct a family of polynomial maps $\mathbb{R}^2\rightarrow\mathbb{R}^2$ such that their images are open semialgebraic sets whose topological exteriors have arbitrarily many connected components, which are parametric semilines.
Algebraic geometry
General
981
988
10.4171/RMI/563
http://www.ems-ph.org/doi/10.4171/RMI/563
Almost classical solutions of Hamilton-Jacobi equations
Robert
Deville
Université de Bordeaux I, TALENCE CEDEX, FRANCE
Jesús
Jaramillo
Universidad Complutense de Madrid, MADRID, SPAIN
Hamilton-Jacobi equations, eikonal equation on manifolds, almost everywhere solutions
We study the existence of everywhere differentiable functions which are almost everywhere solutions of quite general Hamilton-Jacobi equations on open subsets of $\mathbb R^d$ or on $d$-dimensional manifolds whenever $d\geq 2$. In particular, when $M$ is a Riemannian manifold, we prove the existence of a differentiable function $u$ on $M$ which satisfies the Eikonal equation $\Vert \nabla u(x) \Vert_{x}=1$ almost everywhere on $M$.
Real functions
Partial differential equations
Global analysis, analysis on manifolds
General
989
1010
10.4171/RMI/564
http://www.ems-ph.org/doi/10.4171/RMI/564
Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term
Sergio
Polidoro
Università di Bologna, BOLOGNA, ITALY
Maria Alessandra
Ragusa
Università degli Studi di Catania, CATANIA, ITALY
Hypoelliptic operator, Schrödinger equation, Harnack inequality, Green function
We prove a Harnack inequality for the positive solutions of ultraparabolic equations of the type $$ \mathcal {L}_0 u + \mathcal {V} u = 0, $$ where $\mathcal {L}_0$ is a linear second order hypoelliptic operator and $\mathcal {V}$ belongs to a class of functions of Stummel-Kato type. We also obtain the existence of a Green function and an uniqueness result for the Cauchy-Dirichlet problem.
Partial differential equations
Several complex variables and analytic spaces
General
1011
1046
10.4171/RMI/565
http://www.ems-ph.org/doi/10.4171/RMI/565
Multiparameter singular integrals and maximal operators along flat surfaces
Yong-Kum
Cho
Chung-Ang University, SEOUL, SOUTH KOREA
Sunggeum
Hong
Chosun University, GWANGJU, SOUTH KOREA
Joonil
Kim
Yonsei University, SEOUL, SOUTH KOREA
Chan Woo
Yang
Korea University, SEOUL, SOUTH KOREA
Singular Radon transform, multiple Hilbert transform, flat surface
We study double Hilbert transforms and maximal functions along surfaces of the form $(t_1,t_2,\gamma_1(t_1)\gamma_2(t_2))$. The $L^p(\mathbb{R}^3)$ boundedness of the maximal operator is obtained if each $\gamma_i$ is a convex increasing and $\gamma_i(0)=0$. The double Hilbert transform is bounded in $L^p(\mathbb{R}^3)$ if both $\gamma_i$'s above are extended as even functions. If $\gamma_1$ is odd, then we need an additional comparability condition on $\gamma_2$. This result is extended to higher dimensions and the general hyper-surfaces of the form $(t_1,\dots,t_{n},\Gamma(t_1,\dots,t_{n}))$ on $\mathbb{R}^{n+1}$.
Fourier analysis
General
1047
1073
10.4171/RMI/566
http://www.ems-ph.org/doi/10.4171/RMI/566
Majorizing measures and proportional subsets of bounded orthonormal systems
Olivier
Guédon
Université Paris 6 Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
Shahar
Mendelson
Technion - Israel Institute of Technology, HAIFA, ISRAEL
Alain
Pajor
Université de Paris-Est, MARNE-LA-VALLÉE CEDEX 2, FRANCE
Nicole
Tomczak-Jaegermann
University of Alberta, EDMONTON, CANADA
Empirical process, majorizing measure, orthonormal system
In this article we prove that for any orthonormal system $(\varphi_j)_{j=1}^n \subset L_2$ that is bounded in $L_{\infty}$, and any $1 < k < n$, there exists a subset $I$ of cardinality greater than $n-k$ such that on $\mathrm{span}\{\varphi_i\}_{i \in I}$, the $L_1$ norm and the $L_2$ norm are equivalent up to a factor $\mu (\log \mu)^{5/2}$, where $\mu = \sqrt{n/k} \sqrt{\log k}$. The proof is based on a new estimate of the supremum of an empirical process on the unit ball of a Banach space with a good modulus of convexity, via the use of majorizing measures.
Functional analysis
Fourier analysis
General
1075
1095
10.4171/RMI/567
http://www.ems-ph.org/doi/10.4171/RMI/567