Electronic Research Announcements

2014 , Volume 21

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Unboundedness of the Lagrangian Hofer distance in the Euclidean ball
Sobhan Seyfaddini
2014, 21: 1-7 doi: 10.3934/era.2014.21.1 +[Abstract](94) +[PDF](344.8KB)
Let $\mathcal{L}$ denote the space of Lagrangians Hamiltonian isotopic to the standard Lagrangian in the unit ball in $\mathbb{R}^{2n}$. We prove that the Lagrangian Hofer distance on $\mathcal{L}$ is unbounded.
Square function estimates in spaces of homogeneous type and on uniformly rectifiable Euclidean sets
Steve Hofmann, Dorina Mitrea, Marius Mitrea and Andrew J. Morris
2014, 21: 8-18 doi: 10.3934/era.2014.21.8 +[Abstract](121) +[PDF](400.5KB)
We announce a local $T(b)$ theorem, an inductive scheme, and $L^p$ extrapolation results for $L^2$ square function estimates related to the analysis of integral operators that act on Ahlfors-David regular sets of arbitrary codimension in ambient quasi-metric spaces. The inductive scheme is a natural application of the local $T(b)$ theorem and it implies the stability of $L^2$ square function estimates under the so-called big pieces functor. In particular, this analysis implies $L^p$ and Hardy space square function estimates for integral operators on uniformly rectifiable subsets of the Euclidean space.
The spectral gap of graphs and Steklov eigenvalues on surfaces
Bruno Colbois and Alexandre Girouard
2014, 21: 19-27 doi: 10.3934/era.2014.21.19 +[Abstract](141) +[PDF](364.6KB)
Using expander graphs, we construct a sequence $\{\Omega_N\}_{N\in\mathbb{N}}$ of smooth compact surfaces with boundary of perimeter $N$, and with the first non-zero Steklov eigenvalue $\sigma_1(\Omega_N)$ uniformly bounded away from zero. This answers a question which was raised in [10]. The sequence $\sigma_1(\Omega_N) L(\partial\Omega_n)$ grows linearly with the genus of $\Omega_N$, which is the optimal growth rate.
Remarks on 5-dimensional complete intersections
Jianbo Wang
2014, 21: 28-40 doi: 10.3934/era.2014.21.28 +[Abstract](95) +[PDF](385.5KB)
This paper will give some examples of diffeomorphic complex 5-dimensional complete intersections and remarks on these examples. Then a result on the existence of diffeomorphic complete intersections that belong to components of the moduli space of different dimensions will be given as a supplement to the results of P.Brückmann (J. reine angew. Math. 476 (1996), 209--215; 525 (2000), 213--217).
Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies
Amadeu Delshams, Marina Gonchenko and Pere Gutiérrez
2014, 21: 41-61 doi: 10.3934/era.2014.21.41 +[Abstract](106) +[PDF](1137.6KB)
We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a frequency vector $\omega=(1,\Omega)$, where $\Omega$ is a quadratic irrational number, or a 3-dimensional torus with a frequency vector $\omega=(1,\Omega,\Omega^2)$, where $\Omega$ is a cubic irrational number. Applying the Poincaré--Melnikov method, we find exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, and we show that such estimates depend strongly on the arithmetic properties of the frequencies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfilled in 24 cases, which allows us to provide asymptotic estimates in a simple way. In the cubic case, we focus our attention to the case in which $\Omega$ is the so-called cubic golden number (the real root of $x^3+x-1=0$), obtaining also asymptotic estimates. We point out the similitudes and differences between the results obtained for both the quadratic and cubic cases.
A gradient estimate for harmonic functions sharing the same zeros
Dan Mangoubi
2014, 21: 62-71 doi: 10.3934/era.2014.21.62 +[Abstract](113) +[PDF](358.7KB)
Let $u, v$ be two harmonic functions in $\{|z|<2\}\subset\mathbb{C}$ which have exactly the same set $Z$ of zeros. We observe that $\big|\nabla\log |u/v|\big|$ is bounded in the unit disk by a constant which depends on $Z$ only. In case $Z=\emptyset$ this goes back to Li-Yau's gradient estimate for positive harmonic functions. The general boundary Harnack principle gives only Hölder estimates on $\log |u/v|$.
From local to global equilibrium states: Thermodynamic formalism via an inducing scheme
Renaud Leplaideur
2014, 21: 72-79 doi: 10.3934/era.2014.21.72 +[Abstract](86) +[PDF](349.2KB)
We present a method to construct equilibrium states via inducing. This method can be used for some non-uniformly hyperbolic dynamical systems and for non-Hölder continuous potentials. It allows us to prove the existence of phase transition.
Compactly supported Hamiltonian loops with a non-zero Calabi invariant
Asaf Kislev
2014, 21: 80-88 doi: 10.3934/era.2014.21.80 +[Abstract](99) +[PDF](331.3KB)
We give examples of compactly supported Hamiltonian loops with a non-zero Calabi invariant on certain open symplectic manifolds.
Pseudo-Anosov eigenfoliations on Panov planes
Chris Johnson and Martin Schmoll
2014, 21: 89-108 doi: 10.3934/era.2014.21.89 +[Abstract](142) +[PDF](1107.4KB)
We study dynamical properties of direction foliations on the complex plane pulled back from direction foliations on a half-translation torus $T$, i.e., a torus equipped with a strict and integrable quadratic differential. If the torus $T$ admits a pseudo-Anosov map we give a homological criterion for the appearance of dense leaves and leaves with bounded deviation on the universal covering of $T$, called Panov plane. Our result generalizes Dmitri Panov's explicit construction of dense leaves for certain arithmetic half-translation tori [33]. Certain Panov planes are related to the polygonal table of the periodic wind-tree model. In fact, we show that the dynamics on periodic wind-tree billiards can be converted to the dynamics on a pair of singular planes.
    Possible strategies to generalize our main dynamical result to larger sets of directions are discussed. Particularly we include recent results of Frączek and Ulcigrai [17, 18] and Delecroix [6] for the wind-tree model. Implicitly Panov planes appear in Frączek and Schmoll [15], where the authors consider Eaton Lens distributions.
On Helly's theorem in geodesic spaces
Sergei Ivanov
2014, 21: 109-112 doi: 10.3934/era.2014.21.109 +[Abstract](98) +[PDF](264.0KB)
In this note we show that Helly's Intersection Theorem holds for convex sets in uniquely geodesic spaces (in particular, in CAT(0) spaces) without the assumption that the convex sets are open or closed.
On existence of PI-exponents of codimension growth
Mikhail Zaicev
2014, 21: 113-119 doi: 10.3934/era.2014.21.113 +[Abstract](98) +[PDF](323.2KB)
We construct a family of examples of non-associative algebras $\{R_\alpha \,\vert\, 1<\alpha\in\mathbb R\}$ such that $\underline{\exp}(R_\alpha)=1$, $\overline{\exp}(R_\alpha)=\alpha$. In particular, it follows that for any $R_\alpha$, an ordinary PI-exponent of codimension growth does not exist.
Number of extremal subsets in Alexandrov spaces and rigidity
Nina Lebedeva
2014, 21: 120-125 doi: 10.3934/era.2014.21.120 +[Abstract](83) +[PDF](304.0KB)
In this paper we announce the following result. We show that any $n$-dimensional nonnegatively curved Alexandrov space with the maximal possible number of extremal points is isometric to a quotient space of $\mathbb{R}^n$ by an action of a crystallographic group. We describe all such actions. We start with a history, results and open questions concerning estimates on the number of extremal subsets in Alexandrov spaces. Then we give the plan of the proof of our result; the complete proof will published elsewhere.
Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone
Piotr Pokora and Tomasz Szemberg
2014, 21: 126-131 doi: 10.3934/era.2014.21.126 +[Abstract](112) +[PDF](297.9KB)
The purpose of this note is to study the number of elements in Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone.
An arithmetic ball quotient surface whose Albanese variety is not of CM type
Chad Schoen
2014, 21: 132-136 doi: 10.3934/era.2014.21.132 +[Abstract](149) +[PDF](300.8KB)
An example is given of a compact quotient of the unit ball in $\mathbb{C}^2$ by an arithmetic group acting freely such that the Albanese variety is not of CM type. Such examples do not exist for congruence subgroups.
Motivic functions, integrability, and applications to harmonic analysis on $p$-adic groups
Raf Cluckers, Julia Gordon and Immanuel Halupczok
2014, 21: 137-152 doi: 10.3934/era.2014.21.137 +[Abstract](130) +[PDF](467.4KB)
We provide a short and self-contained overview of the techniques based on motivic integration as they are applied in harmonic analysis on $p$-adic groups; our target audience is mainly representation theorists with no background in model theory (and model theorists with an interest in recent applications of motivic integration in representation theory, though we do not provide any representation theory background). We aim to give a fairly comprehensive survey of the results in harmonic analysis that were proved by such techniques in the last ten years, with emphasis on the most recent techniques and applications from [13], [8], and [32, Appendix B].
Canonical Cartan connections on maximally minimal generic submanifolds $\mathbf{M^5 \subset \mathbb{C}^4}$
Masoud Sabzevari, Joël Merker and Samuel Pocchiola
2014, 21: 153-166 doi: 10.3934/era.2014.21.153 +[Abstract](96) +[PDF](490.9KB)
On a real analytic $5$-dimensional CR-generic submanifold $M^5 \subset \mathbb{C}^4$ of codimension $3$ hence of CR dimension $1$, which enjoys the generically satisfied nondegeneracy condition \begin{align*} {\bf 5} &= \text{rank}_\mathbb{C} \big( T^{1,0}M+T^{0,1}M + \big[T^{1,0}M,\,T^{0,1}M\big] \,+ \\&\qquad + \big[T^{1,0}M,\,[T^{1,0}M,T^{0,1}M]\big] + \big[T^{0,1}M,\,[T^{1,0}M,T^{0,1}M]\big] \big), \end{align*} a canonical Cartan connection is constructed after reduction to a certain partially explicit $\{ e\}$-structure of the concerned local biholomorphic equivalence problem.
Groups of Lie type, vertex algebras, and modular moonshine
Robert L. Griess Jr. and Ching Hung Lam
2014, 21: 167-176 doi: 10.3934/era.2014.21.167 +[Abstract](146) +[PDF](379.1KB)
We use recent work on integral forms in vertex operator algebras to construct vertex algebras over general commutative rings and Chevalley groups acting on them as vertex algebra automorphisms. In this way, we get series of vertex algebras over fields whose automorphism groups are essentially those Chevalley groups (actually, an exact statement depends on the field and involves upwards extensions of these groups by outer diagonal and graph automorphisms). In particular, given a prime power $q$, we realize each finite simple group which is a Chevalley or Steinberg variations over $\mathbb{F}_q$ as "most of'' the full automorphism group of a vertex algebra over $\mathbb{F}_q$. These finite simple groups are \[ A_n(q), B_n(q), C_n(q), D_n(q), E_6(q), E_7(q), E_8(q), F_4(q), G_2(q) \] \[ \text{and } ^{2}A_n(q), ^{2}D_n(q), ^{3}D_4(q), ^{2}E_6(q), \] where $q$ is a prime power.
    Also, we define certain reduced VAs. In characteristics 2 and 3, there are exceptionally large automorphism groups. A covering algebra idea of Frohardt and Griess for Lie algebras is applied to the vertex algebra situation.
    We use integral form and covering procedures for vertex algebras to complete the modular moonshine program of Borcherds and Ryba for proving an embedding of the sporadic group $F_3$ of order $2^{15}3^{10}5^3 7^2 13{\cdot }19{\cdot} 31$ in $E_8(3)$.
On the injectivity radius in Hofer's geometry
François Lalonde and Yasha Savelyev
2014, 21: 177-185 doi: 10.3934/era.2014.21.177 +[Abstract](109) +[PDF](329.3KB)
In this note we consider the following conjecture: given any closed symplectic manifold $M$, there is a sufficiently small real positive number $\rho$ such that the open ball of radius $\rho$ in the Hofer metric centered at the identity on the group of Hamiltonian diffeomorphisms of $M$ is contractible, where the retraction takes place in that ball (this is the strong version of the conjecture) or inside the ambient group of Hamiltonian diffeomorphisms of $M$ (this is the weak version of the conjecture). We prove several results that support the weak form of the conjecture.
Globally subanalytic CMC surfaces in $\mathbb{R}^3$
J. L. Barbosa, L. Birbrair, M. do Carmo and A. Fernandes
2014, 21: 186-192 doi: 10.3934/era.2014.21.186 +[Abstract](110) +[PDF](313.0KB)
We prove that globally subanalytic nonsingular CMC surfaces of $\mathbb{R}^3$ are only planes, round spheres, or right circular cylinders.

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