Geometric rigidity for sequences of W^{2,2} conformal immersions

We analyse sequences of discs conformally immersed in $ \mathbb{R }^ n$ with energy $ \int _{ D} |A_k |^ 2 \le \gamma _n$ , where $ \gamma _n = 8\pi $ if $ n=3$ and $ \gamma _n = 4 \pi $ when $n\ge 4$ . We show that if such sequences do not weakly converge to a conformal immersion, then by a sequence of dilations we obtain a complete minimal surface with bounded total curvature, either Enneper’s minimal surface if $ n=3$ or Chen’s minimal graph if $ n \ge 4$ . In the papers, (Kuwert and Li, Comm Anal Geom 20(2), 313–340, 2012; Rivière, Adv Calculus Variations 6(1), 1–31, 2013) it was shown that if a sequence of immersed tori diverges in moduli space then $\liminf _ {k\rightarrow \infty } \mathcal W ( f_k )\ge 8\pi $ . We apply the above analysis to show that in $ \mathbb{R }^3$ if the sequence diverges so that $ \lim _{ k \rightarrow \infty } \mathcal W (f_k) =8\pi $ then there exists a sequence of Möbius transforms $ \sigma _{k}$ such that $ \sigma _k\circ f _k$ converges weakly to a catenoid.     

Global solutions to the Navier–Stokes-{\bar \omega} and related models with rough initial data

We establish the global well-posedness of the Navier–Stokes- ${\bar \omega}$ model with initial data ${u_0 \in H^{1-s}(\mathbb{R}^3)}$ with ${0 < s < \frac{1}{2}}$ which improves the existence results in Fan and Zhou (Appl Math Lett 24:1915–1918, 2011), Layton et al. (Commun Pure Appl Anal 10:1763–1777, 2011) where the initial data are required belonging to ${H^2(\mathbb{R}^3)}$ . We also obtain the similar results for a family of Navier–Stokes-α-like and magnetohydrodynamic-α models.     

Global symplectic coordinates on gradient Kähler–Ricci solitons

A classical result of McDuff [14] asserts that a simply connected complete Kähler manifold $(M,g,\omega )$ with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism $\Psi \ : M \rightarrow \mathbb{R }^{2n}$ (where $n$ is the complex dimension of $M$ ), satisfying the following property (proved by E. Ciriza in [4]): the image $\Psi (T)$ of any complex totally geodesic submanifold $T\subset M$ through the point $p$ such that $\Psi (p)=0$ , is a complex linear subspace of $\mathbb C ^n\simeq \mathbb{R }^{2n}$ . The aim of this paper is to exhibit, for all positive integers $n$ , examples of $n$ -dimensional complete Kähler manifolds with non-negative sectional curvature globally symplectomorphic to $\mathbb{R }^{2n}$ through a symplectomorphism satisfying Ciriza’s property.     

Uniformly Quasiregular Maps on the Compactified Heisenberg Group

We show the existence of a non-injective uniformly quasiregular mapping acting on the one-point compactification $\bar{ {\mathbb{H}}}^{1}={\mathbb{H}}^{1}\cup\{\infty\}$ of the Heisenberg group ?¹ equipped with a sub-Riemannian metric. The corresponding statement for arbitrary quasiregular mappings acting on sphere ${\mathbb{S}}^{n} $ was proven by Martin (Conform. Geom. Dyn. 1:24?C27, 1997). Moreover, we construct uniformly quasiregular mappings on $\bar{ {\mathbb{H}}}^{1}$ with large-dimensional branch sets. We prove that for any uniformly quasiregular map g on $\bar{ {\mathbb{H}}}^{1}$ there exists a measurable CR structure ?? which is equivariant under the semigroup ?? generated by g. This is equivalent to the existence of an equivariant horizontal conformal structure.     

Erratum to: Algèbres de lie quasi-réductives

In Corollary 12(ii) and Theorem 13(v) of [1] we omitted the hypothesis dim $ \mathfrak{z}\leq 1 $ . Moreover, in some places the symbol $ \mathbb{K} $ must be replaced by the symbol $ {{\mathbb{K}}^{\times }} $ .     

On a problem concerning the Banach algebra generated by the maps n\mapsto\lambda^{\binom{n}{k}}

In Jabbari and Namioka (Milan J. Math. 78:503?C522, 2010), the authors characterized the spectrum M(W) of the Weyl algebra W, i.e. the norm closure of the algebra generated by the family of functions $\{n\mapsto x^{n^{k}}; x\in\mathbb{T}, k\in\mathbb{N}\}$ , ( $\mathbb{T}$ the unit circle), with a closed subgroup of $E(\mathbb{T})^{\mathbb{N}}$ where $E(\mathbb{T})$ denotes the family of the endomorphisms of the multiplicative group $\mathbb{T}$ . But the size of M(W) in $E(\mathbb{T})^{\mathbb{N}}$ as well as the induced group operation were left as a problem. In this paper, we will give a solution to this problem.     

On Radial Functions and Distributions and Their Fourier Transforms

We give formulas relating the Fourier transform of a radial function in $\mathbb{R}^{n}$ and the Fourier transform of the same function in $\mathbb{R}^{n+1}$ , completing the analysis of Grafakos and Teschl (J. Fourier Anal. Appl. 19:167–179, 2013) where the case of $\mathbb{R}^{n}$ and $\mathbb{R}^{n+2}$ was considered.     

Boundary Values of Harmonic Functions in Spaces of Triebel–Lizorkin Type

Triebel (J Approx Theory 35:275–297, 1982; 52:162–203, 1988) investigated the boundary values of the harmonic functions in spaces of the Triebel–Lizorkin type ${\mathcal F^{\alpha,q}_{p}}$ on ${\mathbb{R}^{n+1}_+}$ by finding an characterization of the homogeneous Triebel–Lizorkin space ${{\bf \dot{F}}^{\alpha,q}_p}$ via its harmonic extension, where ${0 < p < \infty, 0 < q \leq \infty}$ , and ${\alpha < {\rm min}\{-n/p, -n/q\}}$ . In this article, we extend Triebel’s result to α < 0 and ${0 < p, q \leq \infty}$ by using a discrete version of reproducing formula and discretizing the norms in both ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf{\dot{F}}}^{\alpha,q}_p}$ . Furthermore, for α < 0 and ${1 < p,q \leq \infty}$ , the mapping from harmonic functions in ${\mathcal{F}^{\alpha,q}_{p}}$ to their boundary values forms a topological isomorphism between ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf \dot{F}}^{\alpha,q}_p}$ .     

Covering functions by countably many functions from some families

Let $ \mathcal{A} $ be a nonempty family of functions from $ \mathbb{R} $ to $ \mathbb{R} $ . A function $ f:\mathbb{R}\to \mathbb{R} $ is said to be strongly countably $ \mathcal{A} $ -function if there is a sequence (f _n ) of functions from $ \mathcal{A} $ such that $ \mathrm{Gr}(f)\subset {\cup_n}\mathrm{Gr}\left( {{f_n}} \right) $ (Gr(f) denotes the graph of f). If $ \mathcal{A} $ is the family of all continuous functions, the strongly countable $ \mathcal{A} $ -functions are called strongly countably continuous and were investigated in [Z. Grande and A. Fatz-Grupka, On countably continuous functions, Tatra Mt. Math. Publ., 28:57–63, 2004], [G. Horbaczewska, On strongly countably continuous functions, Tatra Mt. Math. Publ., 42:81–86, 2009], and [T.A. Natkaniec, On additive countably continuous functions, Publ. Math., 79(1–2):1–6, 2011]. In this article, we prove that the families $ \mathcal{A}\left( \mathbb{R} \right) $ of all strongly countably $ \mathcal{A} $ -functions are closed with respect to some operations in dependence of analogous properties of the families $ \mathcal{A} $ , and, in particular, we show some properties of strongly countably differentiable functions, strongly countably approximately continuous functions, and strongly countably quasi-continuous functions.     

Defect Functions of Holomorphic Contractive Operator Functions and the Scattering Suboperator Through the Internal Channels of a System: Part II

Let $\theta (\zeta )$ be a Schur operator function, i.e., it is defined on the unit disk ${\mathbb D}\,{:=}\,\{\zeta \in {\mathbb C}: |\zeta | < 1\}$ and its values are contractive operators acting from one Hilbert space into another one. In the first part of the paper the outer and $*$ -outer Schur operator functions $\varphi (\zeta )$ and $\psi (\zeta )$ which describe respectively the deviations of the function $\theta (\zeta )$ from inner and $*$ -inner operator functions are studied. If $\varphi (\zeta )\ne 0$ , then it means that in the scattering system for which $\theta (\zeta )$ is the transfer function a portion of “information” comes inward the system and does not go outward, i.e., it is left in the internal channels of the system ([11, Sect. 6]). The function $\psi (\zeta )$ has the analogous property. For this reason these functions are called defect ones of the function $\theta (\zeta )$ . The explicit form of the defect functions $\varphi (\zeta )$ and $\psi (\zeta )$ is obtained and the analytic connection of these functions with the function $\theta (\zeta )$ is described ([11, Sect. 3 and Sect. 5]). The operator functions $\left( \begin{matrix} \varphi (\zeta ) \\ \theta (\zeta ) \end{matrix}\right) $ and $(\psi (\zeta ), \theta (\zeta ))$ are Schur functions as well ([11, Sect. 3]). It is important that there exists the unique contractive operator function $\chi (t),t\in \partial {\mathbb D}$ , such that the operator function $\left( \begin{matrix} \chi (t) &{} \varphi (t) \\ \psi (t) &{} \theta (t) \end{matrix}\right) ,t\in \partial {\mathbb D},$ is also contractive (Sect. 6). The second part of the paper is devoted to introducing and studying the properties of the function $\chi (t)$ . Specifically, it is shown that the function $\chi (t)$ is the scattering suboperator through the internal channels of the scattering system for which $\theta (\zeta )$ is the transfer function (Sect. 6).     

Extension of a theorem of Shi and Tam

In this note, we prove the following generalization of a theorem of Shi and Tam (J Differ Geom 62:79–125, 2002): Let (Ω, g) be an n-dimensional (n ≥ 3) compact Riemannian manifold, spin when n?>?7, with non-negative scalar curvature and mean convex boundary. If every boundary component Σ _i has positive scalar curvature and embeds isometrically as a mean convex star-shaped hypersurface ${{\hat \Sigma}_i \subset \mathbb{R}^n}$ , then $$ \int\limits_{\Sigma_i} H \ d \sigma \le \int\limits_{{\hat \Sigma}_i} \hat{H} \ d {\hat \sigma} $$ where H is the mean curvature of Σ _i in (Ω, g), ${\hat{H}}$ is the Euclidean mean curvature of ${{\hat \Sigma}_i}$ in ${\mathbb{R}^n}$ , and where d σ and ${d {\hat \sigma}}$ denote the respective volume forms. Moreover, equality holds for some boundary component Σ _i if, and only if, (Ω, g) is isometric to a domain in ${\mathbb{R}^n}$ . In the proof, we make use of a foliation of the exterior of the ${\hat \Sigma_i}$ ’s in ${\mathbb{R}^n}$ by the ${\frac{H}{R}}$ -flow studied by Gerhardt (J Differ Geom 32:299–314, 1990) and Urbas (Math Z 205(3):355–372, 1990). We also carefully establish the rigidity statement in low dimensions without the spin assumption that was used in Shi and Tam (J Differ Geom 62:79–125, 2002).     

Local behavior and hitting probabilities of the \text{ Airy}_1 process

We obtain a formula for the $n$ -dimensional distributions of the $\text{ Airy}_1$ process in terms of a Fredholm determinant on $L^2(\mathbb{R })$ , as opposed to the standard formula which involves extended kernels, on $L^2(\{1,\dots ,n\}\times \mathbb{R })$ . The formula is analogous to an earlier formula of Prähofer and Spohn (J Stat Phys 108(5–6):1071–1106, 2002) for the $\text{ Airy}_2$ process. Using this formula we are able to prove that the $\text{ Airy}_1$ process is Hölder continuous with exponent $\frac{1}{2}$ —and that it fluctuates locally like a Brownian motion. We also explain how the same methods can be used to obtain the analogous results for the $\text{ Airy}_2$ process. As a consequence of these two results, we derive a formula for the continuum statistics of the $\text{ Airy}_1$ process, analogous to that obtained in Corwin et al. (Commun Math Phys 2011, to appear) for the $\text{ Airy}_2$ process.     

Weyl modules and q-Whittaker functions

Let $G$ be a semi-simple simply connected group over $\mathbb {C}$ . Following Gerasimov et al. (Comm Math Phys 294:97–119, 2010) we use the $q$ -Toda integrable system obtained by quantum group version of the Kostant–Whittaker reduction (cf. Etingof in Am Math Soc Trans Ser 2:9–25, 1999, Sevostyanov in Commun Math Phys 204:1–16, 1999) to define the notion of $q$ -Whittaker functions $\varPsi _{\check{\lambda }}(q,z)$ . This is a family of invariant polynomials on the maximal torus $T\subset G$ (here $z\in T$ ) depending on a dominant weight $\check{\lambda }$ of $G$ whose coefficients are rational functions in a variable $q\in \mathbb {C}^*$ . For a conjecturally the same (but a priori different) definition of the $q$ -Toda system these functions were studied by Ion (Duke Math J 116:1–16, 2003) and by Cherednik (Int Math Res Notices 20:3793–3842, 2009) [we shall denote the $q$ -Whittaker functions from Cherednik (Int Math Res Notices 20:3793–3842, 2009) by $\varPsi '_{\check{\lambda }}(q,z)$ ]. For $G=SL(N)$ these functions were extensively studied in Gerasimov et al. (Comm Math Phys 294:97–119, 2010; Comm Math Phys 294:121–143, 2010; Lett Math Phys 97:1–24, 2011). We show that when $G$ is simply laced, the function $\hat{\varPsi }_{\check{\lambda }}(q,z)=\varPsi _{\check{\lambda }}(q,z)\cdot {\prod \nolimits _{i\in I}\prod \nolimits _{r=1}^{\langle \alpha _i,\check{\uplambda }\rangle }(1-q^r)}$ (here $I$ denotes the set of vertices of the Dynkin diagram of $G$ ) is equal to the character of a certain finite-dimensional $G[[{\mathsf {t}}]]\rtimes \mathbb {C}^*$ -module $D(\check{\lambda })$ (the Demazure module). When $G$ is not simply laced a twisted version of the above statement holds. This result is known for $\varPsi _{\check{\lambda }}$ replaced by $\varPsi '_{\check{\lambda }}$ (cf. Sanderson in J Algebraic Combin 11:269–275, 2000 and Ion in Duke Math J 116:1–16, 2003); however our proofs are algebro-geometric [and rely on our previous work (Braverman, Finkelberg in Semi-infinite Schubert varieties and quantum $K$ -theory of flag manifolds, arXiv/1111.2266, 2011)] and thus they are completely different from Sanderson (J Algebraic Combin 11:269–275, 2000) and Ion (Duke Math J 116:1–16, 2003) [in particular, we give an apparently new algebro-geometric interpretation of the modules $D(\check{\lambda })]$ .     

A limiting case for the divergence equation

We consider the equation $\text{ div}\,\mathbb{Y }=f$ , with $f$ a zero average function on the torus $\mathbb{T }^d$ . In their seminal paper, Bourgain and Brezis [J Am Math Soc 16(2):393–426, 2003 (electronic)] proved the existence of a solution $\mathbb{Y }\in W^{1,d}\cap L^\infty $ for a datum $f\in L^d$ . We extend their result to the critical Sobolev spaces $W^{s,p}$ with $(s+1)p=d$ and $p\ge 2$ . More generally, we prove a similar result in the scale of Triebel–Lizorkin spaces. We also consider the equation $\text{ div} \,\mathbb{Y }=f$ in a bounded domain $\varOmega $ subject to zero Dirichlet boundary condition.     

Quadratic functions with prescribed spectra

We study a class of quadratic p-ary functions ${{\mathcal{F}}_{p,n}}$ from ${\mathbb{F}_{p^n}}$ to ${\mathbb{F}_p, p \geq 2}$ , which are well-known to have plateaued Walsh spectrum; i.e., for each ${b \in \mathbb{F}_{p^n}}$ the Walsh transform ${\hat{f}(b)}$ satisfies ${|\hat{f}(b)|^2 \in \{ 0, p^{(n+s)}\}}$ for some integer 0 ≤ s ≤ n ? 1. For various types of integers n, we determine possible values of s, construct ${{\mathcal{F}}_{p,n}}$ with prescribed spectrum, and present enumeration results. Our work generalizes some of the earlier results, in characteristic two, of Khoo et. al. (Des Codes Cryptogr, 38, 279–295, 2006) and Charpin et al. (IEEE Trans Inf Theory 51, 4286–4298, 2005) on semi-bent functions, and of Fitzgerald (Finite Fields Appl 15, 69–81, 2009) on quadratic forms.     

Uniqueness of the Group Measure Space Decomposition for Popa’s {\mathcal{HT}} Factors

We prove that if ${\Gamma\curvearrowright (X, \mu)}$ is a free ergodic rigid (in the sense of Popa in Ann Math 163:809–889, 2006) probability measure preserving action of a group Γ with positive first ${\ell^2}$ -Betti number, then the II₁ factor ${L^{\infty}(X)\rtimes\Gamma}$ has a unique group measure space Cartan subalgebra, up to unitary conjugacy. We deduce that many ${\mathcal{HT}}$ factors, including the II₁ factors associated with the usual actions ${\Gamma\curvearrowright \mathbb{T^2}}$ and ${\Gamma\curvearrowright}$ ${{\rm SL}_2(\mathbb R)/{\rm SL}_2(\mathbb Z)}$ , where Γ is a non-amenable subgroup of ${{\rm SL}_2(\mathbb Z)}$ , have a unique group measure space decomposition.     

Contragenic Functions of Three Variables

It is shown that harmonic functions from a simply connected domain in $\mathbb R ^3$ to $\mathbb R ^3$ cannot always be expressed as a sum of a monogenic (hyperholomorphic) function and an antimonogenic function, in contrast to the situation for complex numbers or for quaternions. Harmonic functions orthogonal in $L_2$ to all such sums are termed “contragenic” and their properties are studied. A “Bergman kernel” is derived, whose corresponding operator vanishes precisely on the contragenic functions. A graded orthonormal basis is given for the space of contragenic functions in the ball $\mathbb B ^3$ .     

Toric partial density functions and stability of toric varieties

Let $(L, h)\rightarrow (X, \omega )$ denote a polarized toric Kähler manifold. Fix a toric submanifold $Y$ and denote by $\hat{\rho }_{tk}:X\rightarrow \mathbb {R}$ the partial density function corresponding to the partial Bergman kernel projecting smooth sections of $L^k$ onto holomorphic sections of $L^k$ that vanish to order at least $tk$ along $Y$ , for fixed $t>0$ such that $tk\in \mathbb {N}$ . We prove the existence of a distributional expansion of $\hat{\rho }_{tk}$ as $k\rightarrow \infty $ , including the identification of the coefficient of $k^{n-1}$ as a distribution on $X$ . This expansion is used to give a direct proof that if $\omega $ has constant scalar curvature, then $(X, L)$ must be slope semi-stable with respect to $Y$ (cf. Ross and Thomas in J Differ Geom 72(3): 429–466, 2006). Similar results are also obtained for more general partial density functions. These results have analogous applications to the study of toric K-stability of toric varieties.     

Uniqueness of \mathbb{C}^{ * } - and \mathbb{C}_{{\text{ + }}} -actions on Gizatullin Surfaces

A Gizatullin surface is a normal affine surface V over $ \mathbb{C} $ , which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of $ \mathbb{C}^{ * } $ -actions and $ \mathbb{A}^{{\text{1}}} $ -fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with $ \mathbb{C}_{{\text{ + }}} $ -actions on V considered up to a “speed change”. Non-Gizatullin surfaces are known to admit at most one $ \mathbb{A}^{1} $ -fibration V → S up to an isomorphism of the base S. Moreover, an effective $ \mathbb{C}^{ * } $ -action on them, if it does exist, is unique up to conjugation and inversion t $ \mapsto $ t ^?1 of $ \mathbb{C}^{ * } $ . Obviously, uniqueness of $ \mathbb{C}^{ * } $ -actions fails for affine toric surfaces. There is a further interesting family of nontoric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of $ \mathbb{C}^{ * } $ -actions and $ \mathbb{A}^{{\text{1}}} $ -fibrations, see, e.g., [FKZ1]. In the present paper we obtain a criterion as to when $ \mathbb{A}^{{\text{1}}} $ -fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base $ S \cong \mathbb{A}^{{\text{1}}} $ . We exhibit as well large subclasses of Gizatullin $ \mathbb{C}^{ * } $ -surfaces for which a $ \mathbb{C}^{ * } $ -action is essentially unique and for which there are at most two conjugacy classes of $ \mathbb{A}^{{\text{1}}} $ -fibrations over $ \mathbb{A}^{{\text{1}}} $ .