Doubly Connected Minimal Surfaces and Extremal Harmonic Mappings

The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected domains are where one first observes nontrivial conformal invariants. Herbert Gr?tzsch and Johannes C.C.?Nitsche addressed this issue for quasiconformal and harmonic mappings, respectively. Combining these concepts we obtain sharp estimates for quasiconformal harmonic mappings between doubly connected domains. We then apply our results to the Cauchy problem for minimal surfaces, also known as the Bj?rling problem. Specifically, we obtain a sharp estimate of the modulus of a doubly connected minimal surface that evolves from its inner boundary with a given initial slope.     

On the existence of spines for ${\mathbb{Q}}${\mathbb{Q}}-rank 1 groups

Let X = Γ \G/ K be an arithmetic quotient of a symmetric space of non-compact type. In the case that G has -rank 1, we construct Γ-equivariant deformation retractions of D = G/K onto a set D₀. We prove that D₀ is a spine, having dimension equal to the virtual cohomological dimension of Γ. In fact, there is a (k − 1)-parameter family of such deformation retractions, where k is the number of Γ -conjugacy classes of rational parabolic subgroups of G. The construction of the spine also gives a way to construct an exact fundamental domain for Γ.     

On lipschitz continuity of quasiconformalmappings in space

Sufficient conditions for a spatial quasiconformal mapping f to be Lipschitz or weak Lipschitz continuous at a prescribed point are given. The proofs of these results are based on new growth estimates for the conformal modulus under quasiconformal mappings f. The estimates are written in terms of integral means of the co-called pointwise angular dilatation coefficient.     

A note on plurisubharmonic defining functions in \mathbbCn{\mathbb{C}^{n}}

Let , n ≥ 3, be a smoothly bounded domain. Suppose that Ω admits a smooth defining function which is plurisubharmonic on the boundary of Ω. Then a Diederich–Forn?ss exponent can be chosen arbitrarily close to 1, and the closure of Ω admits a Stein neighborhood basis. Research of J. E. Forn?ss was partially supported by an NSF grant. Research of A.-K. Herbig was supported by FWF grant P19147.     

Orthogonal complex structures on domains in \mathbb R4{\mathbb {R}^4}

An orthogonal complex structure on a domain in is a complex structure which is integrable and is compatible with the Euclidean metric. This gives rise to a first order system of partial differential equations which is conformally invariant. We prove two Liouville-type uniqueness theorems for solutions of this system, and use these to give an alternative proof of the classification of compact locally conformally flat Hermitian surfaces first proved by Pontecorvo. We also give a classification of non-degenerate quadrics in under the action of the conformal group SO _°(1, 5). Using this classification, we show that generic quadrics give rise to orthogonal complex structures defined on the complement of unknotted solid tori which are smoothly embedded in . To Nigel Hitchin on the occasion of his 60th birthday. This work was partially supported by MIUR (Metriche Riemanniane e Varietà Differenziabili, PRIN05) and NSF Grant DMS-0503506.     

Low complexity functions and convex sets in \mathbbZk\mathbb{Z}^k

In 1938 Morse and Hedlund proved that a function is periodic if the number of different n-blocks with does not exceed n for some n. In 1940 they studied such functions f with for all positive integers n. These are closely related to Sturmian sequences, which occur in many branches of mathematics, computer science and physics. Recently the authors studied k-dimensional functions with , where is the set of different vectors with for a given configuration . In this paper, we characterize functions satisfying for all configurations . Our proof requires a separation theorem for convex sets of lattice points, which may be of independent interest. Received July 20, 1998     

Harmonic mean curvature lines on surfaces immersed in $ \mathbb{R}^{3} $ \mathbb{R}^{3}

Consider oriented surfaces immersed in . Associated to them, here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature , given by the product of the principal curvatures k ₁, k ₂ is positive. The leaves of the foliations are the lines of harmonic mean curvature, also called characteristic or diagonal lines, along which the normal curvature of the immersion is given by , where is the arithmetic mean curvature. That is, is the harmonic mean of the principal curvatures k ₁, k ₂ of the immersion. The singularities of the foliations are the umbilic points and parabolic curves, where k ₁ = k ₂ and , respectively.Here are determined the structurally stable patterns of harmonic mean curvature lines near the umbilic points, parabolic curves and harmonic mean curvature cycles, the periodic leaves of the foliations. The genericity of these patterns is established.This provides the three essential local ingredients to establish sufficient conditions, likely to be also necessary, for Harmonic Mean Curvature Structural Stability of immersed surfaces. This study, outlined towards the end of the paper, is a natural analog and complement for that carried out previously by the authors for the Arithmetic Mean Curvature and the Asymptotic Structural Stability of immersed surfaces, [13, 14, 17], and also extended recently to the case of the Geometric Mean Curvature Configuration [15].The first author was partially supported by FUNAPE/UFG. Both authors are fellows of CNPq. This work was done under the project PRONEX/FINEP/MCT - Conv. 76.97.1080.00 - Teoria Qualitativa das Equações Diferenciais Ordinárias and CNPq - Grant 476886/2001-5.     

Quasilinear elliptic equations on \mathbbRN{\mathbb{R}^{N}} with singular potentials and bounded nonlinearity

We study the existence and multiplicity of nontrivial radial solutions of the quasilinear equation

Covering and packing in \Bbb Zn{\Bbb Z}^n and \Bbb Rn{\Bbb R}^n, (I)

Given a finite subset A{\cal A} of an additive group \Bbb G{\Bbb G} such as \Bbb Zⁿ{\Bbb Z}^n or \Bbb Rⁿ{\Bbb R}^n , we are interested in efficient covering of \Bbb G{\Bbb G} by translates of A{\cal A} , and efficient packing of translates of A{\cal A} in \Bbb G{\Bbb G} . A set S ì \Bbb G{\cal S} \subset {\Bbb G} provides a covering if the translates A + s{\cal A} + s with s ? Ss \in {\cal S} cover \Bbb G{\Bbb G} (i.e., their union is \Bbb G{\Bbb G} ), and the covering will be efficient if S{\cal S} has small density in \Bbb G{\Bbb G} . On the other hand, a set S ì \Bbb G{\cal S} \subset {\Bbb G} will provide a packing if the translated sets A + s{\cal A} + s with s ? Ss \in {\cal S} are mutually disjoint, and the packing is efficient if S{\cal S} has large density. In the present part (I) we will derive some facts on these concepts when \Bbb G = \Bbb Zⁿ{\Bbb G} = {\Bbb Z}^n , and give estimates for the minimal covering densities and maximal packing densities of finite sets A ì \Bbb Zⁿ{\cal A} \subset {\Bbb Z}^n . In part (II) we will again deal with \Bbb G = \Bbb Zⁿ{\Bbb G} = {\Bbb Z}^n , and study the behaviour of such densities under linear transformations. In part (III) we will turn to \Bbb G = \Bbb Rⁿ{\Bbb G} = {\Bbb R}^n .     

Rational holomorphic maps from \mathbbB2 \mathbb{B}^2 into \mathbbBN \mathbb{B}^N with degree 2

Rational proper holomorphic maps from the unit ball in ℂ² into the unit ball ℂ ^N with degree 2 are studied. Any such map must be equivalent to one of the four types of maps. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Tameness of Pseudovariety Joins Involving \sf R{\sf R}

In this paper, we establish several decidability results for pseudovariety joins of the form \sf Vú\sf W{\sf V}\vee{\sf W} , where \sf V{\sf V} is a subpseudovariety of \sf J{\sf J} or the pseudovariety \sf R{\sf R} . Here, \sf J{\sf J} (resp. \sf R{\sf R} ) denotes the pseudovariety of all J{\cal J} -trivial (resp. ?{\cal R} -trivial) semigroups. In particular, we show that the pseudovariety \sf Vú\sf W{\sf V}\vee{\sf W} is (completely) κ-tame when \sf V{\sf V} is a subpseudovariety of \sf J{\sf J} with decidable κ-word problem and \sf W{\sf W} is (completely) κ-tame. Moreover, if \sf W{\sf W} is a κ-tame pseudovariety which satisfies the pseudoidentity x₁ ⋯ x_ry^ω+1zt^ω = x₁ ⋯ x_ryzt^ω, then we prove that \sf Rú\sf W{\sf R}\vee{\sf W} is also κ-tame. In particular the joins \sf Rú\sf Ab{\sf R}\vee{\sf Ab} , \sf Rú\sf G{\sf R}\vee{\sf G} , \sf Rú\sf OCR{\sf R}\vee{\sf OCR} , and \sf Rú\sf CR{\sf R}\vee{\sf CR} are decidable.     

Hölder continuity of quasiconformal harmonic mappings from the unit ball to a spatial domain with C1 boundary

We prove that every quasiconformal mapping from the harmonic $\beta$-Bloch space between the unit ball and a spatial domain with $C^1$ boundary is globally $\alpha$-Hölder continuous for $\alpha <1?\beta$, with the Hölder coefficient that does not depend neither on the mapping nor on $\beta$. An analogous result also holds for Lipschitz continuous, quasiconformal harmonic mappings for $\alpha <1$. This is an approach towards the extension of some results from the complex plane obtained by Warschawski (1951) for conformal mappings and Kalaj (2022) for quasiconformal harmonic mappings.     

Idempotent multipliers for LP (\Bbb R)(\Bbb R)

The algebra B_p(\Bbb R){\cal B}_p({\Bbb R}), p ? (1,￥)\{2}p\in (1,\infty )\setminus \{2\}, consisting of all measurable sets in \Bbb R{\Bbb R} whose characteristic function is a Fourier p-multiplier, forms an algebra of sets containing many interesting and non-trivial elements (e.g. all intervals and their finite unions, certain periodic sets, arbitrary countable unions of dyadic intervals, etc.). However, B_p(\Bbb R){\cal B}_p({\Bbb R}) fails to be a s\sigma -algebra. It has been shown by V. Lebedev and A. Olevskii [4] that if E ? B_p(\Bbb R)E\in {\cal B}_p({\Bbb R}), then E must coincide a.e. with an open set, a remarkable topological constraint on E. In this note we show if $2 < p < \infty $2 < p < \infty , then there exists E ? B_p(\Bbb R)E\in {\cal B}_p({\Bbb R}) which is not in B_q(\Bbb R){\cal B}_q({\Bbb R}) for any q > pq>p.     

Random self-affine multifractal Sierpinski sponges in \mathbbRd{\mathbb{R}^d}

In this paper we study the Hausdorff and packing dimensions and the Rényi dimensions of random self-affine multifractal Sierpinski sponges in \mathbbR^d{\mathbb{R}^{d}}.     

On certain functional equations related to Jordan triple (q, f){(\theta, \phi)}-derivations on semiprime rings

The main purpose of this paper is to prove the following result. Let R be a 2-torsion free semiprime ring with symmetric Martindale ring of quotients Q _s and let q{\theta} and f{\phi} be automorphisms of R. Suppose T:R? R{T:R\rightarrow R} is an additive mapping satisfying the relation T(xyx)=T(x)q(y)q(x)-f(x)T(y)q(x)+f(x)f(y)T(x){T(xyx)=T(x)\theta (y)\theta (x)-\phi (x)T(y)\theta (x)+\phi (x)\phi (y)T(x)}, for all pairs x,y ? R{x,y\in R}. In this case T is of the form 2T(x)=qq(x)+f(x)q{2T(x)=q\theta (x)+\phi (x)q}, for all x ? R{x\in R} and some fixed element q ? Q_s{q\in Q_{s}}.     

Integrality of L2L^2-Betti numbers

The Atiyah conjecture predicts that the -Betti numbers of a finite CW-complex with torsion-free fundamental group are integers. We establish the Atiyah conjecture, under the condition that it holds for G and that is a normal subgroup, for amalgamated free products . Here F is a free group and is an arbitrary semi-direct product. This includes free products G*F and semi-direct products . We also show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of groups for which it is true. As a corollary it holds for positive 1-relator groups with torsion free abelianization. Putting everything together we establish a new (bigger) class of groups for which the Atiyah conjecture holds, which contains all free groups and in particular is closed under taking subgroups, direct sums, free products, extensions with torsion-free elementary amenable quotient or with free quotient, and under certain direct and inverse limits. Received: 22 August 1998/ Revised: 10 Jannary 2000 / Published online: 28 June 2000     

A graph associated with the p\pi-character degrees of a group

Let G be a group and $\pi$ be a set of primes. We consider the set ${\rm cd}^{\pi}(G)$ of character degrees of G that are divisible only by primes in $\pi$. In particular, we define $\Gamma^{\pi}(G)$ to be the graph whose vertex set is the set of primes dividing degrees in ${\rm cd}^{\pi}(G)$. There is an edge between p and q if pq divides a degree $a \in {\rm cd}^{\pi}(G)$. We show that if G is $\pi$-solvable, then $\Gamma^{\pi}(G)$ has at most two connected components.     

On pseudo-random subsets of \mathbb Z n{\mathbb Z _n}

The notion of pseudo-randomness of subsets of \mathbb Z_n{\mathbb Z_n} is defined, and the measures of pseudo-randomness are introduced. Then a construction (based on the use of hybrid character sums) will be presented for subsets of \mathbb Z_p{\mathbb Z_p} with strong pseudo-random properties.