Rigidity of Area-Minimizing Hyperbolic Surfaces in Three-Manifolds

We prove that if M is a three-manifold with scalar curvature greater than or equal to ?2 and Σ?M is a two-sided compact embedded Riemann surface of genus greater than 1 which is locally area-minimizing, then the area of Σ is greater than or equal to 4π(g(Σ)?1), where g(Σ) denotes the genus of Σ. In the equality case, we prove that the induced metric on Σ has constant Gauss curvature equal to ?1 and locally M splits along Σ. We also obtain a rigidity result for cylinders (I×Σ,dt ²+g _Σ), where I=[a,b]?? and g _Σ is a Riemannian metric on Σ with constant Gauss curvature equal to ?1.     

Trattazione rapida dei problemi al contorno relativi alle equazioni differenziali lineari a derivate parziali

Let Σ andS be two real Hilbert spaces and Σ₀ a subspace of Σ. Moreover, supposeT:S→Σ be a bounded linear operator whose rangeT (S) is contained in Σ₀, andE:S→Σ be a linear operator such that the productE T:S→S is a bounded operator with a closed range. In this framework we present an artifice from which the alternative theorem for the equationE u=q(u?Σ₀,q ?S) follows. It is worthwhile to note thatEu=q may represent a boundary value problem for elliptic equations.     

Functional envelope of Cantor spaces

Given a topological dynamical system(X, T), where X is a compact metric space and T a continuous selfmap of X. Denote by S(X) the space of all continuous selfmaps of X with the compactopen topology. The functional envelope of(X, T) is the system(S(X), FT), where FT is defined by FT(?) = T ? ? for any ? ∈ S(X). We show that(1) If(Σ, T) is respectively weakly mixing, strongly mixing, diagonally transitive, then so is its functional envelope, where Σ is any closed subset of a Cantor set and T a selfmap of Σ;(2) If(S(Σ), F_σ) is transitive then it is Devaney chaos, where(Σ, σ) is a subshift of finite type;(3) If(Σ, T) has shadowing property, then(SU(Σ), FT) has shadowing property,where Σ is any closed subset of a Cantor set and T a selfmap of Σ;(4) If(X, T) is sensitive, where X is an interval or any closed subset of a Cantor set and T : X → X is continuous, then(SU(X), FT) is sensitive;(5) If Σ is a closed subset of a Cantor set with infinite points and T : Σ→Σ is positively expansive then the entropy ent U(FT) of the functional envelope of(Σ, T) is infinity.     

Distribution of the likelihood ratio criterion for testing Σ = Σ0, μ = μ0

The exact null distribution of the likelihood ratio criterion for testing H₀: Σ = Σ₀ and μ = μ₀ against alternatives H₁: Σ ≠ Σ₀ or μ ≠ μ₀ in N_p(μ, Σ) has been obtained as (a) a chi-square series and (b) a beta series. Percentage points have been tabulated for p = 2(1) 6, α = .005, .01, .025, .05, .1, and .25 and various values of sample size N.     

Polynomial identities and maximal subgroups of skew linear groups

Suppose that D is a division ring with center F and N is a non-central normal subgroup of GL _n (D). In this paper we generalize some known results about maximal subgroups of GL _n (D) to maximal subgroups of N. More precisely, we prove that if M is a maximal subgroup of N such that F[M] satisfies a polynomial identity and contains an algebraic element over F or and either n ≥ 2 or n = 1 and M is not abelian, then [D : F] < ∞. This research was partially supported by a grant from IPM (No. 85160047).     

Asymptotics for the standard and the Capelli identities

Let {c _n (St _k )} and {c _n (C _k )} be the sequences of codimensions of the T-ideals generated by the standard polynomial of degreek and by thek-th Capelli polynomial, respectively. We study the asymptotic behaviour of these two sequences over a fieldF of characteristic zero. For the standard polynomial, among other results, we show that the following asymptotic equalities hold:

Minimizability of developable Riemannian foliations

Let (M,F){(M,\mathcal{F})} be a closed manifold with a Riemannian foliation. We show that the secondary characteristic classes of the Molino’s commuting sheaf of (M,F){(M,\mathcal{F})} vanish if (M,F){(M,\mathcal{F})} is developable and π ₁ M is of polynomial growth. By theorems of álvarez López in (álvarez López, Ann. Global Anal. Geom., 10:179–194, 1992) and (álvarez López, Ann. Pol. Math., 64:253–265, 1996), our result implies that (M,F){(M,\mathcal{F})} is minimizable under the same conditions. As a corollary, we show that (M,F){(M,\mathcal{F})} is minimizable if F{\mathcal{F}} is of codimension 2 and π ₁ M is of polynomial growth.     

Multiplikatorsysteme der symplektischen Thetagruppe

Let Γ_θ be the subgroup of Siegel modular groupSp(n, ?) consisting of all matrices \(M = \left( {\begin{array}{*{20}c} {A B} \\ {C D} \\ \end{array} } \right)\) , such that the diagonal elements ofA ^t C andB ^t D are even. A multiplier system of weightr(∈?) is a system of complex numbers ν (M)≠0,M∈Γ_θ, such thatJ (M, Z)=ν(M) det(CZ+D) ^r is an automorphy factor (that isJ (M N, Z)=J (M, N Z) J (N, Z) forM, N ∈Sp(n,?) and $$Z \in S_n = \left\{ {Z = X + i Y \in M^{(n,n)} (\mathbb{C}); X = X^t , Y = Y^t > 0} \right\})$$ . We show that in casen≥2 such a multiplier system exists if and only if 2r∈?. A corollary of this fact is the following. From the cohomology theory of Siegel modular group we derive that in casen≥8 any Γ_θ-invariant divisor is the exact zero divisor of a modular form for Γ_θ. Therefore the zero divisor of classical theta function \(\theta (Z) = \sum\limits_{g \in \mathbb{Z}^n } {e^{\pi iZ[g]} } \) , a modular form of weight 1/2 is irreducible. In the second part of this paper we calculate the commutator factor group of Γ _{n, θ} forn≥2.     

An $ \mathfrak{X} $-crown of a finite soluble group

Let G be a finite soluble group and F_\mathfrakX(G) {\Phi_\mathfrak{X}}(G) an intersection of all those maximal subgroups M of G for which G

Finite Index Operators on Surfaces

We consider differential operators L acting on functions on a Riemannian surface, Σ, of the form $$L = \Delta+ V -a K,$$ where Δ is the Laplacian of Σ, K is the Gaussian curvature, a is a positive constant, and V∈C ^∞(Σ). Such operators L arise as the stability operator of Σ immersed in a Riemannian three-manifold with constant mean curvature (for particular choices of V and a). We assume L is nonpositive acting on functions compactly supported on Σ. If the potential, V:=c+P with c a nonnegative constant, verifies either an integrability condition, i.e., P∈L ¹(Σ) and P is nonpositive, or a decay condition with respect to a point p ₀∈Σ, i.e., |P(q)|≤M/d(p ₀,q) (where d is the distance function in Σ), we control the topology and conformal type of Σ. Moreover, we establish a Distance Lemma. We apply such results to complete oriented stable H-surfaces immersed in a Killing submersion. In particular, for stable H-surfaces in a simply-connected homogeneous space with 4-dimensional isometry group, we obtain:

• There are no complete stable H-surfaces Σ??²×?, H>1/2, so that either $K_{e}^{+}:=\max \left \{0,K_{e}\right \} \in L^{1} (\Sigma)$ or there exist a point p ₀∈Σ and a constant M so that |K _e (q)|≤M/d(p ₀,q); here K _e denotes the extrinsic curvature of Σ.
• Let $\Sigma\subset \mathbb{E}(\kappa, \tau)$ , τ≠0, be an oriented complete stable H-surface so that either ν ²∈L ¹(Σ) and 4H ²+κ≥0, or there exist a point p ₀∈Σ and a constant M so that |ν(q)|²≤M/d(p ₀,q) and 4H ²+κ>0. Then:
• In $\mathbb{S}^{3}_{\text{Berger}}$ , there are no such a stable H-surfaces.
• In Nil₃, H=0 and Σ is either a vertical plane (i.e., a vertical cylinder over a straight line in ?²) or an entire vertical graph.
• In $\widetilde{\mathrm{PSL}(2,\mathbb{R})}$ , $H=\sqrt{-\kappa }/2$ and Σ is either a vertical horocylinder (i.e., a vertical cylinder over a horocycle in ?²(κ)) or an entire graph.

     

Generalization of CS condition

Let R be an associative ring with identity. An R-module M is called an NCS module if C (M)∩S(M) = {0}, where C (M) and S(M) denote the set of all closed submodules and the set of all small submodules of M, respectively. It is clear that the NCS condition is a generalization of the well-known CS condition. Properties of the NCS conditions of modules and rings are explored in this article. In the end, it is proved that a ring R is right Σ-CS if and only if R is right perfect and right countably Σ-NCS. Recall that a ring R is called right Σ-CS if every direct sum of copies of R_R is a CS module. And a ring R is called right countably Σ-NCS if every direct sum of countable copies of R_R is an NCS module.     

An existence-convergence theorem for a class of iterative methods

LetM ₀, characterized byx _k+1=G ₀(x _k),k?0,x ₀ prescribed, be an iterative method for the solution of the operator equationF(x)=0, whereF:X → X is a given operator andX is a Banach space. Let ω:X → X be a given operator, and let the methodM _mbe characterized byx _x+1,m =G _m(x _k,m),k?0,x _0,m prescribed, where $$G_i (x) = G_0 (x) - \sum\limits_{j = 0}^{i - 1} { F'(\omega (x))^{ - 1} F(G_j (x)), i = 1, . . . ,m,} $$ in whichG ₀:X → X is a given operator andF′:X → L(X) is the Fréchet derivative ofF. Sufficient conditions for the existence of a solutionx* _m ofF(x)=0 to which the sequence (x _k,m) generated from methodM _mconverges are given, together with a rate-of-convergence estimate.     

Bounding the number of bases of a matroid

Letb(M) andc(M), respectively, be the number of bases and circuits of a matroidM. For any given minor closed class? of matroids, the following two questions, are investigated in this paper. (1) When is there a polynomial functionp(x) such thatb(M)≤p(c(m)|E(M)|) for every matroidM in?? (2) When is there a polynomial functionp(x) such thatb(M)≤p(|E(M)|) for every matroidM in?? Let us denote byM _Mn the direct sum ofn copies ofU _1,2. We prove that the answer to the first question is affirmative if and only if someM _Mn is not in?. Furthermore, if all the members of? are representable over a fixed finite field, then we prove that the answer to the second question is affirmative if and only if, also, someM _Mn is not in?.     

On the unique continuation problem for CR mappings into nonminimal hypersurfaces

We prove the following results on the unique continuation problem for CR mappings between real smooth hypersurfaces in ? ⁿ . If the CR mappingH extends holomorphically to one side of the source manifoldM near the pointp ₀ εM, the target manifoldM′ contains a holomorphic hypersurface σ′ throughp′₀ =H(p ₀ (i.e.,M′ is nonminimal atp′ ₀), andH(M) ? Σ′ (forcingM to be nonminimal atp ₀), then the transversal component ofH is not flat atp ₀. Furthermore, we show that the assumption thatH extends holomorphically to one side ofM cannot be removed in general. Indeed, we give an example of a smooth CR mappingH, withM, M′ ? ?², real analytic and of infinite type atp ₀ andp′₀ respectively (without being Levi flat), such thatH(M) ? Σ′ but the transversal component ofH is flat atp ₀ (in particular,H is not real analytic!). However, we show that ifM andM′ are assumed to be real analytic, and if the sourceM is “sufficiently far from being Levi flat” in a certain sense (so as to exclude the above mentioned counterexample) then the assumption thatH extends holomorphically to one side ofM can be dropped. Also, in the general case, we prove that the rate of vanishing of the transversal component cannot be too rapid (unlessH(M) ? Σ′), and we relate the possible rate of vanishing to the order of vanishing of the Levi form on a certain holomorphic submanifold ofM.     

Combinatorial constructions for the Zeckendorf sum of digits of polynomial values

Let $p(X)\in\mathbb{Z}[X]$ with $p(\mathbb{N})\subset\mathbb{N}$ be of degree h≥2 and denote by s _F (n) the sum of digits in the Zeckendorf representation of n. We study by combinatorial means three analogues of problems of Gelfond (Acta Arith. 13:259–265, 1967/1968), Stolarsky (Proc. Am. Math. Soc. 71:1–5, 1978) and Lindström (J. Number Theory 65:321–324, 1997) concerning the distribution of s _F on polynomial sequences. First, we show that for m≥2 we have #{n<N:s _F (p(n))≡amodm}? _p,m N ^4/(6h+1) (Gelfond). Secondly, we find the extremal minimal and maximal orders of magnitude of the ratio s _F (p(n))/s _F (n) (Stolarsky). Third, we prove that lim?sup _n→∞ s _F (p(n))/log _φ (p(n))=1/2, where φ denotes the golden ratio (Lindström).     

Absolutely continuous operators on function spaces and vector measures

Let (Ω, Σ, μ) be a finite atomless measure space, and let E be an ideal of L ⁰(μ) such that ${L^\infty(\mu) \subset E \subset L^1(\mu)}$ . We study absolutely continuous linear operators from E to a locally convex Hausdorff space ${(X, \xi)}$ . Moreover, we examine the relationships between μ-absolutely continuous vector measures m : Σ → X and the corresponding integration operators T _m : L ^∞(μ) → X. In particular, we characterize relatively compact sets ${\mathcal{M}}$ in ca _μ (Σ, X) (= the space of all μ-absolutely continuous measures m : Σ → X) for the topology ${\mathcal{T}_s}$ of simple convergence in terms of the topological properties of the corresponding set ${\{T_m : m \in \mathcal{M}\}}$ of absolutely continuous operators. We derive a generalized Vitali–Hahn–Saks type theorem for absolutely continuous operators T : L ^∞(μ) → X.     

The shift on the inverse limit of a covering projection

A group endomorphismα : G → G is said to beweakly shift equivalent to the group endomorphismβ : H → H if there existsh ∈ H such thatα is shift equivalent to Ad[h] °β. Given covering projectionsa : X → X, b : Y → Y of compact, connected, locally path connected, semilocally simply connected metric spaces with fixed pointsx ₀ ∈X,y ₀ ∈Y respectively, the inverse limits $$\begin{array}{l} \sum\nolimits_a { = \lim } (X,a) = \{ (x_i )_{i \in Z^ + } ax_{i + 1} = x_1 ,i \in Z^ + \} , \\ \sum\nolimits_a { = \lim } (Y,b) = \{ (y_i )_{i \in Z^ + } by_{i + 1} = y_1 ,i \in Z^ + \} , \\ \end{array}$$ and the “shift” mapsσ _a : Σ _a → Σ _a ,σ _b : Σ _b → Σ _b defined byσ _a((x _i)i∈Z ⁺)=(x _i+1)i∈Z ⁺ ∈ Σ _a ,σ _b((y _i)i∈Z ⁺)=(y i + 1)i∈Z ⁺ ∈ Σ _b are considered. It is proven that ifσ _a andσ _b are topologically conjugate thena _# :π ₁(X, x ₀) →π ₁(X, x ₀) is weakly shift equivalent tob _# :π ₁(Y, y ₀) →π ₁(Y, y ₀). Furthermore, ifa : X → X andb : Y → Y are expanding endomorphisms of compact differentiable manifolds, weak shift equivalence is a complete invariant of topological conjugacy. The use of this invariant is demonstrated by giving a complete classification of the shifts of expanding maps on the klein bottle. The reader is referred to Section 4 of this work for a detailed statement of results.     

A representation theorem for maxitive measures

A maxitive measure is a nonnegative function η on a σ-algebra Σ and such that η(U_j A_j ) = sup_j η(A_j) for all countable disjoint families of sets (A_j) in Σ. A representation theorem for such measures is established, and next applied to represent Köthe function M-spaces as L_∞-spaces.     

Null curves in {\mathbb{C}^3} and Calabi–Yau conjectures

For any open orientable surface M and convex domain ${\Omega\subset \mathbb{C}^3,}$ there exist a Riemann surface N homeomorphic to M and a complete proper null curve F : N → Ω. This result follows from a general existence theorem with many applications. Among them, the followings:

For any convex domain Ω in ${\mathbb{C}^2}$ there exist a Riemann surface N homeomorphic to M and a complete proper holomorphic immersion F : N → Ω. Furthermore, if ${D \subset \mathbb{R}^2}$ is a convex domain and Ω is the solid right cylinder ${\{x \in \mathbb{C}^2 \,|\, \mbox{Re}(x) \in D\},}$ then F can be chosen so that Re(F) : N → D is proper.

There exist a Riemann surface N homeomorphic to M and a complete bounded holomorphic null immersion ${F:N \to {\rm SL}(2, \mathbb{C}).}$

There exists a complete bounded CMC-1 immersion ${X:M \to \mathbb{H}^3.}$

For any convex domain ${\Omega \subset \mathbb{R}^3}$ there exists a complete proper minimal immersion (X _j ) _j=1,2,3 : M → Ω with vanishing flux. Furthermore, if ${D \subset \mathbb{R}^2}$ is a convex domain and ${\Omega=\{(x_j)_{j=1,2,3} \in \mathbb{R}^3 \,|\, (x_1,x_2) \in D\},}$ then X can be chosen so that (X ₁, X ₂) : M → D is proper.

Any of the above surfaces can be chosen with hyperbolic conformal structure.     

An Erdös-Ko-Rado theorem for the subcubes of a cube

LetP be that partially ordered set whose elements are vectors x=(x ₁, ...,x _n ) withx _i ε {0, ...,k} (i=1, ...,n) and in which the order is given byx≦y iffx _i =y _i orx _i =0 for alli. LetN _i (P)={x εP : |{j:x _j ≠ 0}|=i}. A subsetF ofP is called an Erdös-Ko-Rado family, if for allx, y εF it holdsx ≮y, x ≯ y, and there exists az εN ₁(P) such thatz≦x andz≦y. Let ? be the set of all vectorsf=(f ₀, ...,f _n ) for which there is an Erdös-Ko-Rado familyF inP such that |N _i (P) ∩F|=f _i (i=0, ...,n) and let 〈?〉 be its convex closure in the (n+1)-dimensional Euclidean space. It is proved that fork≧2 (0, ..., 0) and \(\left( {0,...,0,\overbrace {i - component}^{\left( {\begin{array}{*{20}c} {n - 1} \\ {i - 1} \\ \end{array} } \right)}k^{i - 1} ,0,...,0} \right)\) (i=1, ...,n) are the vertices of 〈?〉.