Bijective maps on standard Borel subgroup of symplectic group preserving commutators

Let F be a field, and let \mathbbG\mathbb{G} be the standard Borel subgroup of the symplectic group Sp(2m, F). In this paper, we characterize the bijective maps ϕ: \mathbbG\mathbb{G} → \mathbbG\mathbb{G} satisfying ϕ[x, y] = [ϕ(x), ϕ(y)].     

A stability property of symplectic packing

We prove that for any closed symplectic 4-manifold (M,Ω) with [Ω]∈H ²(M, Q) there exists a number N ₀ such that for every N≥N ₀, (M,Ω) admits full symplectic packing by N equal balls. We also indicate how to compute this N ₀. Our approach is based on Donaldson's symplectic submanifold theorem and on tools from the framework of Taubes theory of Gromov invariants. Oblatum 9-I-1998 & 1-VII-1998 / Published online: 14 January 1999     

ON HARMONIC MAPS INTO SYMPLECTIC GROUPS Sp（N）

50. IntroductionThe construction and the factorization of harmonic maps from R2 (or its simPlyconnecteddomain) into the uIiltary group U(N) were firstly solved by K.Ulilenbeck in [11, wherethe conception of unitons was iniroduced. Since then various developmenis have beencoatributed[2--5]. Recently, by introducing (singular) Darboux transformations, a purelya1gebraic method to construct harmonic maPs and unitons illto U(N) has been shownin t6'7]. This method can be aIso aPplied to the ca…     

Spherical complexes attached to symplectic lattices

To the integral symplectic group Sp(2g,\mathbbZ){{\rm Sp}(2g,\mathbb{Z})} we associate two posets of which we prove that they have the Cohen-Macaulay property. As an application we show that the locus of marked decomposable principally polarized abelian varieties in the Siegel space of genus g has the homotopy type of a bouquet of (g − 2)-spheres. This, in turn, implies that the rational homology of moduli space of (unmarked) principal polarized abelian varieties of genus g modulo the decomposable ones vanishes in degree ≤ g − 2. Another application is an improved stability range for the homology of the symplectic groups over Euclidean rings. But the original motivation comes from envisaged applications to the homology of groups of Torelli type. The proof of our main result rests on a refined nerve theorem for posets that may have an interest in its own right.     

The zeros of functions in Nevanlinna’s area class

Let {Z_n=1{( _n ^∓ ) bea sequence of points in the unit open disk, and letNϕ(U) denote the class of functionsf analytic in the unit disk U such that |f|∈L ( _ϕ ¹ )(U). For ϕ ≡ 1, the necessary and sufficient conditions for the existence off εN(U) and vanishing atz _n is Σ( _n=1 ^∓ ) (1–|Z_n|)² ∞. Also we estimate a large family of canonical products. These results are extended to ϕ(z)=(1-|z|)^ϕ. This represents a part of a Ph.D. thesis conducted at the Technion — Israel Institute of Technology, Department of Mathematics, by Dr. C. A. Horowitz. His help during the preparation of this paper is gratefully acknowledged.     

A question of Gol’dberg concerning entire functions with prescribed zeros

Let (z_j) be a sequence of complex numbers satisfying |z_j|→ ∞ asj → ∞ and denote by n(r) the number of z_j satisfying |z_j|≤ r. Suppose that lim inf_{r → ⇈} log n(r)/ logr > 0. Let ϕ be a positive, non-decreasing function satisfying ∫^∞ (ϕ(t)t logt)⁻¹ dt < ∞. It is proved that there exists an entire functionf whose zeros are the z_j such that log log M(r,f) = o((log n(r))²ϕ(log n(r))) asr → ∞ outside some exceptional set of finite logarithmic measure, and that the integral condition on ϕ is best possible here. These results answer a question by A. A. Gol’dberg.     

Holomorphic functions which are highly nonintegrable at the boundary

Given a bounded convex domainD in ℂ^N with smooth boundary and a positive continuous function ϕ ond, it is proved that there is a holomorphic functionf onD such that |f|ϕ is nonintegrable onM∩D wheneverM is a real submanifold of a neighbourhood of a point ofbD which intersectsbD transversely.     

Bireflectionality of orthogonal and symplectic groups of characteristic 2

Let V be a finite dimensional vector space over a field K of characteristic 2. Let O(V) be the orthogonal group defined by a nondegenerate quadratic form. Then every element in O(V) is a product of two elements of order 2, unless all nonsingular subspaces of V are at most 2-dimensional. If V is a nonsingular symplectic space, then every element in the symplectic group Sp (V) is a product of two elements of order 2, except if dim V = 2 and |K| = 2.     

A remark on symplectic spreads

We first note that each element of a symplectic spread of PG(2n − 1, 2 ^r ) either intersects a suitable nonsingular quadric in a subspace of dimension n − 2 or is contained in it, then we prove that this property characterises symplectic spreads of PG(2n − 1, 2 ^r ). As an application, we show that a translation plane of order q ⁿ , q even, with kernel containing GF(q), is defined by a symplectic spread if and only if it contains a maximal arc of the type constructed by Thas (Europ J Combin 1:189–192, 1980).     

Finiteness of the number of ends of minimal submanifolds in euclidean space

We prove a version of the well-known Denjoy-Ahlfors theorem about the number of asymptotic values of an entire function for properly immersed minimal surfaces of arbitrary codimension in ℝ ^N . The finiteness of the number of ends is proved for minimal submanifolds with finite projective volume. We show, as a corollary, that a minimal surface of codimensionn meeting anyn-plane passing through the origin in at mostk points has no morec(n, N)k ends.