ON MINIMAL SURFACES WHOSE GAUSS MAP IS INVARIANT BY A HOLOMORPHIC FOLIATION

We study real minimal surfaces : M^{2 4} under the hypothesisthat the holomorphic Gauss map of the immersion is invariantby a holomorphic foliation with singularities. We give a sortof Huber–Osserman theorem regarding the algebraicity ofthe holomorphic Gauss map.     

2-Ruled Calibrated 4-Folds in R7 and R8

This article introduces the notion of 2-ruled 4-folds: submanifoldsof Rⁿ fibred over a 2-fold by affine 2-planes. This is motivatedby a paper by Joyce and previous work of the present author.A 2-ruled 4-fold M is r-framed if an oriented basis is smoothlyassigned to each fibre, and then we may write M in terms oforthogonal smooth maps ₁,₂ : S^n–1 and a smooth map : Rⁿ. We focus on 2-ruled Cayley 4-folds in R⁸ as certainother calibrated 4-folds in R⁷ and R⁸ can be considered as specialcases. The main result characterizes non-planar, r-framed, 2-ruledCayley 4-folds, using a coupled system of nonlinear, first-order,partial differential equations that ₁ and ₂ satisfy, and anothersuch equation on which is linear in . We give a means of constructing2-ruled Cayley 4-folds starting from particular 2-ruled Cayleycones using holomorphic vector fields. This is used to giveexplicit examples of U(1)-invariant 2-ruled Cayley 4-folds asymptoticto a U(1)³-invariant 2-ruled Cayley cone. Examples are alsogiven based on ruled calibrated 3-folds in C³ and R⁷ and complexcones in C⁴.     

ASYMPTOTICALLY LINEAR ELLIPTIC PROBLEM ON RN

In this paper, we consider the following semilinear ellipticproblem: where f(x, t) tends to p(x) and q(x) L(^N), respectively, ast 0 and t +. We prove that there exist two numbers l and Lwith L < l such that problem (P) has at least one positivesolution for (– l, –L) and has no positive solutionfor all [–l,–L]. The existence and non-existenceof positive solutions for problem (P) at = –l and =–L are also discussed.     

Global smooth fibrations of R3 by oriented lines

A smooth fibration of ³ by oriented lines is given by a smoothunit vector field V on ³ all of whose integral curves are straightlines. Such a fibration is said to be nondegenerate if dV vanishesonly in the direction of V. Let be the space of oriented linesof ³ endowed with its canonical pseudo-Riemannian neutral metric.We characterize the nondegenerate smooth fibrations of ³ byoriented lines as the closed (in the relative topology) definiteconnected surfaces in . In particular, local conditions on imply the existence of a global fibration. Besides, for anysuch fibration the base space is diffeomorphic to the open discand the directions of the fibers form an open convex set ofthe two-sphere. We characterize as well, in a similar way, thesmooth (possibly degenerate) fibrations.     

REAL HYPERSURFACES IN A EUCLIDEAN COMPLEX SPACE FORM

Let M be an orientable connected and compact real hypersurfaceof the complex space form C^{(n + 1)/2}. If the mean curvature and the function f = g(A, ) of hypersurface M satisfy the inequalityn²² (n² – 1) + f², where is the characteristic vectorfield, A is the shape operator and (n – 1) is the infimumof the Ricci curvatures of hypersurface M, then it is shownthat is a constant and M is the sphere Sⁿ(²) in C^{(n + 1)/2}.     

ON A BRESAR-SEMRL CONJECTURE AND DERIVATIONS OF BANACH ALGEBRAS

In this paper, we answer a question on derivations of densealgebras of linear operators posed by Brear and emrl. Our theoremimplies the following result: let be a complex Banach algebra,and let d and g be continuous derivations of . If dg(x) is quasi-nilpotentfor every x , then dg(x)³ lies in the radical of for everyx . This result was proved by Brear and emrl with the additionalassumption gd = dg.     

MOTIVIC INVARIANTS OF ARTIN STACKS AND 'STACK FUNCTIONS'

An invariant of quasiprojective -varieties X with values ina commutative ring is motivic if (X) = (Y) + (X\ Y) for Y closedin X, and (X x Y) = (X)(Y). Examples include Euler characteristics and virtual Poincaré and Hodge polynomials. We firstdefine a unique extension ' of to finite type Artin -stacks, which is motivic and satisfies '([X/G]) = (X)/(G) when X is a -variety, G a special -groupacting on X, and [X/G] is the quotient stack. This only worksif (G) is invertible in for all special -groups G, which excludes = as (_m) = 0. But we can extend the construction to get roundthis. Then we develop the theory of stack functions on Artin stacks.These are a universal generalization of constructible functionson Artin stacks. There are several versions of the construction:the basic one , and variants ‘twisted’ by motivic invariants. We associate a -vector space or a -module to each Artin stack , with functorial operations of multiplication, pullbacks * and pushforwards _*under 1-morphisms ;, and so on. They will be important tools in the author's series on ‘Configurationsin abelian categories’.     

LABELLING THE CHARACTER TABLES OF SYMMETRIC AND ALTERNATING GROUPS

Let X be a character table of the symmetric group S_n. It isshown that unless n = 4 or n = 6, there is a unique way to assignpartitions of n to the rows and columns of X so that for all and , X is equal to (), the value of the irreducible characterof S_n labelled by on elements of cycle type . Analogous resultsare proved for alternating groups, and for the Brauer charactertables of symmetric and alternating groups.     

Erratum

2-RULED CALIBRATED 4-FOLDS IN ⁷ AND ⁸ JASON LOTAY     

An egg-yolk principle and exponential integrability for quasiregular mappings

Quasiregular mappings f:ⁿⁿ are a natural generalization of analyticfunctions from complex analysis and provide a theory which isrich with new phenomena. In this paper we extend a well-knownresult of Chang and Marshall on exponential integrability ofanalytic functions in the disk, to the case of quasiregularmappings defined in the unit ball of ⁿ. To this end, an ‘egg-yolk’principle is first established for such maps, which extendsa recent result of the first author. Our work leaves open aninteresting problem regarding n-harmonic functions.     

SIMULTANEOUS DIOPHANTINE APPROXIMATION BY SQUARE-FREE NUMBERS

Let ₁, ...,_r R be ‘not very well approximable’,for example, Q-linearly independent real algebraic numbers.Then there are infinitely many positive square-free integersn such that ||n_i|| << n^–(2/3r)+(1 i r), where||·|| denotes distance to the nearest integer.     

COMPRESSIONS, CONVEX GEOMETRY AND THE FREIMAN-BILU THEOREM

We note a link between combinatorial results of Bollobásand Leader concerning sumsets in the grid, the Brunn–Minkowskitheorem and a result of Freiman and Bilu concerning the structureof sets A with small doubling. Our main result is the following. If > 0 and if A is a finitenon-empty subset of a torsion-free abelian group with |A + A| K|A|, then A may be covered by e^KO(1) progressions of dimension log ₂ K + and size at most |A|.     

Lp Lq Estimates for Parabolic Systems in Non-Divergence Form with VMO Coefficients

Consider a parabolic NxN-system of order m on ⁿ with top-ordercoefficients a VMOL. Let 1 < p, q < and let be a Muckenhouptweight. It is proved that systems of this kind possess a uniquesolution u satisfying whereAu = _||m a Du and J = [0,). In particular, choosing = 1, therealization of A in L^p(ⁿ)^N has maximal L^p – L^q regularity.     

CONFORMAL MEASURES ASSOCIATED TO ENDS OF HYPERBOLIC n-MANIFOLDS

Let be a non-elementary Kleinian group acting on the closedn-dimensional unit ball and assume that its Poincaréseries converges at the exponent . Let M be the -quotient ofthe open unit ball. We consider certain families = {E₁, ...,E_p} of open subsets of M such that is compact. The sets E_i are the ends of M and is a completecollection of ends for M. We associate to each end E an -conformalmeasure such that the measures corresponding to different endsare mutually singular if non-trivial. Additionally, each -conformalmeasure for on the limit set () of can be written as a sumof such conformal measures associated to ends E . In dimension3, our results overlap with some results of Bishop and Jones(The law of the iterated logarithm for Kleinian groups, Cont.Math. 211 (1997), 17–50.).     

SMALL C*-ALGEBRAS THAT FAIL TO HAVE SEPARABLE REPRESENTATIONS

It is shown that any -finite, wild AW*-algebra of type II₁ failsto have non-trivial separable representations. Here a wild AW*-algebrameans an AW*-algebra with no direct summand, which is *-isomorphicto a von Neumann algebra. Let be the -finite, wild type II₁AW*-algebra formed by taking the monotone complete tensor productof the Dixmier algebra and a factor of type II₁ acting ona separable Hilbert space. As both and act on separable Hilbertspaces, it seems natural to describe as small and to ask thefollowing question. Does have a non-trivial separable representation?The above described result answers negatively to this question.That is, fails to have non-trivial separable representations.     

OPTIMAL INTEGRABILITY CONDITION FOR THE LOG-SOBOLEV INEQUALITY

Let M denote a connected complete Riemannian manifold (possiblywith a convex boundary), the Riemannian distance function froma fixed point and V C² (M) such that dµ_V e^V d xis a probability measure. For any K 0, we prove that K/2 isthe infimum over all > 0 such that Ric_M – Hess_V –Kand imply the log-Sobolevinequality for the Dirichlet form µ_V(| f |²).     

Euler characteristics of the real points of certain varieties of algebraic tori

Let G be a complex connected reductive group which is definedover , let be its Lie algebra, and let be the variety of maximaltori of G. For (), let be the variety of tori in whose Liealgebra is orthogonal to with respect to the Killing form.We show, using the Fourier–Sato transform of conical sheaveson real vector bundles, that the ‘weighted Euler characteristic’of () is zero unless is nilpotent, in which case it equals(–1)^{(dim )/2}. Here ‘weighted Euler characteristic’means the sum of the Euler characteristics of the connectedcomponents, each weighted by a sign ± 1 which dependson the real structure of the tori in the relevant component.This is a real analogue of a result over finite fields whichis connected with the Steinberg representation of a reductivegroup.     

ON THE LEAST n WITH {chi}(n) != 1

Suppose that a > 1. By using a method of Linnik employinghis Large Sieve one may derive the following result. The numberof primitive Dirichlet characters with conductor Q such that(n) = 1 for all n (log Q)^a is O(Q^2/a+). We improve the exponent2/a for a > 2 by using a refined version by Heath-Brown ofthe Halasz–Montgomery ‘Large Values method’.     

THE AVERAGE VALUE OF DIVISOR SUMS IN ARITHMETIC PROGRESSIONS

Let (n) denote the Fourier coefficients of cusp forms or thenumber of divisors of n. Estimates of the type are shown, uniformly in q X. The methods canbe extended to other arithmetic functions, for example, thenumber of representations of n as a sum of two squares or k-freenumbers. As an application, sums of the type _{n X}(n) (n) forany q-periodic function can be estimated non-trivially.