Harmonic morphisms on heaven spaces

We prove that any (real or complex) analytic horizontally conformalsubmersion from a three-dimensional conformal manifold (M³,c_M) to a two-dimensional conformal manifold (N², c_N) can be,locally, ‘extended’ to a unique harmonic morphismfrom the (eaven)-space (H⁴, g) of (M³, c_N) to (N², c_N). Moreover,any positive harmonic morphism with two-dimensional fibres from(H⁴, g) is obtained in this way.     

Triangle group representations and constructions of regular maps

A regular map of type {m,n} is a 2-cell embedding of a graphin an orientable surface, with the property that for any twodirected edges e and e' there exists an orientation-preservingautomorphism of the embedding that takes e onto e', and in whichthe face length and the vertex valence are m and n, respectively.Such maps are known to be in a one-to-one correspondence withtorsion-free normal subgroups of the triangle groups T(2,m,n).We first show that some of the known existence results aboutregular maps follow from residual finiteness of triangle groups.With the help of representations of triangle groups in speciallinear groups over algebraic extensions of Z we then constructivelydescribe homomorphisms from T(2,m,n)=y,z|y^m=zⁿ=(yz)²=1 intofinite groups of order at most c^r where c=c(m,n), such thatno non-identity word of length at most r in x,y is mapped ontothe identity. As an application, for any hyperbolic pair {m,n}and any r we construct a finite regular map of type {m,n} ofsize at most C^r, such that every non-contractible closed curveon the supporting surface of the map intersects the embeddedgraph in more than r points. We also show that this result isthe best possible up to determining C=C(m,n). For r>m thegraphs of the above regular maps are arc-transitive, of valencen, and of girth m; moreover, if each prime divisor of m is largerthan 2n then these graphs are non-Cayley. 2000 Mathematics SubjectClassification: 05C10, 05C25, 20F99, 20H25.     

Harmonic maps from Kähler manifolds

Some Liouville type theorems for harmonic maps from Kähler manifolds are obtained. The main result is to prove that a harmonic map from a bounded symmetric domain (exceptR _IV(2)) to any Riemannian manifold with finite energy has to be constant.     

The Dimension of the Space of Harmonic 2-Spheres in the 6-Sphere

Using the twistorial approach and some previous results, weprove the conjecture that the dimension of the moduli spaceof harmonic maps of area 4d from the 2-sphere to the 2n-sphereis 2d + n² for the particular case n = 3. 2000 Mathematics SubjectClassification 53A10 (primary), 53C42 (secondary).     

Bifurcations of Periodic Points of Holomorphic Maps From C2 into C2

Let F:Cⁿ Cⁿ be a holomorphic map, F^k be the kth iterate ofF, and p Cⁿ be a periodic point of F of period k. That is,F^k(p) = p, but for any positive integer j with j < k, F^j(p) p. If p is hyperbolic, namely if DF^k(p) has no eigenvalue ofmodulus 1, then it is well known that the dynamical behaviourof F is stable near the periodic orbit = {p, F(p),..., F^k–1(p)}.But if is not hyperbolic, the dynamical behaviour of F near may be very complicated and unstable. In this case, a veryinteresting bifurcational phenomenon may occur even though may be the only periodic orbit in some neighbourhood of : forgiven M N\{1}, there may exist a C^r-arc {F_t: t [0,1]} (wherer N or r = ) in the space H(Cⁿ) of holomorphic maps from Cⁿinto Cⁿ, such that F₀ = F and, for t (0,1], F_t has an Mk-periodicorbit _t with as t 0. Theperiod thus increases by a factor M under a C^r-small perturbation!If such an F_t does exist, then , as well as p, is said to beM-tupling bifurcational. This definition is independent of r. For the above F, there may exist a C^r-arc in H(Cⁿ), with t [0,1], such that and, for t (0,1], has two distinct k-periodic orbits _t,1 and _t,2 with d(_t,i, ) 0 as t 0 for i = 1,2. If such an does exist, then , as well as p, is said to be 1-tupling bifurcational. In recent decades, there have been many papers and remarkableresults which deal with period doubling bifurcations of periodicorbits of parametrized maps. L. Block and D. Hart pointed outthat period M-tupling bifurcations cannot occur for M >2 in the 1-dimensional case. There are examples showing thatfor any M N, period M-tupling bifurcations can occur in higher-dimensionalcases. An M-tupling bifurcational periodic orbit as defined here actsas a critical orbit which leads to period M-tupling bifurcationsin some parametrized maps. The main result of this paper isthe following. Theorem. Let k N and M N, and let F: C² C² be a holomorphicmap with k-periodic point p. Then p is M-tupling bifurcationalif and only if DF^k(p) has a non-zero periodic point of periodM. 1991 Mathematics Subject Classification: 32H50, 58F14.     

Geometric Approximation Of The Fibre Of The Freudenthal Suspension

Let Rat_k(CPⁿ) denote the space of based holomorphic maps ofdegree k from the Riemannian sphere S² to the complex projectivespace CPⁿ. The basepoint condition we assume is that f()=[1,..., 1]. Such holomorphic maps are given by rational functions: Rat_k(CPⁿ) ={(p₀(z), ..., p_n(z)):each p_i(z) is a monic, degree-kpolynomial and such that there are no roots common to all p_i(z)}.(1.1) The study of the topology of Rat_k(CPⁿ) originated in [10]. Later,the stable homotopy type of Rat_k(CPⁿ) was described in [3] interms of configuration spaces and Artin's braid groups. LetW(S²ⁿ) denote the homotopy theoretic fibre of the Freudenthalsuspension E:S²ⁿ S²ⁿ⁺¹. Then we have the following sequenceof fibrations: ²S²ⁿ⁺¹ W(S²ⁿ)S²ⁿ S²ⁿ⁺¹. A theorem in [10] tellsus that the inclusion Rat_k(CPⁿ) ²_kCP^{n 2}S²ⁿ⁺¹ is a homotopy equivalenceup to dimension k(2n–1). Thus if we form the direct limitRat(CPⁿ)= lim_k Rat_k(CPⁿ), we have, in particular, that Rat(CPⁿ)is homotopy equivalent to ²S²ⁿ⁺¹. If we take the results of [3] and [10] into account, we naturallyencounter the following problem: how to construct spaces X_k(CPⁿ),which are natural generalizations of Rat_k(CPⁿ), so that X(CPⁿ)approximates W(S²ⁿ). Moreover, we study the stable homotopytype of X_k(CPⁿ). The purpose of this paper is to give an answer to this problem.The results are stated after the following definition. 1991Mathematics Subject Classification 55P35.     

On Hardy, BMO and Lipschitz spaces of invariantly harmonic functions in the unit ball

The invariantly harmonic functions in the unit ball Bⁿ in Cⁿare those annihilated by the Bergman Laplacian . The Poisson-Szegökernel P(z,) solves the Dirichlet problem for : if f C(Sⁿ),the Poisson-Szegö transform of f, where d is the normalized Lebesgue measure on Sⁿ,is the unique invariantly harmonic function u in Bⁿ, continuousup to the boundary, such that u=f on Sⁿ. The Poisson-Szegötransform establishes, loosely speaking, a one-to-one correspondencebetween function theory in Sⁿ and invariantly harmonic functiontheory in Bⁿ. When n 2, it is natural to consider on Sⁿ functionspaces related to its natural non-isotropic metric, for theseare the spaces arising from complex analysis. In the paper,different characterizations of such spaces of smooth functionsare given in terms of their invariantly harmonic extensions,using maximal functions and area integrals, as in the correspondingEuclidean theory. Particular attention is given to characterizationin terms of purely radial or purely tangential derivatives.The smoothness is measured in two different scales: that ofSobolev spaces and that of Lipschitz spaces, including BMO andBesov spaces. 1991 Mathematics Subject Classification: 32A35,32A37, 32M15, 42B25.     

Regularity of harmonic maps into a static Lorentzian manifold

We study the regularity of harmonic maps from Riemannian manifold into a static Lorentzian manifold. We show that when the domain manifold is two-dimensional, any weakly harmonic map is smooth. We also show that when dimension n of the domain manifold is greater than two, there exists a weakly harmonic map for the Dirichlet problem which is smooth except for a closed set whose (n − 2)-dimensional Hausdorff measure is zero.     

Hopf C*-Algebras

In this paper we define and study Hopf C^*-algebras. Roughlyspeaking, a Hopf C^*-algebra is a C^*-algebra A with a comultiplication: A M(A A) such that the maps a b (a)(1 b) and a (a 1)(b)have their range in A A and are injective after being extendedto a larger natural domain, the Haagerup tensor product A _hA. In a purely algebraic setting, these conditions on are closelyrelated to the existence of a counit and antipode. In this topologicalcontext, things turn out to be much more subtle, but neverthelessone can show the existence of a suitable counit and antipodeunder these conditions. The basic example is the C^*-algebra C₀(G) of continuous complexfunctions tending to zero at infinity on a locally compact groupwhere the comultiplication is obtained by dualizing the groupmultiplication. But also the reduced group C^*-algebra of a locally compact group with thewell-known comultiplication falls in this category. In factall locally compact quantum groups in the sense of Kustermansand the first author (such as the compact and discrete ones)as well as most of the known examples are included. This theory differs from other similar approaches in that thereis no Haar measure assumed. 2000 Mathematics Subject Classification: 46L65, 46L07, 46L89.     

Damping Oscillatory Integrals with Polynomial Phase and Convolution Operators with the Affine Arclength Measure on Polynomial Curves in Rn

McMichael proved that the convolution with the (euclidean) arclengthmeasure supported on the curve t (t, t², ..., tⁿ), 0 < t< 1, maps L^p(Rⁿ) boundedly into L^p'(Rⁿ) if and only if 2n(n+1)/(n²+n+2) p 2. In proving this, a uniform estimate on damping oscillatoryintegrals with polynomial phase was crucial. In this paper,a remarkably simple proof of the same estimate on oscillatoryintegrals is presented. In addition, it is shown that the convolutionoperator with the affine arclength measure on any polynomialcurve in Rⁿ maps L^p(Rⁿ) boundedly into L^p'(Rⁿ) if p = 2n(n+1)/(n²+n+2).     

The existence of harmonic maps from Finsler manifolds of Riemannian manifolds

In this paper,we consider the existence of harmonic maps from a Finsler manifold and study the characterisation of harmonic maps,in the spirit of Ishihara.Using heat quation method we show that any map from a compact Finsler manifold M to a compact Riemannian manifold with non-positive sectional curvature can be deformed into a harmonic map which has minimum energy in its homotopy class.     

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.     

Minimal Lagrangian 2-Tori in CP2 Come in Real Families of Every Dimension

It is shown that for every non-negative integer n, there isa real n-dimensional family of minimal Lagrangian tori in CP²,and hence of special Lagrangian cones in C³ whose link is atorus. The proof utilises the fact that such tori arise fromintegrable systems, and can be described using algebro-geometric(spectral curve) data.     

Topologie Des (M - 2)-Courbes Reelles Symetriques

Let X be a non-singular real algebraic curve in CP² of evendegree. In this paper a refinement is proved of a theorem ofKharlamov about (M – 2)-curves that are invariants underthe projective involution. In particular, if the (M –2)-symmetric curve X satisfies the Arnold congruence, then eitherX or its twin is a separating curve. 2000 Mathematics SubjectClassification 14P25.     

A Criterion for Finite Topological Determinacy of Map-Germs

Let f, g: (Rⁿ, 0) (R^p, 0) be two C map-germs. Then f and gare C⁰-equivalent if there exist homeomorphism-germs h and lof (Rⁿ, 0) and (R^p, 0) respectively such that g = l f h^–1.Let k be a positive integer. A germ f is k-C⁰-determined ifevery germ g with j^k g(0) = j^k f(0) is C⁰-equivalent to f. Moreover,we say that f is finitely topologically determined if f is k-C⁰-determinedfor some finite k. We prove a theorem giving a sufficient conditionfor a germ to be finitely topologically determined. We explainthis condition below. Let N and P be two C manifolds. Consider the jet bundle J^k(N,P) with fiber J^k(n, p). Let z in J^k(n, p) and let f be suchthat z = j^kf(0). Define Whether (f) < k depends only on z, not on f. We can thereforedefine the set Let W^k(N, P) be the subbundle of J^k(N, P) with fiber W^k(n, p).Mather has constructed a finite Whitney (b)-regular stratificationS^k(n, p) of J^k(n, p) – W^k(n, p) such that all strata aresemialgebraic and K-invariant, having the property that if S^k(N,P) denotes the corresponding stratification of J^k(N, P) –W^k(N, P) and f C(N, P) is a C map such that j^kf is multitransverseto S^k(N, P), j^kf(N) W^k(N, P) = and N is compact (or f is proper),then f is topologically stable. For a map-germ f: (Rⁿ, 0) (R^p, 0), we define a certain ojasiewiczinequality. The inequality implies that there exists a representativef: U R^p such that j^kf(U – 0) W^k (Rⁿ, R^p = and suchthat j^kf is multitransverse to S^k (Rⁿ, R^p) at any finite setof points S U – 0. Moreover, the inequality controlsthe rate j^kf becomes non-transverse as we approach 0. We showthat if f satisfies this inequality, then f is finitely topologicallydetermined. 1991 Mathematics Subject Classification: 58C27.     

UNIVERSAL SPACES OF TWO-CELL COMPLEXES AND THEIR EXPONENT BOUNDS

Let P²ⁿ⁺¹ be a two-cell complex which is formed by attachinga (2n + 1)-cell to a 2m-sphere by a suspension map. We constructa universal space U for P²ⁿ⁺¹ in the category of homotopy associative,homotopy commutative H-spaces. By universal, we mean that Uis homotopy associative, homotopy commutative and has the propertythat any map f: P²ⁿ⁺¹ Y to a homotopy associative, homotopycommutative H-space Y extends to a uniquely determined H-map: U Y. We then prove upper and lowerbounds of the H-homotopy exponent of U. In the case of a modp^r, Moore space U is the homotopy fibre S²ⁿ⁺¹{p^r} of the p^r-powermap on S²ⁿ⁺¹, and we reproduce Neisendorfer's result that S²ⁿ⁺¹{p^r}is homotopy associative, homotopy commutative and that the p^r-powermap on S²ⁿ⁺¹{p^r} is null homotopic.     

From Trace States to States on the K0 Group of a Simple C*-Algebra

It is shown that any continuous affine surjection from a metrizableChoquet simplex onto a compact convex set occurs as the restrictionmap from the tracial state space onto the state space of theK₀ group of a separable unital simple C^*-algebra which is theinductive limit of a sequence of subhomogeneous C^*-algebras     

Scattering Matrix for Manifolds with Conical Ends

Let M be a manifold with conical ends. (For precise definitionssee the next section; we only mention here that the cross-sectionK can have a nonempty boundary.) We study the scattering forthe Laplace operator on M. The first question that we are interestedin is the structure of the absolute scattering matrix S(s).If M is a compact perturbation of Rⁿ, then it is well-knownthat S(s) is a smooth perturbation of the antipodal map on asphere, that is, S(s)f(·)=f(–·) (mod C) On the other hand, if M is a manifold with a scattering metric(see [8] for the exact definition), it has been proved in [9]that S(s) is a Fourier integral operator on K, of order 0, associatedto the canonical diffeomorphism given by the geodesic flow atdistance . In our case it is possible to prove that S(s) isin fact equal to the wave operator at a time t = plus C terms.See Theorem 3.1 for the precise formulation. This result isnot too difficult and is obtained using only the separationof variables and the asymptotics of the Bessel functions. Our second result is deeper and concerns the scattering phasep(s) (the logarithm of the determinant of the (relative) scatteringmatrix).     

A Functional Relation for Accessory Parameters for Genus 2 Algebraic Curves with an Order 4 Automorphism

Let C be a genus 2 algebraic curve defined by an equation ofthe form y² = x(x² – 1)(x – a)(x – 1/a). Asis well known, the five accessory parameters for such an equationcan all be expressed in terms of a and the accessory parameter b corresponding to a. The main result of the paper is thatif a' = 1 – a², which in general yields a non-isomorphiccurve C', then b'a'(a'² – 1) = – – ba(a²– 1). This is proven by it being shown how the uniformizing functionfrom the unit disk to C' can be explicitly described in termsof the uniformizing function for C.     

Teichmuller Distance is not C2+{varepsilon}

The Teichmüller space of a finite-type surface is considered.It is shown that Teichmüller distance is not C^{2 +} forany > 0. Furthermore, Teichmüller distance is not C^{2+ g} for any gauge function g with . 2000 Mathematics Subject Classification 30F60.