Differentiable structures with zero entropy on simply connected 4-manifolds

We show that a closed 4-dimensional simply connected topological manifoldM admits a differentiable structure with aC Riemannian metric whose geodesic flow has zero topological entropy if and only ifM is homeomorphic toS ⁴, ²,S ²×S ², or ²#².     

Certain 4-manifolds with non-negative sectional curvature

In this paper, we study certain compact 4-manifolds with non-negative sectional curvature K. If s is the scalar curvature and W. is the self-dual part of Weyl tensor, then it will be shown that there is no metric g on S ^• × S ^• with both (i) K > 0 and (ii) ÷ s − W _• ⩾ 0. We also investigate other aspects of 4-manifolds with non-negative sectional curvature. One of our results implies a theorem of Hamilton: “If a simply-connected, closed 4-manifold M^• admits a metric g of non-negative curvature operator, then M^• is one of S^•, ℂP^• and S^•×S^•”. Our method is different from Hamilton’s and is much simpler. A new version of the second variational formula for minimal surfaces in 4-manifolds is proved.      

Existence and uniqueness of packings with specified combinatorics

Generalizations of the Andreev-Thurston circle packing theorem are proved. One such result is the following. Let G=G(V) be a planar graph, and for each vertex v ∈ V, let ℱ _v be a proper 3-manifold of smooth topological disks in S ²,with the property that the pattern of intersection of any two sets A, B ∈ ℱ _v is topologically the pattern of intersection of two circles (i.e., there is a homeomorphism h:S ²→S ² taking A and B to circles). Then there is a packing P=(P _v :v ∈V)whose nerve is G, and which satisfies P _v ∈ ℱ _ν for v ∈ V. (‘The nerve is G’ means that two sets, P _v ,P _u ,touch, if, and only if, u ↔ v is an edge in G.) In the case whereG is the 1-skeleton of a triangulation, we also give a precise uniqueness statement. Various examples and applications are discussed.     

Minimal, rigid foliations by curves on ℂℙ n

We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space ℂℙ ⁿ for every dimension n≥2 and every degree d≥2. Precisely, we construct a foliation ℱ which is induced by a homogeneous vector field of degree d, has a finite singular set and all the regular leaves are dense in the whole of ℂℙ ⁿ . Moreover, ℱ satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if ℱ is conjugate to another holomorphic foliation by a homeomorphism sufficiently close to the identity, then these foliations are also conjugate by a projective transformation. Finally, all these properties are persistent for small perturbations of ℱ.?This is done by considering pseudo-groups generated on the unit ball 𝔹 ⁿ ⊆ℂ ⁿ by small perturbations of elements in Diff(ℂ ⁿ ,0). Under open conditions on the generators, we prove the existence of many pseudo-flows in their closure (for the C ⁰-topology) acting transitively on the ball. Dynamical features as minimality, ergodicity, positive entropy and rigidity may easily be derived from this approach. Finally, some of these pseudo-groups are realized in the transverse dynamics of polynomial vector fields in ℂℙ ⁿ . Received March 7, 2002 / final version received November 26, 2002?Published online February 7, 2003 Most of this work has been carried out during a visit of the first author to IMPA/RJ and a visit of the second author to the University of Lille 1. We would like to thank these institutes for hospitality and express our gratitude to CNPq-Brazil and CNRS-France for the financial support which made these visits possible. We are also indebted to Paulo Sad, Marcel Nicolau and the referee whose comments helped us to improve on the preliminary version. Finally, the second author has partially conducted this research for the Clay Mathematics Institute.     

Counterexamples to the Kneser conjecture in dimension four

We construct a connected closed orientable smooth four-manifold whose fundamental group is the free product of two non-trivial groups such that it is not homotopy equivalent toM ₀#M ₁ unlessM ₀ orM ₁ is homeomorphic toS ⁴. LetN be the nucleus of the minimal elliptic Enrique surfaceV ₁(2, 2) and putM=N∪ _∂NN. The fundamental group ofM splits as ℤ/2 * ℤ/2. We prove thatM#k(S ²×S²) is diffeomorphic toM ₀#M ₁ for non-simply connected closed smooth four-manifoldsM ₀ andM ₁ if and only ifk≥8. On the other hand we show thatM is homeomorphic toM ₀#M ₁ for closed topological four-manifoldsM ₀ andM ₁ withπ ₁(M_i)=ℤ/2.     

The lagrange intersections for (ℂP n , ℝP n )

Summary If (M, ω) is a compact symplectic manifold andL ⊂M a compact Lagrangian submanifold and if φ is a Hamiltonian diffeomorphism ofM then the V. Arnold conjecture states (possibly under additional conditions) that the number of intersection section points ofL and φ (L) can be estimated by #{Lϒφ (L)}≥ cuplength +1. We shall prove this conjecture for the special case (L, M)=(ℝP ⁿ , ℂP ⁿ ) with the standard symplectic structure.     

Dihedral manifold approximate fibrations over the circle

Consider the cyclic group C ₂ of order two acting by complex-conjugation on the unit circle S ¹. The main result is that a finitely dominated manifold W of dimension > 4 admits a cocompact, free, discontinuous action by the infinite dihedral group D _∞ if and only if W is the infinite cyclic cover of a free C ₂-manifold M such that M admits a C ₂-equivariant manifold approximate fibration to S ¹. The novelty in this setting is the existence of codimension-one, invariant submanifolds of M and W. Along the way, we develop an equivariant sucking principle for orthogonal actions of finite groups on Euclidean space.     

One-Parameter Families of Modules for Tame Algebras and Bocses

Let Λ be a finite-dimensional algebra over an algebraically closed field k. We denote by mod Λ the category of finitely generated left Λ-modules. Consider the family ℱ(u) of the indecomposables M∈mod Λ such that , where is the subspace of morphisms which factorize through semisimple modules. If P,Q are projectives in mod Λ, ℱ(u)(P,Q) is the family of those modules M∈ℱ(u) such that a minimal projective presentation is of the formf_M: P→Q. We prove that if Λ is of tame representation type then each ℱ(P,Q) has only a finite number of isomorphism classes or is parametrized by μ(u,P,Q) one-parameter families. We give an upper bound for this number in terms of u,P and Q. Then we give some sufficient conditions for tame of polynomial growth type. For the proof we consider similar results for bocses. Presented by Y. Drozd Mathematics Subject Classifications (2000) 16G60, 16G70, 16G20.     

Preimages of CR submanifolds with generic holomorphic maps

We investigate the notion of CR transversality of a generic holomorphic map f: ℂ ⁿ → ℂ ^m to a smooth CR submanifold M of ℂ ^m . We construct a stratification of the set of non-CR transversal points in the preimage M′ = f ⁻¹ (M) by smooth submanifolds, consisting of points where the CR dimension of M′ is constant. We show the existence of a Whitney stratification for sets which are locally diffeomorphic to the product of an open set and an analytic set. Work on this paper was supported by ARRS, Republic of Slovenia.     

On Minimal Entropy and Stability

We study n-manifolds Y whose fundamental groups are subexponential extensions of the fundamental group of some closed locally symmetric manifold X of negative curvature. We show that, in this case, MinEnt(Y)ⁿ is an integral multiple of MinEnt(X)ⁿ, and the value MinEnt(Y) is generally not attained (unless if Y is diffeomorphic to X). This gives a new class of manifolds for which the minimal entropy problem is completely solved. Several examples (even complex projective), obtained by gluings and by taking plane intersections in complex projective space, are described. Some problems about topological stability, related to the minimal entropy problem, are also discussed.     

Generation of fusion systems of characteristic 2-type

We prove that if ℱ is a saturated fusion system on a finite 2-group S, then either ℱ is known, or ℱ is generated by the normalizers of two canonically defined ℱ-characteristic subgroups of S. There are various corollaries for finite groups of characteristic 2-type.     

The signature of generalized flag manifolds and applications

Let G be a linear algebraic group over C and P be a parabolic subgroup. We determine the signatures of the flag manifold G/P. As an application, we prove that the nonsingular hypersurfaces of degree 2 in CP^n are prime if n satisfies certain conditions.     

Feuilletages holomorphes locaux et analyticité partielle d'applications C ∞

 Let f : M → M′ be a smooth CR mapping between a generic real analytic submanifold M ⊂ ℂ ⁿ , n > 1, and a real analytic subset M′ ⊂ ℂ ^n′ . We prove that if M is minimal and if M′ does not contain any complex curves, then f is analytic on a dense open subset of M. More generally, we establish an upper estimate of the partial analyticity of f, which depends on the maximal dimension of local holomorphic foliations contained in M ^′ . Received: 7 August 2001 Mathematics Subject Classification (2000): 32V25, 32V40, 32H99     

On the first pontrjagin class of homotopy complex projective spaces

Let M ²ⁿ be a closed smooth manifold homotopy equivalent to the complex projective space ℂP(n). It is known that the first Pontrjagin class p ₁(M) of M ²ⁿ has the form (n+1+24α(M))u ² for some integer α(M) where u is a generator of H ²(M; ℤ). We prove that α(M) is even when n is even but not divisible by 64.     

Stable ℱ-harmonic maps between Finsler manifolds

In this paper, we derive the first and second variation formulas for JC-harmonic maps between Finsler manifolds, and when F″≤ 0 and n ≥ 3, we prove that there is no nondegenerate stable F-harmonic map between a Riemannian unit sphere Sn and any compact Finsler manifold.     

Limits of functions and elliptic operators

We show that a subspaceS of the space of real analytical functions on a manifold that satisfies certain regularity properties is contained in the set of solutions of a linear elliptic differential equation. The regularity properties are thatS is closed inL ² (M) and that if a sequence of functions fn in ƒ_n converges inL ²(M), then so do the partial derivatives of the functions ƒ_n.     

A note on the volume flux group of four-manifolds

We show that closed orientable smooth four-manifolds with non-trivial volume flux group and fundamental group of subexponential growth type are finitely covered by a manifold homeomorphic to S³×S¹, S²×T² or a nil-manifold. We also show that if a compact complex surface has non-trivial volume flux group then it has zero minimal volume.     

The Problem of Prescribed Critical Functions

Let (M,g) be a compact Riemannian manifold on dimension n ≥ 4 not conformally diffeomorphic to the sphere Sⁿ. We prove that a smooth function f on M is a critical function for a metric g conformal to g if and only if there exists x ∈ M such that f(x) > 0.Mathematics Subject Classifications (2000): 53C21, 46E35, 26D10.     

ℱ-subnormal and ℱ-subabnormal chains in finite groups

The class ℱ_pc of all finite groupsG is defined such thatX≤G implies that there exists an ℱ-subnormal subgroupS ofG containingX such thatX is ℱ-subabnormal inS, where ℱ is a saturated formation, closed under taken subgroups. Groups in ℱ_pc are characterized by ℱ-projectors and ℱ-covering subgroups. Project supported by the National Natural Science Foundation of China (Grant No. 19760001) and the Natural Science Foundation of Guangxi Autonomous Region.     

Real Paley-Wiener type theorems for the Dunkl transform on {\mathcal{S}}'(\mathbb{R}^{d})

We prove real Paley-Wiener type theorems for the Dunkl transform ℱ _D on the space of tempered distributions. Let T∈S′(ℝ ^d ) and Δ _κ the Dunkl Laplacian operator. First, we establish that the support of ℱ _D (T) is included in the Euclidean ball , M>0, if and only if for all R>M we have lim  _n→+∞ R ⁻²ⁿ Δ _κ ⁿ T=0 in S′(ℝ ^d ). Second, we prove that the support of ℱ _D (T) is included in ℝ ^d ∖B(0,M), M>0, if and only if for all R<M, we have lim  _n→+∞ R ²ⁿ  ℱ _D ⁻¹(‖y‖⁻²ⁿ ℱ _D (T))=0 in S′(ℝ ^d ). Finally, we study real Paley-Wiener theorems associated with -slowly increasing function.