Poincaré series of resolutions of surface singularities

Let be a resolution of singularities of a normal surface singularity , with integral exceptional divisors . We consider the Poincaré series

where

We show that if has characteristic zero and is a semi-abelian variety, then the Poincaré series is rational. However, we give examples to show that this series can be irrational if either of these conditions fails.

     

Poincaré-Hopf inequalities

In this article the main theorem establishes the necessity and sufficiency of the Poincaré-Hopf inequalities in order for the Morse inequalities to hold. The convex hull of the collection of all Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data determines a Morse polytope defined on the nonnegative orthant. Using results from network flow theory, a scheme is provided for constructing all possible Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data. Geometrical properties of this polytope are described.

     

Poincaré series on bounded symmetric domains

We show that any holomorphic automorphic form of sufficiently large weight on an irreducible bounded symmetric domain in , , is the Poincaré series of a polynomial in ,..., and give an upper bound for the degree of this polynomial. We also give an explicit construction of a basis in the space of holomorphic automorphic forms.

     

From an iteration formula to Poincaré's Isochronous Center Theorem for holomorphic vector fields

We first generalize a classical iteration formula for one variable holomorphic mappings to a formula for higher dimensional holomorphic mappings. Then, as an application, we give a short and intuitive proof of a classical theorem, due to H. Poincaré, for the condition under which a singularity of a holomorphic vector field is an isochronous center.

     

Conjugate points and closed geodesic arcs on convex surfaces

This paper discusses conjugate points on the geodesics of convex surfaces. It establishes their relationship with the cut locus. It shows the possibility of having many geodesics with conjugate points at very large distances from each other. It also shows that on many surfaces there are arbitrarily many closed geodesic arcs originating and ending at a common point. To achieve these goals, Baire category methods are employed.     

Maximal Poincaré polynomials and minimal Morse functions

In this paper we introduce the maximum Poincaré polynomial of a compact manifold , and prove its uniqueness. We show that its coefficients are topological invariants of the manifolds which, in some cases, correspond to known ones. We also investigate its realizability via a Morse function on .

     

Poincaré duality algebras and rings of coinvariants

Let be a faithful representation of a finite group over the field . Via the group acts on and hence on the algebra of homogenous polynomial functions on the vector space . R. Kane (1994) formulated the following result based on the work of R. Steinberg (1964): If the field has characteristic 0, then is a Poincaré duality algebra if and only if is a pseudoreflection group. The purpose of this note is to extend this result to the case (i.e. the order of is relatively prime to the characteristic of ).

     

Global Poincaré Inequalities on the Heisenberg Group and Applications

Let f be in the localized nonisotropic Sobolev space Wloc^1,p （H^n） on the n-dimensional Heisenberg group H^n = C^n ×R, where 1≤ p ≤ Q and Q = 2n ＋ 2 is the homogeneous dimension of H^n. Suppose that the subelliptic gradient is gloablly L^p integrable, i.e., fH^n ｜△H^n f｜^p du is finite. We prove a Poincaré inequality for f on the entire space H^n. Using this inequality we prove that the function f subtracting a certain constant is in the nonisotropic Sobolev space formed by the completion of C0^∞（H^n） under the norm of （∫H^n ｜f｜ Qp/Q-p）^Q-p/Qp ＋ （∫ H^n ｜△H^n f｜^p）^1/p. We will also prove that the best constants and extremals for such Poincaré inequalities on H^n are the same as those for Sobolev inequalities on H^n. Using the results of Jerison and Lee on the sharp constant and extremals for L^2 to L（2Q/Q-2） Sobolev inequality on the Heisenberg group, we thus arrive at the explicit best constant for the aforementioned Poincaré inequality on H^n when p=2. We also derive the lower bound of the best constants for local Poincaré inequalities over metric balls on the Heisenberg group H^n.     

Convergence of harmonic maps on the Poincaré disk

Let be a sequence of locally quasiconformal harmonic maps on the unit disk with respect to the Poincaré metric. Suppose that the energy densities of are uniformly bounded from below by a positive constant and locally uniformly bounded from above. Then there is a subsequence of that locally uniformly converges on , and the limit function is either a locally quasiconformal harmonic map of the Poincaré disk or a constant. Especially, if the limit function is not a constant, the subsequence can be chosen to satisfy some stronger conditions. As an application, it is proved that every point of the space , a subspace of the universal Teichmüller space, can be represented by a quasiconformal harmonic map that is an asymptotic hyperbolic isometry.

     

The Poincaré polynomial of an mp arrangement

Let be an mp arrangement in a complex algebraic variety with corresponding complement and intersection poset . Examples of such arrangements are hyperplane arrangements and toral arrangements, i.e., collections of codimension 1 subtori, in an algebraic torus. Suppose a finite group acts on as a group of automorphisms and stabilizes the arrangement setwise. We give a formula for the graded character of on the cohomology of in terms of the graded character of on the cohomology of certain subvarieties in .

     

Non-accumulation of critical points of the Poincaré time on hyperbolic polycycles

We call Poincaré time the time associated to the Poincaré (or first return) map of a vector field. In this paper we prove the non-accumulation of isolated critical points of the Poincaré time on hyperbolic polycycles of polynomial vector fields. The result is obtained by proving that the Poincaré time of a hyperbolic polycycle either has an unbounded principal part or is an almost regular function. The result relies heavily on the proof of Il'yashenko's theorem on non-accumulation of limit cycles on hyperbolic polycycles.

     

The Poincaré metric and isoperimetric inequalities for hyperbolic polygons

We prove several isoperimetric inequalities for the conformal radius (or equivalently for the Poincaré density) of polygons on the hyperbolic plane. Our results include, as limit cases, the isoperimetric inequality for the conformal radius of Euclidean -gons conjectured by G. Pólya and G. Szegö in 1951 and a similar inequality for the hyperbolic -gons of the maximal hyperbolic area conjectured by J. Hersch. Both conjectures have been proved in previous papers by the third author.

Our approach uses the method based on a special triangulation of polygons and weighted inequalities for the reduced modules of trilaterals developed by A. Yu. Solynin. We also employ the dissymmetrization transformation of V. N. Dubinin. As an important part of our proofs, we obtain monotonicity and convexity results for special combinations of the Euler gamma functions, which appear to have a significant interest in their own right.

     

Poincaré duality in P.A. Smith theory

Let , or more generally be a finite -group, where is an odd prime. If acts on a space whose cohomology ring fulfills Poincaré duality (with appropriate coefficients ), we prove a mod congruence between the total Betti number of and a number which depends only on the -module structure of . This improves the well known mod congruences that hold for actions on general spaces.

     

Sur l'inégalité de Poincaré, à support compact, pour un ou plusieurs champs de vecteurs

Nous démontrons, dans cette note, une inégalité de type Poincaré pour un ou plusieurs champs de vecteurs , et des fonctions régulières à support contenu dans un voisinage d'une hypersurface , sous une hypothèse naturelle de contact entre et la famille . La constante intervenant dans cette inégalité est précisément reliée à l'épaisseur du voisinage autour de et à l'ordre du contact entre et .

     

Poincaré polynomials of moduli spaces of stable maps into flag manifolds

By using the Bialynicki-Birula decomposition and holomorphic Lefschetz formula, we calculate the Poincaré polynomials of the moduli spaces in low degrees.     

An -dimensional space that admits a Poincaré inequality but has no manifold points

For each integer we construct a compact, geodesic metric space which has topological dimension , is Ahlfors -regular, satisfies the Poincaré inequality, possesses as a unique tangent cone at almost every point, but has no manifold points.

     

Global Algebraic Poincaré–Bendixson Annulus for the van der Pol System

Differential Equations - We construct analytically two algebraic closed curves forming a Poincaré–Bendixson annulus for the van der Pol system for all values of its parameter. The inner...     

The second closed geodesic on a complex projective plane

We show the existence of at least two geometrically distinct closed geodesics on a complex projective plane with a bumpy and non-reversible Finsler metric.