Duality and the Poincaré–Hopf Inequalities

In this article we obtain a duality result for an n-manifold N with boundary ∂N = N + ⊔N ⁻ a disjoint union, where N ⁺ and N ⁻ are arbitrarily chosen parts in ∂N and need not be compact. This duality result is used to generalize the Poincaré–Hopf inequalities in a non-compact setting.     

Poincaré's recurrence theorem and the unitarity of the S-matrix

A scattering process can be described by suitably closing the system and considering the first return map from the entrance onto itself. This scattering map may be singular and discontinuous, but it will be measured preserving as a consequence of the recurrence theorem applied to any region of a simpler map. In the case of a billiard this is the Birkhoff map. The semiclassical quantization of the Birkhoff map can be subdivided into an entrance and a repeller. The construction of a scattering operator then follows in exact analogy to the classical process. Generically, the approximate unitarity of the semiclassical Birkhoff map is inherited by the S-matrix, even for highly resonant scattering where direct quantization of the scattering map breaks down.     

Poincaré–Sobolev equations in the hyperbolic space

We study the a priori estimates, existence/nonexistence of radial sign changing solution, and the Palais–Smale characterisation of the problem ${-\Delta_{{\mathbb B}^{N}}u - \lambda u = |u|^{p-1}u, u\in H^1({\mathbb B}^{N})}$ in the hyperbolic space ${{\mathbb B}^{N}}$ where ${1 < p\leq\frac{N+2}{N-2}}$ . We will also prove the existence of sign changing solution to the Hardy–Sobolev–Mazya equation and the critical Grushin problem.     

A quantitative version of Steinhaus’ theorem for compact, connected, rank-one symmetric spaces

Let  $d_1,\,d_2$ , ... be a sequence of positive numbers that converges to zero. A generalization of Steinhaus’ theorem due to Weil implies that, if a subset of a homogeneous Riemannian manifold has no pair of points at distances  $d_1,\,d_2$ , ... from each other, then it has to have measure zero. We present a quantitative version of this result for compact, connected, rank-one symmetric spaces, by showing how to choose distances so that the measure of a subset not containing pairs of points at these distances decays exponentially in the number of distances.     

On the Existence of Solutions of Poisson Equation and Poincaré–Lelong Equation

One of the main goals of this paper is to solve the Poincaré–Lelong equation on a class of Kähler manifolds with nonnegative holomorphic bisectional curvature, $\mathrm{Ric}(x)\geq \left(a\ln\ln\left(10+r(x)\right)\right)\Big/\big.\left(\left(1+r^2(x)\right)\ln(10+r(x))\right)One of the main goals of this paper is to solve the Poincaré–Lelong equation on a class of K?hler manifolds with nonnegative holomorphic bisectional curvature, for some a > 67(n + 4)². We will also study the Poisson equation on complete noncompact manifolds which satisfy volume doubling and Poincaré inequality.     

Nielsen theory, Floer homology and a generalisation of the Poincaré-Birkhoff theorem

The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology for symplectomorphisms of surfaces and a calculation of Seidel’s symplectic Floer homology for different mapping classes. We also describe symplectic zeta functions and an asymptotic symplectic invariant. A generalisation of the Poincaré-Birkhoff fixed point theorem and Arnold conjecture is proposed. Dedicated to Vladimir Igorevich Arnold     

A theorem connected with results of Steinhaus and Smítal

We will prove that if A and B are subsets of the real line, each having positive outer Lebesgue measure, then A + B, the set of all numbers a + b with a ϵ A and b ϵ B, is “full,” in the sense of outer Lebesgue measure, in some interval K. This result is related to theorems of Steinhaus and Smítal.     

Combinatorial cell complexes and Poincaré duality

We define and study a class of finite topological spaces, which model the cell structure of a space obtained by gluing finitely many Euclidean convex polyhedral cells along congruent faces. We call these finite topological spaces, combinatorial cell complexes (or c.c.c). We define orientability, homology and cohomology of c.c.c’s and develop enough algebraic topology in this setting to prove the Poincaré duality theorem for a c.c.c satisfying suitable regularity conditions. The definitions and proofs are completely finitary and combinatorial in nature.     

Picard’s theorem and the Rickman construction

Nevanlinna theory (value-distribution theory) has its genesis in Picard’s discovery that a function analytic in the plane which omits two values is constant. Nearly a century later, attention turned to the analogous situation in Rn, n≥3, where entire functions are necesarily replaced by entire quasiregular mappings. This expository article centers on one of Seppo Rickman’s main contributions to this issue, including an outline of his famous example showing that the omitted set in R3, while finite, can be much larger than possible in the plane.     

Addendum to the paper “on a theorem of Miranda”

An existence theorem is proved for closed convex surfaces whose principal radii of curvature regarded as functions of the unit normal vectorn satisfy the equation

On the CR Poincaré–Lelong Equation,Yamabe Steady Solitons and Structures of Complete Noncompact Sasakian Manifolds

In this paper, we solve the so-called CR Poincaré–Lelong equation by solving the CR Poisson equation on a complete noncompact CR(2n + 1)-manifold with nonegative pseudohermitian bisectional curvature tensors and vanishing torsion which is an odd dimensional counterpart of K?hler geometry. With applications of this solution plus the CR Liouvelle property, we study the structures of complete noncompact Sasakian manifolds and CR Yamabe steady solitons.     

Poincaré–Treshchev Mechanism in Multi-scale,Nearly Integrable Hamiltonian Systems

This paper is a continuation to our work (Xu et al. in Ann Henri Poincaré 18(1):53–83, 2017) concerning the persistence of lower-dimensional tori on resonant surfaces of a multi-scale, nearly integrable Hamiltonian system. This type of systems, being properly degenerate, arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem ties closely to the stability of the systems. For such a system, under certain non-degenerate conditions of Rüssmann type, the majority persistence of non-resonant tori and the existence of a nearly full measure set of Poincaré non-degenerate, lower-dimensional, quasi-periodic invariant tori on a resonant surface corresponding to the highest order of scale is proved in Han et al. (Ann Henri Poincaré 10(8):1419–1436, 2010) and Xu et al. (2017), respectively. In this work, we consider a resonant surface corresponding to any intermediate order of scale and show the existence of a nearly full measure set of Poincaré non-degenerate, lower-dimensional, quasi-periodic invariant tori on the resonant surface. The proof is based on a normal form reduction which consists of a finite step of KAM iterations in pushing the non-integrable perturbation to a sufficiently high order and the splitting of resonant tori on the resonant surface according to the Poincaré–Treshchev mechanism.     

Functional mechanics: Evolution of the moments of distribution function and the Poincaré recurrence theorem

One of modern approaches to the problem of coordination of classical mechanics and statistical physics — functional mechanics is considered. Deviations from classical trajectories are calculated and evolution of themoments of distribution function is constructed. The relation between the received results and absence of the Poincaré-Zermelo paradox in functional mechanics is discussed. Destruction of periodicity of movement in functional mechanics is shown and decrement of attenuation for classical invariants of movement on a trajectory of functional mechanical averages is calculated.