本刊英文版Vol.30(2014),No.9论文摘要(英文)

<正>Least Number of Periodic Points of Self-maps of Lie Groups Jerzy JEZIERSKI Abstract There are two algebraic lower bounds of the number of n-periodic points of a self-map f:M→M of a compact smooth manifold of dimension at least 3:NF_n(f)=min{#Fix(g~n);g~f;g is continuous}and NJD_n(f)=min{#Fix(g~n);g~f;g is smooth}.In general,NJD_n(f)may be much greater than NF_n(f).If M is a torus,then the invariants are     

Optimal packings of two to four equal circles on any flat torus

We find explicit formulas for the radii and locations of the circles in all the optimally dense packings of two, three or four equal circles on any flat torus, defined to be the quotient of the Euclidean plane by the lattice generated by two independent vectors. We prove the optimality of the arrangements using techniques from rigidity theory and topological graph theory.     

Regularity of viscosity solutions near KAM torus

In this paper, we give weak regularity theorems on P of u~ε(x, P), where u~ε(x, P)is the viscosity solution of the cell problem H_ε(P D_xu~ε, x)=H_ε(P).     

Compact Manifolds Covered by a Torus

Let X be a compact complex manifold which is the image of a complex torus by a holomorphic surjective map A→X. We prove that X is Kähler and that up to a finite étale cover, X is a product of projective spaces by a torus.     

A characterization of the Ejiri torus in <Emphasis Type="Italic">S</Emphasis><Superscript>5</Superscript>

We conjecture that a Willmore torus having Willmore functional between 2π ² and 2π ² \(\sqrt 3 \) is either conformally equivalent to the Clifford torus, or conformally equivalent to the Ejiri torus. Ejiri’s torus in S ⁵ is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any real space form. Li and Vrancken classified all Willmore surfaces of tensor product in S ⁿ by reducing them into elastic curves in S ³, and the Ejiri torus appeared as a special example. In this paper, we first prove that among all Willmore tori of tensor product, the Willmore functional of the Ejiri torus in S ⁵ attains the minimum 2π ² \(\sqrt 3 \), which indicates our conjecture holds true for Willmore surfaces of tensor product. Then we show that all Willmore tori of tensor product are unstable when the co-dimension is big enough. We also show that the Ejiri torus is unstable even in S ⁵. Moreover, similar to Li and Vrancken, we classify all constrained Willmore surfaces of tensor product by reducing them with elastic curves in S ³. All constrained Willmore tori obtained this way are also shown to be unstable when the co-dimension is big enough.     

The biinvariant diagonal class for Hamiltonian torus actions

Suppose that an algebraic torus G acts algebraically on a projective manifold X with generically trivial stabilizers. Then the Zariski closure of the set of pairs {(x,y)∈X×X|y=gx for some g∈G} defines a nonzero equivariant cohomology class . We give an analogue of this construction in the case where X is a compact symplectic manifold endowed with a Hamiltonian action of a torus, whose complexification plays the role of G. We also prove that the Kirwan map sends the class [ΔG] to the class of the diagonal in each symplectic quotient. This allows to define a canonical right inverse of the Kirwan map.     

Equi-Gaussian curvature folding

In this paper we introduce a new type of folding called equi-Gaussian curvature folding of connected Riemannian 2-manifolds. We prove that the composition and the cartesian product of such foldings is again an equi-Gaussian curvature folding. In case of equi-Gaussian curvature foldings, f: M → P _n , of an orientable surface M onto a polygon P _n we prove that

Skew loops in flat tori

We produce skew loops—loops having no pair of parallel tangent lines—homotopic to any loop in a flat torus or other quotient of R ⁿ . The interesting case here is n = 3. More subtly for any n, we characterize the homotopy classes that will contain a skew loop having a specified loop as tangent indicatrix. A fellowship from the Lady Davis foundation helped support this work.     

Representation of torus homeotopies by simple polyhedra with boundary

In 1991, Turaev and Viro constructed a quantum topological linear representation of mapping class groups of closed surfaces. To the mappings of a surface into itself, they assigned simple polyhedra whose boundaries consisted of two simple graphs cutting the surface into cells. The computational complexity of the Turaev-Viro representations strongly depends on the choice of suitable sets of simple polyhedra. In this paper, simple polyhedra for the torus are constructed. One of the reasons why they are convenient is that they all are obtained by gluing along boundary of copies of the same simple polyhedron. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 533–539, October, 1999.     

Synthesis,Properties and Formation of (RCp)Co‐ and (RCp)Rh‐Stabilized[4.8]3Cyclacene Derivatives

Metal‐stabilized belts : A torus, 3 , consisting of three four‐ and three eight‐membered conjugated rings and stabilized by (RCp)Co‐ and (RCp)Rh‐ units, was generated by irradiation of [(RCp)Co(CO)₂] and [(RCp)Rh(C₂H₄)₂], respectively, and 1 .