Closed Geodesics on Incomplete Surfaces

We consider the problem of finding embedded closed geodesics on the two-sphere with an incomplete metric defined outside a point. Various techniques including curve shortening methods are used The authors acknowledge the support of the Australian Research Council     

Homotopy Types of Stabilizers and Orbits of Morse Functions on Surfaces

Let M be a smooth compact surface, orientable or not, with boundary or without it, P either the real line ℝ ¹ or the circle S ¹, and D(M) the group of diffeomorphisms of M acting on C^∞(M, P) by the rule h⋅ f = f ∘ h ⁻¹ for h ∊ D(M) and f ∊ C^∞ (M,P). Let f: M → P be an arbitrary Morse mapping, Σ _f the set of critical points of f, D(M,Σ _f ) the subgroup of D(M) preserving Σ _f , and S(f), S (f,Σ _f ), O(f), and O(f,Σ _f ) the stabilizers and the orbits of f with respect to D(M) and D(M,Σ _f ). In fact S(f) = S(f,Σ _f ).In this paper we calculate the homotopy types of S(f), O(f) and O(f,Σ _f ). It is proved that except for few cases the connected components of S(f) and O(f,Σ _f ) are contractible, π _k O(f) = π _k M for k ≥ 3, π₂ O(f) = 0, and π₁ O(f) is an extension of π₁ D(M) ⊕ Z ^k (for some k ≥ 0) with a (finite) subgroup of the group of automorphisms of the Kronrod-Reeb graph of f.We also generalize the methods of F. Sergeraert to give conditions for a finite codimension orbit of a tame smooth action of a tame Lie group on a tame Fréchet manifold to be a tame Fréchet manifold itself. In particular, we obtain that O(f) and O(f, Σ _f ) are tame Fréchet manifolds. Communicated by Peter Michor Vienna Mathematics Subject Classifications (2000): 37C05, 57S05, 57R45.     

On a Criterion for Catalan's Conjecture

We give a new proof of a theorem of P. Mihailescu which states that the equation x ^p – y ^q = 1 is unsolvable with x, y integral and p, q odd primes, unless the congruences p ^q p (mod q ²) and q ^p q (mod p ²) hold.     

On the ergodicity conditions for stochastically recursive sequences

The ergodicity criterion for stochastically recursive sequences is given in terms of imbedded subsequences. Several examples are considered.     

The Fibered Isomorphism Conjecture for Complex Manifolds

In this paper we show that the Fibered Isomorphism Conjecture of Farrell and Jones, corresponding to the stable topological pseudoisotopy functor, is true for the fundamental groups of a class of complex manifolds. A consequence of this result is that the Whitehead group, reduced projective class groups and the negative K-groups of the fundamental groups of these manifolds vanish whenever the fundamental group is torsion free. We also prove the same results for a class of real manifolds including a large class of 3-manifolds which has a finite sheeted cover fibering over the circle.     

On codimension one foliations with Morse singularities on three-manifolds

A closed, connected oriented three-manifold supporting a codimension one oriented smooth foliation with Morse singularities having more centers than saddles and without saddle connections is diffeomorphic to the three-sphere. The use of the Reeb Stability theorem in place of the Poincaré-Bendixson theorem paves the way to a three-dimensional version, for foliations with singularities of Morse type, of a classical result of Haefliger. Finally, we give an example of a codimension one C^∞ foliation in the closed ball , with only one singularity which is of saddle type 2-2 and transverse to the boundary S³=∂B⁴.     

次二次Hamilton系统的对偶Morse指标及其闭特征的稳定性

本文通过Ｇｅｌｅｒｋｉｎ逼近方法,在没有任何凸的条件下,研究了次二次Ｈａｍｉｌ－ｔｏｎ系统的ｋ对偶Ｍｏｒｓｅ指标理论．作为应用,在本文研究了Ｒ２ｎ中的凸超曲面上的闭特征的稳定性,证明了在一个较宽松的夹条件下,这类超曲面上至少有一条椭圆闭特征．     

On Lang's Conjecture for Surfaces of General Type

Abstract

Let m be a nonnegative integer. We explicitly bound the number of rational curves with arithmetic genus no greater than m on a surface of general type by geometric invariants of the surface. We also briefly discuss the possibility of bounding all rational curves lying on a surface of general type in ?³.     

Fundamental Theorems of Morse Theory for Optimization on Manifolds with Corners

This note extends the fundamental theorems of Morse theory for stable stationary solutions to optimization problems on manifolds with corners.     

On Operators on Polynomials Preserving Real-Rootedness and the Neggers-Stanley Conjecture

We refine a technique used in a paper by Schur on real-rooted polynomials. This amounts to an extension of a theorem of Wagner on Hadamard products of Pólya frequency sequences. We also apply our results to polynomials for which the Neggers-Stanley Conjecture is known to hold. More precisely, we settle interlacing properties for E-polynomials of series-parallel posets and column-strict labelled Ferrers posets.     

Poles of Zeta Functions on Normal Surfaces

Let (S, 0) be a normal surface germ and Let f a non-constantregular function on Let (S, 0) with Let f(0) = 0. Using anyadditive invariant on complex algebraic varieties one can associatea zeta function to these data, where the topological and motiviczeta functions are the roughest and the finest zeta functions,respectively. In this paper we are interested in a geometricdetermination of the poles of these functions. The second authorhas already provided such a determination for the topologicalzeta function in the case of non-singular surfaces. Here wegive a complete answer for all normal surfaces, at least onthe motivic level. The topological zeta function however seemsto be too rough for this purpose, although for negative poles,which are the only ones in the non-singular case, we are ableto prove exactly the same result as for non-singular surfaces. We also give and verify a (natural) definition for when a rationalnumber is a pole of the motivic zeta function. 2000 MathematicsSubject Classification 14B05, 14E15, 14J17 (primary), 32S50(secondary).     

On the Generalized Ray-Singer Conjecture for Covering Spaces

We find a relation between L² -combinatorial torsion and analytic torsion associated with a small spectrum of the Witten-type Laplacian on a covering space. The result is used as one crucial step in a proof of the generalized Ray-Singer conjecture for covering spaces.     

E~3中闭曲面上的一个积分公式及其应用

利用Minkowski公式,得到了等距浸入在E3中的闭曲面上的一个有趣积分公式,应用此公式得到了闭曲面的一个特征.     

Properties of the Absolute That Affect Smoothness of Flows on Closed Surfaces

Let be a closed orientable surface of genus , endowed with the structure of a Riemann manifold of constant negative curvature. For the universal covering , there is the notion of absolute, each of whose points determines an asymptotic direction of a bundle of parallel equidirected geodesics. In the paper it is proved that there is a set on the absolute having the cardinality of the continuum and such that if an arbitrary flow on has a semitrajectory whose covering has asymptotic direction defined by a point from , then this flow is not analytical and has infinitely many stationary points.     

On Brennan's Conjecture for a Special Class of Functions

In this paper, we prove Brennan's conjecture for conformal mappings f of the disk {z : | z| < 1} assuming that the Taylor coefficients of the function log(zf′(z)/f(z)) at zero are nonnegative. We also obtain inequalities for the integral means over the circle |z| = r of the squared modulus of the function zf′(z)/f(z).     

Induced orbit data and induced unitary representations for complex groups

We use induced orbit covers to define induced orbit data.By studying the space of regular functions on orbit cover,we know that the induced representation has close connection with the induced orbit datum under the meaning of Vogan’s conjecture.Therefore,when verifying Vogan’s conjecture,many cases can be reduced to the case of rigid orbit data.     

广义单瞬时生-灭过程的Kendall猜想

A new structure with the special property that instantaneous state and catastrophes is imposed to ordinary birth-death processes is considered. Kendall's conjecture for the processes is proved to be right.     

A Morse Index Theorem for Elliptic Operators on Bounded Domains

Given a selfadjoint, elliptic operator L, one would like to know how the spectrum changes as the spatial domain Ω ? ? ⁿ is deformed. For a family of domains {Ω _t } _{t∈[a, b]} we prove that the Morse index of L on Ω _a differs from the Morse index of L on Ω _b by the Maslov index of a path of Lagrangian subspaces on the boundary of Ω. This is particularly useful when Ω _a is a domain for which the Morse index is known, e.g. a region with very small volume. Then the Maslov index computes the difference of Morse indices for the “original” problem (on Ω _b ) and the “simplified” problem (on Ω _a ). This generalizes previous multi-dimensional Morse index theorems that were only available on star-shaped domains or for Dirichlet boundary conditions. We also discuss how one can compute the Maslov index using crossing forms, and present some applications to the spectral theory of Dirichlet and Neumann boundary value problems.