A gap theorem for hypersurfaces of the sphere with constant scalar curvature one

We consider closed hypersurfaces of the sphere with scalar curvature one, prove a gap theorem for a modified second fundamental form and determine the hypersurfaces that are at the end points of the gap. As an application we characterize the closed, two-sided index one hypersurfaces with scalar curvature one in the real projective space. Received: October 12, 2001     

GENERALIZED SIMPLE NONCOMMUTATIVE TORI

The generalized noncommutative torus Tkp of rank n was defined in [4] by the crossed product Am/k ×a3 Z ×a4 … ×an Z, where the actions ai of Z on the fibre Mk(C) of a rational rotation algebra Am/k are trivial, and C*(kZ × kZ) ×a3 Z ×a4 ... ×an Z is a completely irrational noncommutative torus Ap of rank n. It is shown in this paper that Tkp is strongly Morita equivalent to Ap, and that Tkp (?) Mp∞ is isomorphic to Ap (?) Mk(C) (?) Mp∞ if and only if the set of prime factors of k is a subset of the set of prime factors of p.     

Fillable contact structures on torus bundles over circles

We will construct an example of a strongly symplectically fillable contact structure on a torus bundle over the circle with parabolic monodromy.

     

Formal Markoff maps are positive

This note defines a family of Laurent polynomials indexed in which generalize the Markoff numbers and relate to the character variety of the one-cusped torus. We describe which monomials appear in each polynomial and prove all the coefficients are positive integers. We also conjecture a generalization of that positivity result.      

n+2维最大秩幂零n-Lie代数及其导子代数

根据最大秩幂零n-Lie代数的概念及有关结论, 证明不可分解非Abel最大秩幂零的n 2维n-Lie代数在同构意义下只有一类, 给出了具体的乘法表, 并讨论了它的导子代数及其内导子代数.     

The discriminant of an algebraic torus

For a torus T defined over a global field K, we revisit an analytic class number formula obtained by Shyr in the 1970s as a generalization of Dirichlet?s class number formula. We prove a local-global presentation of the quasi-discriminant of T, which enters into this formula, in terms of cocharacters of T. This presentation can serve as a more natural definition of this invariant.     

Lie Tori of Type BC 2 and Structurable Quasitori

Structurable quasitori are introduced and shown to give all Lie tori of type BC ₂ via the Kantor construction. Structurable quasitori, including Jordan quasitori, are classified. The methods include a use of generalized quadrangles of order (2,t).     

Quantum hyperbolic invariants for diffeomorphisms of small surfaces

An earlier article [Bonahon, F., Liu, X. B.: Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms. Geom. Topol., 11, 889-937 (2007)] introduced new invariants for pseudo-Anosov diffeomorphisms of surface, based on the representation theory of the quantum Teichmu¨ller space. We explicitly compute these quantum hyperbolic invariants in the case of the 1-puncture torus and the 4-puncture sphere.