International Journal of Mathematics

Based on the structure of hyperfinite II₁ factor, we study its Dirichlet forms which can be constructed from Dirichlet forms on M_2ⁿ(ℂ).

We show that a general prime Fano three-fold X of genus 8 can be reconstructed from the pair (Γ, L), where Γ is its Fano curve of lines and L = O_Γ(1) is the theta-characteristic which gives a natural embedding Γ ⊂ ℙ⁵.

We generalize the Ruh–Vilms problem by characterizing the submanifolds in Euclidean spaces with proper biharmonic Gauss map and we construct examples of such hypersurfaces.

It is proved in this article, that in the framework of Riemannian geometry, the existence of large sets of antipodes (i.e. farthest points) for diametral points of a smooth surface has very strong consequences on the topology and the metric of this surface. Roughly speaking, if the sets of antipodes of diametral points are closed curves, then the surface is nothing but the real projective plane.

Let G be a locally compact abelian group with dual group Ĝ. The Hausdorff–Young theorem states that if f ∈ L^p(G), where 1 ≤ p ≤ 2, then its Fourier transform belongs to L^q(Ĝ) (where (1/p) + (1/q) = 1) and . Kunze and Terp extended this to unimodular and locally compact groups, respectively. We further generalize this result to an arbitrary locally compact quantum group 𝔾 by defining a Fourier transform and showing that this Fourier transform satisfies the Hausdorff–Young inequality.

We consider principal bundles over G/P, where P is a parabolic subgroup of a semi-simple and simply connected linear algebraic group G defined over ℂ. We prove that a holomorphic principal H-bundle E_H → G/P, where H is a complex reductive group, and is homogeneous if the adjoint vector bundle ad(E_H) is homogeneous. Fix a faithful H-module V. We also show that E_H is homogeneous if the vector bundle E_H ×^H V associated to it for the H-module V is homogeneous.

We generalize the notions of the Futaki invariant and extremal vector field of a compact Kähler manifold to the general almost-Kähler case and show the periodicity of the extremal vector field when the symplectic form represents an integral cohomology class modulo torsion. We also give an explicit formula of the Hermitian scalar curvature in Darboux coordinates which allows us to obtain examples of non-integrable extremal almost-Kähler metrics saturating LeBrun's estimates.

We prove a classification result for properly outer actions σ of discrete amenable groups G on strongly amenable subfactors of type II, N ⊂ M, a class of subfactors that were shown to be completely classified by their standard invariant , in [27]. The result shows that the action σ is completely classified in terms of the action it induces on . As an application of this, we obtain that inclusions of type III_λ factors, 0 < λ < 1, having discrete decomposition and strongly amenable graph, are completely classified by their standard invariant.