International Journal of Mathematics

Let G be a connected semisimple real-rank one Lie group with finite center. We consider intertwining operators on tensor products of spherical principal series representations of G. This allows us to construct an invariant trilinear form indexed by a complex multiparameter and defined on the space of smooth functions on the product of three spheres in 𝔽ⁿ, where 𝔽 is either ℝ, ℂ, ℍ, or 𝕆 with n = 2. We then study the analytic continuation of the trilinear form with respect to (ν₁, ν₂, ν₃), where we locate the hyperplanes containing the poles. Using a result due to Johnson and Wallach on the so-called "partial intertwining operator", we obtain an expression for the generalized Bernstein–Reznikov integral in terms of hypergeometric functions.

Let (M, g) be an Asymptotically Locally Hyperbolic (ALH) manifold which is the interior of a conformally compact manifold and (∂M, [γ]) its conformal infinity. Suppose that the Ricci tensor of (M, g) dominates that of the hyperbolic space and that its scalar curvature satisfies a certain decay condition at infinity. If the Yamabe invariant of (∂M, [γ]) is non-negative, we prove that there exists a conformal metric on M with non-negative scalar curvature and whose boundary ∂M has either positive or zero constant inner mean curvature. In the spin case, we make use of a previous estimate obtained by X. Zhang and the authors for the Dirac operator of the induced metric on ∂M. As a consequence, we generalize and simplify the proof of the result by Andersson and Dahl in [Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global Anal. Geom.16 (1998) 1–27] about the rigidity of the hyperbolic space when the prescribed conformal infinity ∂M is a round sphere. We also provide non-existence results for conformally compact ALH spin metrics when ∂M is conformal to a Riemannian manifold with special holonomy.

The purpose of this paper is to investigate the convergence of the volume-preserving mean curvature flow of closed hypersurfaces in space forms. Assume that the initial hypersurface satisfies suitable integral curvature pinching conditions. We prove that along the volume-preserving mean curvature flow the initial hypersurface will be deformed to a totally umbilical sphere.

We describe the primitive ideal space of the C*-algebra of a row-finite k-graph with no sources when every ideal is gauge invariant. We characterize which spectral spaces can occur, and compute the primitive ideal space of two examples. In order to do this we prove some new results on aperiodicity. Our computations indicate that when every ideal is gauge invariant, the primitive ideal space only depends on the 1-skeleton of the k-graph in question.

Let X be a smooth prime Fano threefold of genus 7 and let Γ be its homologically projectively dual curve. We prove that, for d ≥ 6, an irreducible component of the moduli scheme M_X(2, 1, d) of rank-2 stable sheaves on X with c₁ = 1, c₂ = d is birational to a generically smooth (2d - 9)-dimensional component of the Brill–Noether variety of stable vector bundles on Γ of rank d - 5 and degree 5d-24 with at least 2d - 10 independent global sections. The space M_X(2, 1, 6) is proved to be isomorphic to , and to be a smooth irreducible threefold if X is general enough.

For a nonautonomous dynamics defined by a sequence of linear operators, we introduce the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility in l^p spaces, both for the space of perturbations and the space of solutions. This allows unifying the notions of uniform and nonuniform exponential behavior. Moreover, we consider the general case of a noninvertible dynamics. As a nontrivial application we show that the conditional stability of a nonuniform exponential dichotomy persists under sufficiently small linear perturbations.

Let 0 < ε ≤ ½ be fixed. We prove that on a bounded pseudoconvex domain D ⋐ ℂⁿ the Bergman metric grows at least like times the euclidean metric, provided that on D there exists a family (φ_δ)_δ of smooth plurisubharmonic functions with a self-bounded complex gradient (uniformly in δ), such that for any δ the Levi form of φ_δ has eigenvalues ≥ δ^-2ε on the set {z ∈ D | δ_D(z) < δ}. Here, δ_D denotes the boundary-distance function on D.

Let ℳ_i be a family of II₁-factors, containing a common II₁-subfactor 𝒩, such that [ℳ_i : 𝒩] ∈ ℕ₀ for all i. Furthermore, let ϕ: ℕ₀ → ℂ. We show that if a Hankel matrix related to ϕ is trace-class, then there exists a unique completely bounded map M_ϕ on the amalgamated free product of the ℳ_i with amalgamation over 𝒩, which acts as a radial multiplier. Hereby, we extend a result of Haagerup and the author for radial multipliers on reduced free products of unital C*- and von Neumann algebras.

The notion of a complex tangent arises for embeddings of real manifolds into complex spaces. It is of particular interest when studying embeddings of real n-dimensional manifolds into ℂⁿ. The generic topological structure of the set complex tangents to such embeddings Mⁿ ↪ ℂⁿ takes the form of a (stratified) (n-2)-dimensional submanifold of Mⁿ. In this paper, we generalize our results from our previous work for the 3-dimensional sphere and the Heisenberg group to obtain results regarding the possible topological configurations of the sets of complex tangents to embeddings of odd-dimensional spheres S^2n-1 ↪ ℂ^2n-1 by first considering the situation for the higher-dimensional analogues of the Heisenberg group.