International Journal of Mathematics

The irreducible decomposition of scalar holomorphic discrete series representations when restricted to semisimple symmetric pairs (G, H) is explicitly known by Schmid [Die Randwerte holomorphe funktionen auf hermetisch symmetrischen Raumen, Invent. Math.9 (1969–1970) 61–80] for H compact and by Kobayashi [Multiplicity-Free Theorems of the Restrictions of Unitary Highest Weight Modules with Respect to Reductive Symmetric Pairs, Progress in Mathematics, Vol. 255 (Birhäuser, 2007), pp. 45–109] for H non-compact. In this paper, we deal with the symmetric pair (U(n, n), SO*(2n)), and extend the Kobayashi–Schmid formula to certain non-tempered unitary representations which are realized in Dolbeault cohomology groups over open Grassmannian manifolds with indefinite metric. The resulting branching rule is multiplicity-free and discretely decomposable, which fits in the framework of the general theory of discrete decomposable restrictions by Kobayashi [Discrete decomposability of the restriction of A_𝔮(λ) with respect to reductive subgroups II — micro-local analysis and asymptotic K-support, Ann. Math.147 (1998), 709–729].

We generalize the representation theorem of Junge, Neufang and Ruan in [A representation theorem for locally compact quantum groups, Internat. J. Math.20(3) (2009) 377–400], and some of the important results which were used in its proof, to the case of actions of locally compact quantum groups on von Neumann algebras.

Using degeneration to scrolls, we give an easy proof of non-existence of curves of low genera on general surfaces in ℙ³ of degree d ≥ 5. We also show that there exist Kobayashi hyperbolic surfaces in ℙ³ of degree d = 7 (a result so far unknown), and give a new construction of such surfaces of degree d = 6. Our method yields also some lower bounds for geometric genera of surfaces lying on general hypersurfaces of degree 3d ≥ 15 in ℙ⁴.

Let G be a finite group. A subgroup H of G is called weakly normal in G if H^g ≤ N_G(H) implies g ∈ N_G(H) for all g ∈ G. A finite group G is called an -group if all cyclic subgroups of G of order prime or 4 are weakly normal in G. In this paper, the structure of finite groups all of whose second maximal subgroups satisfy -property has been characterized.

Chen famously conjectured that every submanifold of Euclidean space with harmonic mean curvature vector is minimal. In this note, we establish a much more general statement for a large class of submanifolds satisfying a growth condition at infinity. We discuss in particular two popular competing natural interpretations of the conjecture when the Euclidean background space is replaced by an arbitrary Riemannian manifold. Introducing the notion of ε-superbiharmonic submanifolds, which contains each of the previous notions as special cases, we prove that ε-superbiharmonic submanifolds of a complete Riemannian manifold which satisfy a growth condition at infinity are minimal.

Motivated by Gage [On an area-preserving evolution equation for plane curves, in Nonlinear Problems in Geometry, ed. D. M. DeTurck, Contemporary Mathematics, Vol. 51 (American Mathematical Society, Providence, RI, 1986), pp. 51–62] and Ma–Cheng [A non-local area preserving curve flow, preprint (2009), arXiv:0907.1430v2, [math.DG]], in this paper, an area-preserving flow for convex plane curves is presented. This flow will decrease the perimeter of the evolving curve and make the curve more and more circular during the evolution process. And finally, as t goes to infinity, the limiting curve will be a finite circle in the C^∞ metric.

Realizing the possibility suggested by Hardouin [Iterative q-difference Galois theory, J. Reine Angew. Math.644 (2010) 101–144], we show that her own Picard–Vessiot (PV) theory for iterative q-difference rings is covered by the (consequently, more general) framework, settled by Amano and Masuoka [Picard–Vessiot extensions of artinian simple module algebras, J. Algebra285 (2005) 743–767], of artinian simple module algebras over a cocommutative pointed Hopf algebra. An essential point is to represent iterative q-difference modules over an iterative q-difference ring R, by modules over a certain cocommutative ×_R-bialgebra. Recall that the notion of ×_R-bialgebras was defined by Sweedler [Groups of simple algebras, Publ. Math. Inst. Hautes Études Sci.44 (1974) 79–189], as a generalization of bialgebras.

We showed that there is a complete analogue of a representation of the quantum plane where |q| = 1, with the classical ax+b group. We showed that the Fourier transform of the representation of on has a limit (in the dual corepresentation) toward the Mellin transform of the unitary representation of the ax+b group, and furthermore the intertwiners of the tensor products representation has a limit toward the intertwiners of the Mellin transform of the classical ax+b representation. We also wrote explicitly the multiplicative unitary defining the quantum ax+b semigroup and showed that it defines the corepresentation that is dual to the representation of above, and also correspond precisely to the classical family of unitary representation of the ax+b group.

It is proved that on any smoothly bounded domain Ω ⋐ ℝⁿ, n ≥ 2, the output of the harmonic Bergman projection belongs to the Sobolev space of order k ∈ ℕ as long as all tangential derivatives of order up to k of the input function belong to L²(Ω).