Narrow Results

The authors give a complete classification of projective threefolds admitting a holomorphic conformal structure. A corollary is the complete list of projective threefolds, whose tangent bundle is a symmetric square.

Kobayashi's multiplicity-free theorem asserts that irreducible unitary highest-weight representations π are multiplicity-free when restricted to every symmetric pair if π is of scalar type. The aim of this paper is to find the "classical limit" of this multiplicity-free theorem in terms of the geometry of two coadjoint orbits, for which the correspondence is predicted by the Kirillov–Kostant–Duflo orbit method.

For this, we study the Corwin–Greenleaf multiplicity function for Hermitian symmetric spaces G/K. First, we prove that for any G-coadjoint orbit and any K-coadjoint orbit if . Here, 𝔤 = 𝔨 + 𝔭 is the Cartan decomposition of the Lie algebra 𝔤 of G.

Second, we find a necessary and sufficient condition for by means of strongly orthogonal roots. This criterion may be regarded as the "classical limit" of a special case of the Hua–Kostant–Schmid–Kobayashi branching laws of holomorphic discrete series representations with respect to symmetric pairs.

Suppose an irreducible unitary representation π of a Lie group G is obtained as a geometric quantization of a coadjoint orbit in the Kirillov–Kostant–Duflo orbit philosophy. Let H be a closed subgroup of G, and we compare the following two conditions.

(1) The restriction π|_H is discretely decomposable in the sense of Kobayashi.

(2) The momentum map is proper.

In this article, we prove that (1) is equivalent to (2) when π is any holomorphic discrete series representation of scalar type of a semisimple Lie group G and (G, H) is any symmetric pair.

Let G_ℂ be a complex simple Lie group, G_U a compact real form, and the natural projection between the dual of the Lie algebras. We prove that, for any coadjoint orbit of G_U, the intersection of with a coadjoint orbit of G_ℂ is either an empty set or a single orbit of G_U if is isomorphic to a complex symmetric space.

We classify all projective manifolds with flat holomorphic conformal structure. The Kähler–Einstein case was treated by Kobayashi and Ochiai, there exists a very short list of possible manifolds. In the non-Kähler–Einstein case a classification was only known in small dimensions: by Kobayashi and Ochiai for complex surfaces and by the authors for projective threefolds. This paper completes the general case.

Let $G/K$ be a non-compact irreducible Hermitian symmetric space of rank $r$ and let $NAK$ be an Iwasawa decomposition of $G$. The group $N$ acts on $G/K$ by biholomorphisms and the real $r$-dimensional submanifold $A\cdot eK$ intersects every $N$-orbit transversally in a single point. Moreover $A\cdot eK$ is contained in a complex $r$-dimensional submanifold of $G/K$ biholomorphic to ${ℍ}^r$, the product of $r$ copies of the upper half-plane in $ℂ$. This fact leads to a one-to-one correspondence between $N$-invariant domains in $G/K$ and tube domains in ${ℍ}^r$. In this setting we prove an analogue of Bochner’s tube theorem. Namely, an $N$-invariant domain $D$ in $G/K$ is Stein if and only if the base $\mathrm{\Omega }$ of the associated tube domain is convex and “cone invariant”.

We also prove the univalence of $N$-invariant holomorphically separable Riemann domains over $G/K$. This yields a precise description of the envelope of holomorphy of an arbitrary $N$-invariant domain in $G/K$. Finally, we obtain a characterization of several classes of $N$-invariant plurisubharmonic functions on $D$ in terms of the corresponding classes of convex functions on $\mathrm{\Omega }$. As an application we present an explicit Lie group theoretical description of all $N$-invariant potentials of the Killing metric on $G/K$ and of the associated moment maps.

We investigate the structure of real hypersurfaces with isometric Reeb flow in Kähler manifolds. As an application we classify real hypersurfaces with isometric Reeb flow in irreducible Hermitian symmetric spaces of compact type.

In this paper, a generalized Darboux transformation is obtained for Fordy–Kulish NLS (nonlinear Schrödinger) systems on general Hermitian symmetric spaces in order to rigorously obtain rogue wave solutions for these systems. In particular, we express the generalized algebraic relations in a simple and elegant compact form. As an illustration, we derive multi-soliton, breather-type and mainly rogue wave solutions of triangular patterns for single- and multi-component NLS systems on $C{P}^1$ and $SP(2)/U(2),$ respectively. We also analyze the modulation instability of proper plane wave solutions. In order to get visual intuition for the dynamics of the result and solutions for the running examples, the associated simulations of profiles are furnished as well.

In this paper, we construct noncompact homogeneous Lagrangian submanifolds in some Hermitian symmetric spaces of noncompact type. Our noncompact homogeneous Lagrangian submanifolds are obtained as connected closed subgroups of the solvable part of the Iwasawa decomposition.

In this paper we discuss a method to construct a family of Lagrangian submanifolds in an (n + 1)-dimensional complex projective space ℂPⁿ⁺¹ from an arbitrary given Lagrangian submanifold in an n-dimensional complex hyperquadric Q_n(ℂ) ⊂ ℂPⁿ⁺¹. We use the moment map and symplectic quotient techniques involving with the isoparametric function on ℂPⁿ⁺¹ associated to a rank 2 Riemannian symmetric pair (SO(n+4), SO(2)×SO(n+2)). Moreover by using this method we also show similar results for Lagrangian submanifolds in Q_n+1(ℂ) and Q_n(ℂ), and Lagrangian submanifolds in ℂPⁿ⁺¹ and ℂP_n. This paper is an improved version and a continuation of our previous article [18].