Noncompact Symmetric Spaces Research Articles

Wave equation on general noncompact symmetric spaces

abstract: We establish sharp pointwise kernel estimates and dispersive properties for the wave equation on noncompact symmetric spaces of general rank. This is achieved by combining the stationary phase method and the Hadamard parametrix, and in particular, by introducing a subtle spectral decomposition, which allows us to overcome a well-known difficulty in higher rank analysis, namely the fact that the Plancherel density is not a differential symbol in general. Consequently, we deduce the Strichartz inequality for a large family of admissible pairs and prove global well-posedness results for the corresponding semi-linear equation with low regularity data as on hyperbolic spaces.

Sharp Adams Type Inequalities for the Fractional Laplace–Beltrami Operator on Noncompact Symmetric Spaces

Sharp Adams Type Inequalities for the Fractional Laplace–Beltrami Operator on Noncompact Symmetric Spaces

A unified Fourier slice method to derive ridgelet transform for a variety of depth-2 neural networks

To investigate neural network parameters, it is easier to study the distribution of parameters than to study the parameters in each neuron. The ridgelet transform is a pseudo-inverse operator that maps a given function f to the parameter distribution γ so that a network NN[γ] reproduces f, i.e. NN[γ]=f. For depth-2 fully-connected networks on a Euclidean space, the ridgelet transform has been discovered up to the closed-form expression, thus we could describe how the parameters are distributed. However, for a variety of modern neural network architectures, the closed-form expression has not been known. In this paper, we explain a systematic method using Fourier expressions to derive ridgelet transforms for a variety of modern networks such as networks on finite fields Fp, group convolutional networks on abstract Hilbert space H, fully-connected networks on noncompact symmetric spaces G/K, and pooling layers, or the d-plane ridgelet transform.

The snapshot problem for the wave equation

By definition, a wave is a C∞ solution u(x,t) of the wave equation on Rn, and a snapshot of the wave u at time t is the function ut on Rn given by ut(x)=u(x,t). We show that there are infinitely many waves with given C∞ snapshots f0 and f1 at times t=0 and t=1 respectively, but that all such waves have the same snapshots at integer times. We present a necessary condition for the uniqueness, and a compatibility condition for the existence, of a wave u to have three given snapshots at three different times, and we show how this compatibility condition leads to the problem of small denominators and Liouville numbers. We extend our results to wave equations on noncompact symmetric spaces. Finally, we consider the two-snapshot problem and corresponding small denominator results for the wave equation on the n-sphere.

Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces

This work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces G/K of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi-K-invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi-K-invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on G/K. In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} asymptotic convergence without the assumption of bi-K-invariance.

Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds

This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.

Geometry of Hermitian symmetric spaces under the action of a maximal unipotent group

Let [Formula: see text] be a non-compact irreducible Hermitian symmetric space of rank [Formula: see text] and let [Formula: see text] be an Iwasawa decomposition of [Formula: see text]. The group [Formula: see text] acts on [Formula: see text] by biholomorphisms and the real [Formula: see text]-dimensional submanifold [Formula: see text] intersects every [Formula: see text]-orbit transversally in a single point. Moreover [Formula: see text] is contained in a complex [Formula: see text]-dimensional submanifold of [Formula: see text] biholomorphic to [Formula: see text], the product of [Formula: see text] copies of the upper half-plane in [Formula: see text]. This fact leads to a one-to-one correspondence between [Formula: see text]-invariant domains in [Formula: see text] and tube domains in [Formula: see text]. In this setting we prove an analogue of Bochner’s tube theorem. Namely, an [Formula: see text]-invariant domain [Formula: see text] in [Formula: see text] is Stein if and only if the base [Formula: see text] of the associated tube domain is convex and “cone invariant”. We also prove the univalence of [Formula: see text]-invariant holomorphically separable Riemann domains over [Formula: see text]. This yields a precise description of the envelope of holomorphy of an arbitrary [Formula: see text]-invariant domain in [Formula: see text]. Finally, we obtain a characterization of several classes of [Formula: see text]-invariant plurisubharmonic functions on [Formula: see text] in terms of the corresponding classes of convex functions on [Formula: see text]. As an application we present an explicit Lie group theoretical description of all [Formula: see text]-invariant potentials of the Killing metric on [Formula: see text] and of the associated moment maps.

Classification of homogeneous hypersurfaces in some noncompact symmetric spaces of rank two

We classify, up to isometric congruence, the homogeneous hypersurfaces in the Riemannian symmetric spaces textrm{SL}(3,mathbb {H})/textrm{Sp}(3), textrm{SO}(5,mathbb {C})/textrm{SO}(5), and textrm{Gr}^*(2,mathbb {C}^{n+4}) = textrm{SU}(n+2,2)/textrm{S}(textrm{U}(n+2)textrm{U}(2)), , n geqslant 1.

Horofunction Compactifications and Duality

We study the global topology and geometry of the horofunction compactification of classes of symmetric spaces under Finsler distances in three settings: bounded symmetric domains of the form B=B_1times cdots times B_r, where B_i is an open Euclidean ball in {mathbb {C}}^{n_i}, with the Kobayashi distance, symmetric cones with the Hilbert distance, and Euclidean Jordan algebras with the spectral norm. For these spaces we show, that the horofunction compactification is naturally homeomorphic to the closed unit ball of the dual norm of the Finsler metric in the tangent space at the basepoint. In each case we give an explicit homeomorphism. For finite dimensional normed spaces the link between the geometry of the horofunction compactification and the dual unit ball was suggested by Kapovich and Leeb, which we confirm for Euclidean Jordan algebras with the spectral norm. Our results also show that this duality phenomenon not only occurs in normed spaces, but also in a variety of noncompact type symmetric spaces with invariant Finsler metrics.

Schrödinger equation on non-compact symmetric spaces

We establish sharp-in-time kernel and dispersive estimates for the Schrödinger equation on non-compact Riemannian symmetric spaces of any rank. Due to the particular geometry at infinity and the Kunze-Stein phenomenon, these properties are more pronounced in large time and enable us to prove the global-in-time Strichartz inequality for a larger family of admissible couples than in the Euclidean case. Consequently, we obtain the global well-posedness for the corresponding semilinear equation with lower regularity data and some scattering properties for small powers which are known to fail in the Euclidean setting. The crucial kernel estimates are achieved by combining the stationary phase method based on a subtle barycentric decomposition, a subordination formula of the Schrödinger group to the wave propagator and an improved Hadamard parametrix.

Asymptotic behavior of solutions to the heat equation on noncompact symmetric spaces

This paper is twofold. The first part aims to study the long-time asymptotic behavior of solutions to the heat equation on Riemannian symmetric spaces G/K of noncompact type and of general rank. We show that any solution to the heat equation with bi-K-invariant L1 initial data behaves asymptotically as the mass times the fundamental solution, and provide a counterexample in the non bi-K-invariant case. These answer problems recently raised by J.L. Vázquez. In the second part, we investigate the long-time asymptotic behavior of solutions to the heat equation associated with the so-called distinguished Laplacian on G/K. Interestingly, we observe in this case phenomena which are similar to the Euclidean setting, namely L1 asymptotic convergence with no bi-K-invariance condition and strong L∞ convergence.

Full Laplace spectrum of distance spheres in symmetric spaces of rank one

We use Lie-theoretic methods to explicitly compute the full spectrum of the Laplace--Beltrami operator on homogeneous spheres which occur as geodesic distance spheres in (compact or noncompact) symmetric spaces of rank one, and provide a single unified formula for all cases. As an application, we find all resonant radii for distance spheres in the compact case, i.e., radii where there is bifurcation of embedded constant mean curvature spheres, and show that distance spheres are stable and locally rigid in the noncompact case.

Symplectic geometry of Cartan–Hartogs domains

This paper studies the geometry of Cartan–Hartogs domains from the symplectic point of view. Inspired by duality between compact and noncompact Hermitian symmetric spaces, we construct a dual counterpart of Cartan–Hartogs domains and give explicit expression of global Darboux coordinates for both Cartan–Hartogs domains and their dual. Further, we compute their symplectic capacity and show that a Cartan–Hartogs domain admits a symplectic duality if and only if it reduces to be a complex hyperbolic space.

Explicit Proper p-Harmonic Functions on the Riemannian Symmetric Spaces mathbf{SU} (n)/mathbf{SO} (n), mathbf{Sp} (n)/mathbf{U} (n), mathbf{SO} (2n)/mathbf{U} (n), mathbf{SU} (2n)/mathbf{Sp} (n)

In this work we construct explicit complex-valued p-harmonic functions on the compact Riemannian symmetric spaces mathbf{SU} (n)/mathbf{SO} (n), mathbf{Sp} (n)/mathbf{U} (n), mathbf{SO} (2n)/mathbf{U} (n), mathbf{SU} (2n)/mathbf{Sp} (n). We also describe how the same can be manufactured on their non-compact symmetric dual spaces.

Marcinkiewicz-type multipliers on products of noncompact symmetric spaces

In this paper we prove a Marcinkiewicz-type multiplier result for the spherical Fourier transform on products of rank one noncompact symmetric spaces.

Wave equation on certain noncompact symmetric spaces

In this paper, we prove sharp pointwise kernel estimates and dispersive properties for the linear wave equation on noncompact Riemannian symmetric spaces G/K of any rank with G complex. As a consequence, we deduce Strichartz inequalities for a large family of admissible pairs and prove global well-posedness results for the corresponding semilinear equation with low regularity data as on hyperbolic spaces.

Invariant plurisubharmonic functions on non-compact Hermitian symmetric spaces

Let ,G/K, be an irreducible non-compact Hermitian symmetric space and let ,D, be a ,K-invariant domain in ,G/K. In this paper we characterize several classes of ,K-invariant plurisubharmonic functions on ,D, in terms of their restrictions to a slice intersecting all ,K-orbits. As applications we show that ,K-invariant plurisubharmonic functions on ,D, are necessarily continuous and we reproduce the classification of Stein ,K-invariant domains in ,G/K, obtained by Bedford and Dadok. (J Geom Anal 1:1–17, 1991).

Codimension one Ricci soliton subgroups of solvable Iwasawa groups

Recently, Jablonski proved that, to a large extent, a simply connected solvable Lie group endowed with a left-invariant Ricci soliton metric can be isometrically embedded into the solvable Iwasawa group of a non-compact symmetric space. Motivated by this result, we classify codimension one subgroups of the solvable Iwasawa groups of irreducible symmetric spaces of non-compact type whose induced metrics are Ricci solitons. We also obtain the classifications of codimension one Ricci soliton subgroups of Damek-Ricci spaces and generalized Heisenberg groups.

Einstein hypersurfaces of Damek-Ricci spaces

Einstein hypersurfaces are “very rare” in rank-one symmetric spaces. Damek-Ricci spaces may be viewed as the closest and the most natural generalisations of noncompact rank-one symmetric spaces. We prove that no Damek-Ricci space admits an Einstein hypersurface.

Homogeneous non-degenerate 3-(\u03b1,\u03b4)-Sasaki manifolds and submersions over quaternionic K\xe4hler spaces

We show that every 3-(alpha ,delta )-Sasaki manifold of dimension 4n + 3 admits a locally defined Riemannian submersion over a quaternionic Kähler manifold of scalar curvature 16n(n+2)alpha delta. In the non-degenerate case we describe all homogeneous 3-(alpha ,delta )-Sasaki manifolds fibering over symmetric Wolf spaces and over their non-compact dual symmetric spaces. If alpha delta > 0, this yields a complete classification of homogeneous 3-(alpha ,delta )-Sasaki manifolds. For alpha delta < 0, we provide a general construction of homogeneous 3-(alpha , delta )-Sasaki manifolds fibering over non-symmetric Alekseevsky spaces, the lowest possible dimension of such a manifold being 19.