Symmetric Spaces Of Noncompact Type Research Articles

Functional Inequalities on Symmetric Spaces of Noncompact Type and Applications

The aim of this paper is to begin a systematic study of functional inequalities on symmetric spaces of noncompact type of higher rank. Our first main goal of this study is to establish the Stein–Weiss inequality, also known as a weighted Hardy–Littlewood–Sobolev inequality, for the Riesz potential on symmetric spaces of noncompact type. This is achieved by performing delicate estimates of ground spherical function with the use of polyhedral distance on symmetric spaces and by combining the integral Hardy inequality developed by Ruzhansky and Verma with the sharp Bessel-Green-Riesz kernel estimates on symmetric spaces of noncompact type obtained by Anker and Ji. As a consequence of the Stein–Weiss inequality, we deduce Hardy–Sobolev, Hardy–Littlewood–Sobolev, Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities on symmetric spaces of noncompact type. The second main purpose of this paper is to show the applications of aforementioned inequalities for studying nonlinear PDEs on symmetric spaces. Specifically, we show that the Gagliardo-Nirenberg inequality can be used to establish small data global existence results for the semilinear wave equations with damping and mass terms for the Laplace–Beltrami operator on symmetric spaces.

Infinite energy harmonic maps from quasi-compact Kähler surfaces

Abstract We construct infinite energy harmonic maps from a quasi-compact Kähler surface with a Poincaré-type metric into an NPC space. This is the first step in the construction of pluriharmonic maps from quasiprojective varieties into symmetric spaces of non-compact type, Euclidean and hyperbolic buildings and Teichmüller space.

Isoparametric hypersurfaces in symmetric spaces of non-compact type and higher rank

We construct inhomogeneous isoparametric families of hypersurfaces with non-austere focal set on each symmetric space of non-compact type and rank ${\geq }3$. If the rank is ${\geq }4$, there are infinitely many such examples. Our construction yields the first examples of isoparametric families on any Riemannian manifold known to have a non-austere focal set. They can be obtained from a new general extension method of submanifolds from Euclidean spaces to symmetric spaces of non-compact type. This method preserves the mean curvature and isoparametricity, among other geometric properties.

Approaching the isoperimetric problem in HCm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{{\\mathbb {C}}}^m$$\\end{document} via the hyperbolic log-convex density conjecture

We prove that geodesic balls centered at some base point are uniquely isoperimetric sets in the real hyperbolic space HRn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{{\\mathbb {R}}}^n$$\\end{document} endowed with a smooth, radial, strictly log-convex density on the volume and perimeter. This is an analogue of the result by G. R. Chambers for log-convex densities on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document}. As an application we prove that in any rank one symmetric space of non-compact type, geodesic balls are uniquely isoperimetric in a class of sets enjoying a suitable notion of radial symmetry.

Unique continuation inequalities for Schrödinger equation on Riemannian symmetric spaces of noncompact type

Unique continuation inequalities for Schrödinger equation on Riemannian symmetric spaces of noncompact type

$$L^p$$-boundedness of pseudo-differential operators on rank one Riemannian symmetric spaces of noncompact type

$$L^p$$-boundedness of pseudo-differential operators on rank one Riemannian symmetric spaces of noncompact type

Simplicial volume of closed locally homogeneous Riemannian manifolds

In this paper, we show that every closed, locally homogeneous Riemannian manifold with positive simplicial volume must be homeomorphic to a locally symmetric space of non-compact type, giving a converse to a result by Lafont and Schmidt within the scope of closed, locally homogeneous Riemannian manifolds. This characterizes all closed locally homogeneous Riemannian manifolds with nonzero simplicial volume.

Cohomogeneity one actions on symmetric spaces of noncompact type and higher rank

We develop a new structural result for cohomogeneity one actions on (not necessarily irreducible) symmetric spaces of noncompact type and arbitrary rank. We apply this result to classify cohomogeneity one actions on the spaces SLn(R)/SOn, n≥2, up to orbit equivalence. We also reduce the classification problem on a reducible space to the classification on each one of its irreducible factors, which in particular allows to classify cohomogeneity one actions on any finite product of hyperbolic spaces.

Sub-Feller semigroups generated by pseudodifferential operators on symmetric spaces of noncompact type

We consider global pseudodifferential operators on symmetric spaces of noncompact type, defined using spherical functions. The associated symbols have a natural probabilistic form that extend the notion of the characteristic exponent appearing in Gangolli's Lévy–Khinchine formula to a function of two variables. The Hille–Yosida–Ray theorem is used to obtain conditions on such a symbol so that the corresponding pseudodifferential operator has an extension that generates a sub-Feller semigroup, generalising existing results for Euclidean space.

Convergence of normalized Betti numbers in nonpositive curvature

We study the convergence of volume-normalized Betti numbers in Benjamini–Schramm convergent sequences of nonpositively curved manifolds with finite volume. In particular, we show that if X is an irreducible symmetric space of noncompact type, X≠H3, and (Mn) is any Benjamini–Schramm convergent sequence of finite-volume X-manifolds, then the normalized Betti numbers bk(Mn)∕vol(Mn) converge for all k. As a corollary, if X has higher rank and (Mn) is any sequence of distinct, finite-volume X-manifolds, then the normalized Betti numbers of Mn converge to the L2-Betti numbers of X. This extends our earlier work with Nikolov, Raimbault, and Samet, where we proved the same convergence result for uniformly thick sequences of compact X-manifolds. One of the novelties of the current work is that it applies to all quotients M=Γ∖X where Γ is arithmetic; in particular, it applies when Γ is isotropic.

Totally geodesic submanifolds in exceptional symmetric spaces

We classify maximal totally geodesic submanifolds in exceptional symmetric spaces up to isometry. Moreover, we introduce an invariant for certain totally geodesic embeddings of semisimple symmetric spaces, which we call the Dynkin index. We prove a result analogous to the index conjecture: for every irreducible symmetric space of non-compact type, there exists a totally geodesic submanifold which is of minimal codimension and whose non-flat irreducible factors have Dynkin index equal to one.

Einstein hypersurfaces in irreducible symmetric spaces

We give a full classification of Einstein hypersurfaces in irreducible Riemannian symmetric spaces of rank greater than 1 (the classification in the rank-one case was previously known). There are two classes of such hypersurfaces. The first class consists of codimension one Einstein solvmanifolds in irreducible symmetric spaces of noncompact type constructed via the Iwasawa decomposition. The second class consists of two exceptional families in low-dimensional symmetric spaces \({\overline{M}}=\text {SU}(3)/\text {SO}(3)\) and \({\overline{M}}=\text {SL}(3)/\text {SO}(3)\). Any Einstein hypersurface M in such space \({\overline{M}}\) is developable: it is foliated by totally geodesic spheres (respectively, by totally geodesic hyperbolic planes) of \({\overline{M}}\), with the space of leaves being a special Legendrian surface in \(S^5\) (respectively, a proper affine sphere in a \({\mathbb {R}}^3\)).

Foliated Hopf hypersurfaces in complex hyperbolic quadrics

This paper deals with a limiting case motivated by contact geometry. The limiting case of a tensorial characterization of contact hypersurfaces in Kähler manifolds leads to Hopf hypersurfaces whose maximal complex subbundle of the tangent bundle is integrable. It is known that in non-flat complex space forms and in complex quadrics such real hypersurfaces do not exist, but the existence problem in other irreducible Kähler manifolds is open. In this paper we construct explicitly a one-parameter family of homogeneous Hopf hypersurfaces, whose maximal complex subbundle of the tangent bundle is integrable, in a Hermitian symmetric space of non-compact type and rank two. These are the first known examples of such real hypersurfaces in irreducible Kähler manifolds.

Injective metrics on buildings and symmetric spaces

In this article, we show that the Goldman–Iwahori metric on the space of all norms on a fixed vector space satisfies the Helly property for balls. On the non-Archimedean side, we deduce that most classical Bruhat–Tits buildings may be endowed with a natural piecewise ℓ ∞ $\ell ^\infty$ metric which is injective. We also prove that most classical semisimple groups over non-Archimedean local fields act properly and cocompactly on Helly graphs. This gives another proof of biautomaticity for their uniform lattices. On the Archimedean side, we deduce that most classical symmetric spaces of non-compact type may be endowed with a natural invariant Finsler metric, restricting to an ℓ ∞ $\ell ^\infty$ metric on each flat, which is coarsely injective. We also prove that most classical semisimple groups over Archimedean local fields act properly and cocompactly on injective metric spaces. We identify the injective hull of the symmetric space of GL ( n , R ) $\operatorname{GL}(n,\mathbb {R})$ as the space of all norms on R n $\mathbb {R}^n$ . The only exception is the special linear group: if n = 3 $n=3$ or n ⩾ 5 $n \geqslant 5$ and K $\mathbb {K}$ is a local field, we show that SL ( n , K ) $\operatorname{SL}(n,\mathbb {K})$ does not act properly and coboundedly on an injective metric space.

Hypersurfaces of constant higher-order mean curvature in $$M\times {\mathbb{R}}$$

We consider hypersurfaces of products $$M\times {\mathbb{R}}$$ with constant rth mean curvature $${H_{r}} \ge 0$$ (to be called $${{H}_{r}}$$ -hypersurfaces), where M is an arbitrary Riemannian n-manifold. We develop a general method for constructing them and employ it to produce many examples for a variety of manifolds M, including all simply connected space forms and the hyperbolic spaces $${\mathbb{H}}_{{\mathbb{F}}}^{m}$$ (rank one symmetric spaces of noncompact type). We construct and classify complete rotational $${{H}_{r}}(\ge 0)$$ -hypersurfaces in $${\mathbb{H}}_{{\mathbb{F}}}^{m}\times {\mathbb{R}}$$ and in $${\mathbb{S}}^{n}\times {\mathbb{R}}$$ as well. They include spheres, Delaunay-type annuli and, in the case of $${\mathbb{H}}_{{\mathbb{F}}}^{m}\times {\mathbb{R}},$$ entire graphs. We also construct and classify complete $${{H}_{r}}(\ge 0)$$ -hypersurfaces of $${\mathbb{H}}_{{\mathbb{F}}}^{m}\times {\mathbb{R}}$$ which are invariant by either parabolic isometries or hyperbolic translations. We establish a Jellett–Liebmann-type theorem by showing that a compact, connected and strictly convex $${{H}_{r}}$$ -hypersurface of $${\mathbb{H}}^{n}\times {\mathbb{R}}$$ or $${\mathbb{S}}^{n}\times {\mathbb{R}}$$ $$(n\ge 3)$$ is a rotational embedded sphere. Other uniqueness results for complete $${{H}_{r}}$$ -hypersurfaces of these ambient spaces are obtained.

Complexified Hermitian Symmetric Spaces, Hyperkähler Structures, and Real Group Actions

There is a known hyperkähler structure on any complexified Hermitian symmetric space G/K, whose construction relies on identifying G/K with both a (co)adjoint orbit and the cotangent bundle to the compact Hermitian symmetric space Gu/K0. Via a family of explicit diffeomorphisms, we show that almost all of the complex structures are equivalent to the one on G/K; via a family of related diffeomorphisms, we show that almost all of the symplectic structures are equivalent to the one on \(T^{*}\left (G_{u}/K_{0}\right )\). We highlight the intermediate Kähler structures, which share a holomorphic action of G related to the one on G/K, but moment geometry related to that of \(T^{*}\left (G_{u}/K_{0}\right )\). As an application, for the real form G0 ⊂ G corresponding to G0/K0, the Hermitian symmetric space of noncompact type, we give a strategy for study of the action on G/K using the moment-critical subsets for the intermediate structures. We give explicit computations for SL(2).

An extension problem and Hardy's inequality for the fractional Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type

In this paper we study an extension problem for the Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type and use the solution to prove Hardy-type inequalities for fractional powers of the Laplace-Beltrami operator. Next, we study the mapping properties of the extension operator. In the last part we prove Poincaré-Sobolev inequalities on these spaces.

Dynamics of $ L^p $ multipliers on harmonic manifolds

<abstract><p>Let $ X $ be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of nonpositive curvature, and in particular all known examples of non-compact harmonic manifolds except for the flat spaces. We use the Fourier transform from <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup> to investigate the dynamics on $ L^p(X) $ for $ p > 2 $ of certain bounded linear operators $ T : L^p(X) \to L^p(X) $ which we call "$ L^p $-multipliers" in accordance with standard terminology. Examples of $ L^p $-multipliers are given by the operator of convolution with an $ L^1 $ radial function, or more generally convolution with a finite radial measure. In particular elements of the heat semigroup $ e^{t\Delta} $ act as multipliers. Given $ 2 < p < \infty $, we show that for any $ L^p $-multiplier $ T $ which is not a scalar multiple of the identity, there is an open set of values of $ \nu \in {\mathbb C} $ for which the operator $ \frac{1}{\nu} T $ is chaotic on $ L^p(X) $ in the sense of Devaney, i.e., topologically transitive and with periodic points dense. Moreover such operators are topologically mixing. We also show that there is a constant $ c_p > 0 $ such that for any $ c \in {\mathbb C} $ with $ \operatorname{Re} c > c_p $, the action of the shifted heat semigroup $ e^{ct} e^{t\Delta} $ on $ L^p(X) $ is chaotic. These results generalize the corresponding results for rank one symmetric spaces of noncompact type and harmonic $ NA $ groups (or Damek-Ricci spaces).</p></abstract>

On a theorem of Chernoff on rank one Riemannian symmetric spaces

In 1975, P.R. Chernoff used iterates of the Laplacian on Rn to prove an L2 version of the Denjoy-Carleman theorem which provides a sufficient condition for a smooth function on Rn to be quasi-analytic. In this paper we prove an exact analogue of Chernoff's theorem for all rank one Riemannian symmetric spaces of noncompact type using iterates of the associated Laplace-Beltrami operators. Moreover, we also prove an analogue of Chernoff's theorem for the sphere which is a rank one compact symmetric space.

Equivariant harmonic maps depend real analytically on the representation

We consider harmonic maps into symmetric spaces of non-compact type that are equivariant for representations that induce a free and proper action on the symmetric space. We show that under suitable non-degeneracy conditions such equivariant harmonic maps depend in a real analytic fashion on the representation they are associated to. The main tool in the proof is the construction of a family of deformation maps which are used to transform equivariant harmonic maps into maps mapping into a fixed target space so that a real analytic version of the results in [4] can be applied.