Symmetric Spaces Of Noncompact Type Research Articles

Nonzero degree tangential maps between dual symmetric spaces

We construct a tangential map from a locally symmetric space of noncompact type to its dual compact type twin. By comparing the induced map in cohomology to a map defined by Matsushima, we conclude that in the equal rank case the map has a nonzero degree.

Un théorème de rigidité non-métrique pour les variétés localement symétriques hermitiennes

Let X be an irreducible Hermitian symmetric space of non-compact type of dimension greater than 1 and G be the group of biholomorphisms of X ; let {\rm M} = \Gamma \backslash X be a quotient of X by a torsion-free discrete subgroup \Gamma of G such that M is of finite volume in the canonical metric. Then, due to the G-equivariant Borel embedding of X into its compact dual Xc, the locally symmetric structure of M can be considered as a special kind of a (G_{\Bbb C} , X_c) -structure on M, a maximal atlas of Xc-valued charts with locally constant transition maps in the complexified group {\rm G}_{\Bbb C} . By Mostow's rigidity theorem the locally symmetric structure of M is unique. We prove that the ({\rm G}_{\Bbb C} , X_c) -structure of M is the unique one compatible with its complex structure. In the rank one case this result is due to Mok and Yeung.

Ergodic theory and rigidity on the symmetric space of non-compact type

In this paper we investigate the rigidity of symmetric spaces of non-compact type using ergodic theory such as Patterson–Sullivan measure and the marked length spectrum along with the cross ratio on the limit set. In particular, we prove that the marked length spectrum determines the Zariski dense subgroup up to conjugacy in the isometry group of the product of rank-one symmetric spaces. As an application, we show that two convex cocompact, negatively curved, locally symmetric manifolds are isometric if the Thurston distance is zero and the critical exponents of the Poincaré series are the same, and the same is true if the geodesic stretch is equal to one.

Corank and asymptotic filling-invariants for symmetric spaces

Let X be a Riemannian symmetric space of noncompact type. We prove that there exists an embedded submanifold \( Y \subset X \) which is quasi-isometric to a manifold with strictly negative sectional curvature, which intersects a given flat F in a geodesic line and which satisfies dim(Y) — 1 = dim(X) — rank(X). This yields an estimate of the hyperbolic corank of X. As another application we show that certain asymptotic filling invariants of X are exponential.

Hermitian-Einstein metrics for vector bundles on complete Kähler manifolds

In this paper, we prove the existence of Hermitian-Einstein metrics for holomorphic vector bundles on a class of complete Kähler manifolds which include Hermitian symmetric spaces of noncompact type without Euclidean factor, strictly pseudoconvex domains with Bergman metrics and the universal cover of Gromov hyperbolic manifolds etc. We also solve the Dirichlet problem at infinity for the Hermitian-Einstein equations on holomorphic vector bundles over strictly pseudoconvex domains.

Geometric flow and rigidity on symmetric spaces of noncompact type

In this paper we show that, under a suitable condition, every nonsingular geometric flow on a manifold which is modeled on the Furstenberg boundary of X X , where X X is a symmetric space of non-compact type, induces a torus action, and, in particular, if the manifold is a rational homology sphere, then the flow has a closed orbit.

Very Ampleness of Line Bundles and Canonical Embedding of Coverings of Manifolds

Let L be an ample line bundle on a Kahler manifolds of nonpositive sectional curvature with K as the canonical line bundle. We give an estimate of m such that K+mL is very ample in terms of the injectivity radius. This implies that m can be chosen arbitrarily small once we go deep enough into a tower of covering of the manifold. The same argument gives an effective Kodaira Embedding Theorem for compact Kahler manifolds in terms of sectional curvature and the injectivity radius. In case of locally Hermitian symmetric space of noncompact type or if the sectional curvature is strictly negative, we prove that K itself is very ample on a large covering of the manifold.

Piecewise Euclidean structures and Eberlein’s Rigidity Theorem in the singular case

In this article, we generalize Eberlein’s Rigidity Theorem to the singular case, namely, one of the spaces is only assumed to be a CAT(0) topological manifold. As a corollary, we get that any compact irreducible but locally reducible locally symmetric space of noncompact type does not admit a nonpositively curved (in the Aleksandrov sense) piecewise Euclidean structure. Any hyperbolic manifold, on the other hand, does admit such a structure.

Negatively curved homogeneous Osserman spaces

We show that solvable Lie groups of Iwasawa type satisfying the Osserman condition are symmetric spaces of noncompact type and rank one. As a consequence, the Osserman conjecture holds for non-flat homogeneous manifolds of nonpositive curvature.

Deformation of Schottky groups in complex hyperbolic space

Let G = P U ( 1 , d ) G=PU(1,d) be the group of holomorphic isometries of complex hyperbolic space H C d \mathbf {H}^d_\mathbf {C} . The latter is a Kähler manifold with constant negative holomorphic sectional curvature. We call a finitely generated discrete group Γ = ⟨ g 1 , … , g n ⟩ ⊂ G \Gamma = \langle g_1,\dots , g_n \rangle \subset G a marked classical Schottky group of rank n n if there is a fundamental polyhedron for G G whose sides are equidistant hypersurfaces which are disjoint and not asymptotic, and for which g 1 , … , g n g_1, \dots , g_n are side-pairing transformations. We consider smooth families of such groups Γ t = ⟨ g 1 , t , … , g n , t ⟩ \Gamma _t = \langle g_{1,t}, \dots , g_{n,t} \rangle with g j , t g_{j,t} depending smoothly ( C 1 C^1 ) on t t whose fundamental polyhedra also vary smoothly. The groups Γ t \Gamma _t are all algebraically isomorphic to the free group in n n generators, i.e. there are canonical isomorphisms ϕ t : Γ 0 → Γ t \phi _t: \Gamma _0\to \Gamma _t . We shall construct a homeomorphism Ψ t \Psi _t of H ¯ C d = H C d ∪ ∂ H C d \overline {\mathbf {H}}^d_\mathbf {C} = \mathbf {H}^d_\mathbf {C}\cup \partial \mathbf {H}^d_\mathbf {C} which is equivariant with respect to these groups: ϕ t ( g ) ∘ Ψ t = Ψ t ∘ g ∀ g ∈ Γ 0 , 0 ≤ t ≤ 1 \begin{equation*} \phi _t(g) \circ \Psi _t = \Psi _t \circ g \quad \; \forall g\in \Gamma _0, \quad 0\leq t\leq 1 \end{equation*} which is quasiconformal on ∂ H C d \partial \mathbf {H}^d_\mathbf {C} with respect to the Heisenberg metric, and which is symplectic in the interior. As a corollary, the limit sets of such Schottky groups of equal rank are quasiconformally equivalent to each other. The main tool for the construction is a time-dependent Hamiltonian vector field used to define a diffeomorphism, mapping D 0 D_0 onto D t D_t , where D t D_t is a fundamental domain of Γ t \Gamma _t . In two steps, this is extended equivariantly to H ¯ C d \overline {\mathbf {H}}^d_\mathbf {C} . The method yields similar results for real hyperbolic space, while the analog for the other rank-one symmetric spaces of noncompact type cannot hold.

Patterson-Sullivan Theory in Higher Rank Symmetric Spaces

Let X = G/K be a Riemannian symmetric space of noncompact type and $ \Gamma $ a discrete “generic” subgroup of G with critical exponent $ \delta(\Gamma) $ . Denote by $ X_{reg} (\infty) $ the set of regular elements of the geometric boundary $ X(\infty) $ of X. We show that the support of all $ \Gamma $ -invariant conformal densities of dimension $ \delta(\Gamma) $ on $ X_{reg} (\infty) $ (e.g. Patterson-Sullivan densities) lie in a same and single regular G-orbit on $ X(\infty) $ . This provides information on the large-scale growth of $ \Gamma $ -orbits in X. If in addition we assume $ \Gamma $ to be of divergence type, then there is a unique density of the previous type. Furthermore, we explicitly determine $ \delta(\Gamma) $ and this G-orbit for lattices, and show that they are of divergence type.

Closures of totally geodesic immersions into locally symmetric spaces of noncompact type

It is established that if M 1 \mathcal {M}_1 and M 2 \mathcal {M}_2 are connected locally symmetric spaces of noncompact type where M 2 \mathcal {M}_2 has finite volume, and ϕ : M 1 → M 2 \phi :\mathcal {M}_1 \to \mathcal {M}_2 is a totally geodesic immersion, then the closure of ϕ ( M 1 ) \phi (\mathcal {M}_1) in M 2 \mathcal {M}_2 is an immersed “algebraic” submanifold. It is also shown that if in addition, the real ranks of M 1 \mathcal {M}_1 and M 2 \mathcal {M}_2 are equal, then the the closure of ϕ ( M 1 ) \phi (\mathcal {M}_1) in M 2 \mathcal {M}_2 is a totally geodesic submanifold of M 2 . \mathcal {M}_2. The proof is a straightforward application of Ratner’s Theorem combined with the structure theory of symmetric spaces.

Heat and Harmonic Extensions for Function Spaces of Hardy–Sobolev–Besov Type on Symmetric Spaces and Lie Groups

Function spaces of Hardy–Sobolev–Besov type on symmetric spaces of noncompact type and unimodular Lie groups are investigated. The spaces were originally defined by uniform localization. In the paper we give a characterization of the spaceFsp,q(X) andBsp,q(X) in terms of heat and Poisson semigroups, for 1⩽p,q⩽∞ ands∈R. The main tool we use, is an atomic decomposition of function spaces on manifolds.

Factorization Theorems on Symmetric Spaces of Noncompact Type

We prove the analogs of the Khinchin factorization theorems for K-invariant probability measures on symmetric spaces X=G/K with G semisimple noncompact. We use the Kendall theory of delphic semigroups and some properties of the spherical Fourier transform and spherical functions on X.

Coordinates of the representation space in the semisimple Lie group of rank one

In this paper we show that the space of irreducible representations from a finitely presented group into the group of isometries of a rank one symmetric space of non-compact type, embeds into ℝn for some n, where the coordinates are the translation lengths of isometries in the representation. The ingredients of the proof consist of the two facts that the representation is determined by its marked length spectrum and that the nested sequence of algebraic subvarieties is stabilised at a finite step by the Noetherian property of the polynomial ring. As a minor application, we use this fact to simplify McMullen's proof about the exponential algebraic convergence of Thurston's double limit to the geometrically infinite manifold in the space of discrete faithful representations of π1(S) in Iso+.

Uniqueness of conformal densities and the semiflow of Weyl chambers

Let X = G/K be a symmetric space of noncompact type and Γ a discrete “generic” subgroup of G with critical exponent δ(Γ). We show that, if Γ is of divergence type, then there is a unique Γ-invariant conformal density of dimension δ (Γ) (hence a Patterson-Sullivan density) on the set of regular elements of the geometric boundary of X. This problem is directly related to the recurrence of the semiflow of Weyl chambers on X/gG.

Invariant subspaces in some function spaces on symmetric spaces. II

Let G be a semisimple connected Lie group with finite centre, K a maximal compact subgroup of G, and M=G/K a Riemannian symmetric space of non-compact type. We study the problem of describing the structure of closed linear subspaces in various function spaces on M that are invariant under the quasiregular representation of the group G. We consider the case when M is a symplectic symmetric space of rank 1.

THE CONFORMAL ANOMALY IN GENERAL RANK 1 SYMMETRIC SPACES AND ASSOCIATED OPERATOR PRODUCT

We compute the one-loop effective action and the conformal anomaly associated with the product ⊗pℒp of the Laplace type operators ℒp, p=1,2, acting in irreducible rank 1 symmetric spaces of non-compact type. The explicit form of the zeta functions and the conformal anomaly of the stress-energy momentum tensor is derived.

Pointwise Fourier Inversion on Rank One Symmetric Spaces and Related Topics

This paper develops necessary and sufficient conditions for pointwise inversion of Fourier transforms on rank one symmetric spaces of non-compact type for functions in the piecewise smooth category. This extends results of Pinsky for isotropic Riemannian manifolds of constant curvature. Methodologically, a similar result on the Jacobi transform and a transplantation scheme from even dimensional spaces to certain odd dimensional real hyperbolic spaces naturally enter the picture.

Tilings and finite energy retractions of locally symmetric spaces

Let \( \Gamma \backslash \overline{X} \) be the Borel-Serre compactifiction of an arithmetic quotient \( \Gamma \backslash X \) of a symmetric space of noncompact type. We construct natural tilings \( \Gamma \backslash \overline{X} = \coprod _P \Gamma \backslash \overline{X}_P \) (depending on a parameter b) which generalize the Arthur-Langlands partition of \( \Gamma \backslash X \). This is applied to yield a natural piecewise analytic deformation retraction of \( \Gamma \backslash \overline{X} \) onto a compact submanifold with corners \( \Gamma \backslash X _0 \subset \Gamma \backslash X \). In fact, we prove that \( \Gamma \backslash X _0 \) is a realization (under a natural piecewise analytic diffeomorphism) of \( \Gamma \backslash \overline{X} \) inside the interior \( \Gamma \backslash X \). For application to the theory of harmonic maps and geometric rigidity, we prove this retraction and diffeomorphism have finite energy except for a few low ranks examples. We also use tilings to give an explicit description of a cofinal family of neighborhoods of a face of \( \Gamma \backslash \overline{X} \), and study the dependance of tilings on the parameter b and the degeneration of tilings.