Noncompact Type Research Articles

Random walk on a building of type Ãr and Brownian motion of the Weyl chamber

In this paper we study a random walk on an affine building of type $\tilde{A}_r$, whose radial part, when suitably normalized, converges to the Brownian motion of the Weyl chamber. This gives a new discrete approximation of this process, alternative to the one of Biane \cite{Bia2}. This extends also the link at the probabilistic level between Riemannian symmetric spaces of the noncompact type and their discrete counterpart, which had been previously discovered by Bougerol and Jeulin in rank one \cite{BJ}. The main ingredients of the proof are a combinatorial formula on the building and the estimate of the transition density proved in \cite{AST}.

Fundamental classes not representable by products

We prove that rationally essential manifolds with suitably large fundamental groups do not admit any maps of non-zero degree from products of closed manifolds of positive dimension. Particular examples include all manifolds of non-positive sectional curvature of rank one and all irreducible locally symmetric spaces of non-compact type. For closed manifolds from certain classes, say non-positively curved ones, or certain surface bundles over surfaces, we show that they do admit maps of non-zero degree from non-trivial products if and only if they are virtually diffeomorphic to products.

Geometry of domains with the uniform squeezing property

We introduce the notion of domains with uniform squeezing property, study various analytic and geometric properties of such domains and show that they cover many interesting examples, including Teichmüller spaces and Hermitian symmetric spaces of non-compact type. The properties supported by such manifolds include pseudoconvexity, hyperconvexity, Kähler-hyperbolicity, vanishing of cohomology groups and quasi-isometry of various invariant metrics. It also leads to nice geometric properties for manifolds covered by bounded domains and a simple criterion to provide positive examples to a problem of Serre about Stein properties of holomorphic fiber bundles.

Convex functions on symmetric spaces, side lengths of polygons and the stability inequalities for weighted configurations at infinity

In a symmetric space of noncompact type X = G/K oriented geodesic segments correspond modulo isometries to vectors in the Euclidean Weyl chamber. We can hence assign vector valued lengths to segments. Our main result is a system of homoge- neous linear inequalities, which we call the generalized triangle inequalities or stability inequalities, describing the restrictions on the vector valued side lengths of oriented polygons. It is based on the mod 2 Schubert calculus in the real Grassmannians G=P for maximal parabolic subgroups P.

Resonances and residue operators for symmetric spaces of rank one

Let X = G / K be a rank-one Riemannian symmetric space of the noncompact type and let −Δ be the Laplace–Beltrami operator on X. We show that the resolvent operator R ( z ) of Δ can be meromorphically continued across the spectrum and explicitly determine the poles, i.e. the resonances. Further we describe the residue operators in terms of finite-dimensional spherical representations of G. The result answers a question posed by M. Zworski in [M. Zworski, What are the residues of the resolvent of the Laplacian on non-compact symmetric spaces? Seminar held at the IRTG Summer School 2006, Schloss Reisensburg, 2006. Available at http://math.berkeley.edu/~zworski/reisensburg.pdf]. The rank of the residue operators is derived from a restricted root version of the Weyl dimension formula for spherical highest weight representations which we prove for arbitrary symmetric spaces of the noncompact type.

On the classification of infinite-dimensional irreducible Hermitian-symmetric affine coadjoint orbits

In the finite-dimensional setting, every Hermitian-symmetric space of compact type is a coadjoint orbit of a finite-dimensional Lie group. It is natural to ask whether every infinite-dimensional Hermitian-symmetric space of compact type, which is a particular example of an Hilbert manifold, is transitively acted upon by a Hilbert Lie group of isometries. In this paper we give the classification of infinite-dimensional irreducible Hermitian-symmetric affine coadjoint orbits of connected simple L *-groups of compact type using the notion of simple roots of non-compact type. The key step is, given an infinite-dimensional symmetric pair (, ), where is a simple L *-algebra of compact type and a subalgebra of , to construct an increasing sequence of finite-dimensional subalgebras n of together with an increasing sequence of finite-dimensional subalgebras n of such that , , and such that the pairs ( n , n ) are symmetric. Comparing with the classification of Hermitian-symmetric spaces given by W. Kaup, it follows that any Hermitian-symmetric space of compact or non-compact type is an affine-coadjoint orbit of an Hilbert Lie group.

Chaotic C-distribution semigroups

We introduce and analyze hypercyclic and chaotic properties of C-distribution semigroups and global integrated C-semigroups. The obtained results are incorporated in the study of the dynamics of the backwards Lp heat semigroups (p > 2) on symmetric spaces of non-compact type.

Ends of locally symmetric spaces with maximal bottom spectrum

Let X be a symmetric space of non-compact type and Γ\ X a locally symmetric space. Then the bottom spectrum λ 1 (Γ\ X ) satisfies the inequality λ 1 (Γ\ X ) ≦ λ 1 ( X ). We show that if equality λ 1 (Γ\ X ) = λ 1 ( X ) holds, then Γ\ X has either one end, which is necessarily of infinite volume, or two ends, one of infinite volume and another of finite volume. In the latter case, Γ\ X is isometric to ℝ 1 × N endowed with a multi-warped metric, where N is compact.

Degree theorems and Lipschitz simplicial volume for nonpositively curved manifolds of finite volume

We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we derive an extension of Gromov's volume comparison theorem to products of negatively curved manifolds or locally symmetric spaces of non-compact type. In contrast, we provide vanishing results for the ordinary simplicial volume; for instance, we show that the ordinary simplicial volume of non-compact locally symmetric spaces with finite volume of Q-rank at least 3 is zero.

The L p Spectrum of Locally Symmetric Spaces with Small Fundamental Group

We determine the Lp spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces M whose universal covering X is a symmetric space of non-compact type with rank one. More precisely, we show that the Lp spectra of M and X coincide if the fundamental group of M is small and if the injectivity radius of M is bounded away from zero. In the L2 case, the restriction on the injectivity radius is not needed.

Homogeneous Poisson Structures on Loop Spaces of Symmetric Spaces

This paper is a sequel to (Caine A., Pickrell D., Int. Math. Res. Not., to appear, arXiv:0710.4484), where we studied the Hamiltonian systems which arise from the Evens- Lu construction of homogeneous Poisson structures on both compact and noncompact type symmetric spaces. In this paper we consider loop space analogues. Many of the results extend in a relatively routine way to the loop space setting, but new issues emerge. The main point of this paper is to spell out the meaning of the results, especially in the SU(2) case. Applications include integral formulas and factorizations for Toeplitz determinants.

Horospherical transform on Riemannian symmetric manifolds of noncompact type

We discuss I. M. Gelfand’s project of rebuilding the representation theory of semisimple Lie groups on the basis of integral geometry. The basic examples are related to harmonic analysis and the horospherical transform on symmetric manifolds. Specifically, we consider the inversion of this transform on Riemannian symmetric manifolds of noncompact type. In the known explicit inversion formulas, the nonlocal part essentially depends on the type of the root system. We suggest a universal modification of this operator.

Coupled Vortex Equations on Complete Kähler Manifolds

AbstractIn this paper, we first investigate the Dirichlet problem for coupled vortex equations. Secondly, we give existence results for solutions of the coupled vortex equations on a class of complete noncompact Kähler manifolds which include simply-connected strictly negative curved manifolds, Hermitian symmetric spaces of noncompact type and strictly pseudo-convex domains equipped with the Bergmann metric.

Growth of conjugacy classes of Schottky groups in higher rank symmetric spaces

Let X be a globally symmetric space of noncompact type and rank greater that one, and \({\Gamma \subset Isom(X)}\) a Schottky group of axial isometries. Then \({M := X/\Gamma}\) is a locally symmetric Riemannian manifold of infinite volume. The goal of this note is to give an asymptotic estimate for the number of primitive closed geodesics in M modulo free homotopy with period less than t.

Fisher information metric and Poisson kernels

A complete Riemannian manifold X with negative curvature satisfying − b 2 ⩽ K X ⩽ − a 2 < 0 for some constants a , b , is naturally mapped in the space of probability measures on the ideal boundary ∂ X by assigning the Poisson kernels. We show that this map is embedding and the pull-back metric of the Fisher information metric by this embedding coincides with the original metric of X up to constant provided X is a rank one symmetric space of non-compact type. Furthermore, we give a geometric meaning of the embedding.

Weighted Strichartz estimates for the Schrödinger and wave equations on Damek–Ricci spaces

We prove Strichartz estimates for radial solutions of the Schrodinger and wave equations on Damek–Ricci spaces, and in particular on symmetric spaces of noncompact type and rank one, using the perturbative theory with potentials. The curvature of the noncompact manifold has an influence on the dispersive properties, and indeed we obtain Strichartz estimates with weights at spatial infinity, which are stronger than the standard ones in the flat case.

Topology of Complete Manifolds with Radial Curvature Bounded from Below

We investigate the topology of a complete Riemannian manifold whose radial curvature at the base point is bounded from below by that of a von Mangoldt surface of revolution. Sphere theorem is generalized to a wide class of metrics, and it is proven that such a manifold of a noncompact type has finitely many ends.

Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary

Let X be a symmetric space of noncompact type, and let Γ be a lattice in the isometry group of X. We study the distribution of orbits of Γ acting on the symmetric space X and its geometric boundary X(∞), generalizing the main equidistribution result of Margulis's thesis [M, Theorem 6] to higher-rank symmetric spaces. More precisely, for any y∈X and b∈X(∞), we investigate the distribution of the set {(yγ,bγ−1):γ∈Γ} in X×X(∞). It is proved, in particular, that the orbits of Γ in the Furstenberg boundary are equidistributed and that the orbits of Γ in X are equidistributed in “sectors” defined with respect to a Cartan decomposition. Our main tools are the strong wavefront lemma and the equidistribution of solvable flows on homogeneous spaces, which we obtain using Shah's result [S, Corollary 1.2] based on Ratner's measure-classification theorem [R1, Theorem 1]

L p -Spectral Theory of Locally Symmetric Spaces with $\mathbb{Q}$ -Rank One

We study the L p-spectrum of the Laplace–Beltrami operator on certain complete locally symmetric spaces \(M=\Gamma\backslash X\) with finite volume and arithmetic fundamental group Γ whose universal covering X is a symmetric space of non-compact type. We also show, how the obtained results for locally symmetric spaces can be generalized to manifolds with cusps of rank one.

Complex hyperpolar actions with a totally geodesic orbit

We first show that homogeneous submanifolds with abelian normal bundle in a symmetric space of non-compact type occur as principal orbits of complex hyperpolar actions on the symmetric space. Next we show that all complex hyperpolar actions with a reflective orbit are orbit equivalent to Hermann type actions. Furthermore, we classify complex hyperpolar actions with a totally geodesic orbit in the case where the ambient symmetric space is irreducible. Also, we list up the cohomogeneities of Hermann type actions on irreducible symmetric spaces.