Symmetric Spaces Of Noncompact Type Research Articles

Dimension rigidity of lattices in semisimple Lie groups

We prove that if \Gamma is a lattice in the group of isometries of a symmetric space of non-compact type without euclidean factors, then the virtual cohomological dimension of \Gamma equals its proper geometric dimension.

Mean curvature flow of certain kind of isoparametric foliations on non-compact symmetric spaces

In this paper, we investigate the mean curvature flows starting from all non-minimal leaves of the isoparametric foliation given by a certain kind of solvable group action on a symmetric space of non-compact type. We prove that the mean curvature flow starting from each non-minimal leaf of the foliation exists in infinite time, if the foliation admits no minimal leaf, then the flow asymptotes the self-similar flow starting from another leaf, and if the foliation admits a minimal leaf (in this case, it is shown that there exists the only one minimal leaf), then the flow converges to the minimal leaf of the foliation in $C^{\infty}$-topology.

On splitting rank of non-compact-type symmetric spaces and bounded cohomology

Let [Formula: see text] be a higher rank symmetric space of non-compact type where [Formula: see text]. We define the splitting rank of [Formula: see text], denoted by [Formula: see text], to be the maximal dimension of a totally geodesic submanifold [Formula: see text] which splits off an isometric [Formula: see text]-factor. We compute explicitly the splitting rank for each irreducible symmetric space. For an arbitrary (not necessarily irreducible) symmetric space, we show that the comparison map [Formula: see text] is surjective in degrees [Formula: see text], provided [Formula: see text] has no direct factors of [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. This generalizes the result of [J.-F. Lafont and S. Wang, Barycentric straightening and bounded cohomology, to appear in J. Eur. Math. Soc.] regarding Dupont’s problem.

Complex Riemannian foliations of open Kähler manifolds

Classification results for complex Riemannian foliations are obtained. For open subsets of irreducible Hermitian symmetric spaces of compact type, where one has explicit control over the curvature tensor, we completely classify such foliations by studying the infinitesimal model associated with the canonical connection. We also establish results for symmetric spaces of noncompact type and a general rigidity result for any irreducible Kähler manifold.

On the stability of $$L^p$$ L p -norms of Riemannian curvature at rank one symmetric spaces

We study stability and local minimizing property of $$L^p$$ -norms of Riemannian curvature tensor denoted by $$\mathcal {R}_p$$ by variational methods. We compute the Hessian of $$\mathcal {R}_p$$ at compact rank 1 symmetric spaces and prove that they are stable for $$\mathcal {R}_p$$ for certain values of $$p\ge 2$$ . A similar result also holds for compact quotients of rank 1 symmetric spaces of non-compact type. Consequently, we obtain stability of $$L^{\frac{n}{2}}$$ -norm of Weyl curvature at these metrics using results from Gursky and Viaclovsky (J Reine Angew Math 400:37–91, 2015).

Eigenfunction of the Laplacian as a Degenerate Case of a Function with its Fourier Transform Supported in an Annulus

The aim of this paper is to reach a characterization of the eigenfunction of the Laplace–Beltrami operator as a degenerate case of the inverse Paley–Wiener theorem (for functions whose Fourier transform is supported on a compact annulus) on the rank-1 Riemannian symmetric spaces of noncompact type. The most distinguished prototypes of these spaces are the hyperbolic spaces.

Uncertainty principles of Ingham and Paley–Wiener on semisimple Lie groups

Classical results due to Ingham and Paley-Wiener characterize the existence of nonzero functions supported on certain subsets of the real line in terms of the pointwise decay of the Fourier transforms. Viewing these results as uncertainty principles for Fourier transforms, we prove certain analogues of these results on connected, noncompact, semisimple Lie groups with finite center. We also use these results to show unique continuation property of solutions to the initial value problem for time-dependent Schr\"odinger equations on Riemmanian symmetric spaces of noncompact type.

Finiteness of totally geodesic exceptional divisors in Hermitian locally symmetric spaces

We prove that on a smooth complex surface which is a compact quotient of the bidisc or of the 2-ball, there is at most a finite number of totally geodesic curves with negative self-intersection. More generally, there are only finitely many exceptional totally geodesic divisors in a compact Hermitian locally symmetric space of noncompact type of dimension at least 2. This is deduced from a convergence result for currents of integration along totally geodesic subvarieties in compact Hermitian locally symmetric spaces, which itself follows from an equidistribution theorem for totally geodesic submanifolds in a locally symmetric space of finite volume.

EXTERIOR DIFFERENTIAL FORMS ON RIEMANNIAN SYMMETRIC SPACES

In the present paper we give a rough classification of exterior differential forms on a Riemannian manifold. We define conformal Killing, closed conformal Killing, coclosed conformal Killing and harmonic forms due to this classification and consider these forms on a Riemannian globally symmetric space and, in particular, on a rank-one Riemannian symmetric space. We prove vanishing theorems for conformal Killing L 2-forms on a Riemannian globally symmetric space of noncompact type. Namely, we prove that every closed or co-closed conformal Killing L 2-form is a parallel form on an arbitrary such manifold. If the volume of it is infinite, then every closed or co-closed conformal Killing L 2-form is identically zero. In addition, we prove vanishing theorems for harmonic forms on some Riemannian globally symmetric spaces of compact type. Namely, we prove that all harmonic one-formsvanish everywhere and every harmonic r -form  r  2 is parallel on an arbitrary such manifold. Our proofs are based on the Bochnertechnique and its generalized version that are most elegant and important analytical methods in differential geometry “in the large”.

Spectral multipliers on exponentially locally doubling metric measure spaces

Let $$(X, \rho )$$ be a geodesic space endowed with a positive Borel measure $$\mu $$ which satisfies an exponentially locally doubling condition. Assume that L is a nonnegative self-adjoint operator on $$L^{2}(X,\text {d}\mu )$$ whose heat kernel obeys a local Gaussian upper bound. In this paper, we prove that if $$\varPhi $$ is a bounded even holomorphic function in a suitable strip of the complex plane, and satisfies the Mihlin-type condition of appropriate order at infinity, then the operator $$\varPhi (\sqrt{L})$$ extends to an operator bounded on $$L^{p}(X,\text {d}\mu )$$ for $$1<p<\infty $$ and of weak type (1, 1). This partially extends some existing results concerning spherical multipliers on symmetric spaces of noncompact type and spectral multipliers on Riemannian manifolds with bounded geometry.

Bourgain–Brezis inequalities on symmetric spaces of non-compact type

Let M be a global Riemannian symmetric space of non-compact type. We prove a duality estimate, for pairings of divergence-free L1 vector fields, with vector fields in a critical Sobolev space on M:|∫M〈f,ϕ〉dV|≤C‖f‖L1(dV)‖∇ϕ‖Lm(dV). This estimate provides a remedy for the failure of a critical Sobolev embedding on such symmetric spaces.

Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compact type

First we investigate the evolutions of the radius function and its gradient along the volume-preserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space under certain condition for the radius function. Next, we prove that the tubeness is preserved along the flow in the case where the ambient space is a rank one symmetric space of non-compact type (other than a (real) hyperbolic space) and the reflective submanifold is an invariant submanifold and the radius function of the initial tube is radial. Furthermore, in this case, we prove that the flow reaches to the invariant submanifold or it exists in infinite time and converges to another tube of constant mean curvature over the invariant submanifold in the $C^{\infty}$-topology in infinite time.

On the Canonical Line Bundle of a Locally Hermitian Symmetric Space

We study three problems related to towers of coverings of Hermitian symmetric spaces of non-compact type. The first one is on the possibility that the index of the canonical line bundle gets arbitrarily large on such a tower of coverings. The second one is on some vanishing and non-vanishing results on bundle-valued forms on a Kahler hyperbolic manifolds. The third one is to explain that on any tower of coverings of locally Hermitian symmetric spaces of Lie algebra type DIII, some fractional power of the canonical line bundle aK with rational \(0<a<1\) are very ample as one goes high enough in the tower.

Approximate lattices

In this article we introduce and study uniform and non-uniform approximate lattices in locally compact second countable (lcsc) groups. These are approximate subgroups (in the sense of Tao) which simultaneously generalize lattices in lcsc group and mathematical quasi-crystals (a.k.a. Meyer sets) in lcsc abelian groups. We show that envelopes of strong approximate lattices are unimodular, and that approximate lattices in nilpotent groups are uniform. We also establish several results relating properties of approximate lattices and their envelopes. For example, we prove a version of the Milnor-Schwarz lemma for uniform approximate lattices in compactly-generated lcsc groups, which we then use to relate metric amenability of uniform approximate lattices to amenability of the envelope. Finally we extend a theorem of Kleiner and Leeb to show that the isometry groups of higher rank symmetric spaces of non-compact type are QI rigid with respect to finitely-generated approximate groups.

Tits geometry and positive curvature

There is a well-known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns–Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least $3$ and irreducible topological spherical buildings of rank at least $3$. We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type $\mathsf{M}$ associated with any polar action of cohomogeneity at least $2$ on any simply connected closed positively curved manifold. Although this chamber system is typically not a Tits geometry of type $\mathsf{M}$, its universal Tits cover indeed is a building in all but two exceptional cases. We construct a topology on this universal cover making it into a compact spherical building in the sense of Burns and Spatzier. Using this structure, we classify up to equivariant diffeomorphism all polar actions on (simply connected) positively curved manifolds of cohomogeneity at least $2$.

Maximal totally geodesic submanifolds and index of symmetric spaces

Let $M$ be an irreducible Riemannian symmetric space. The index $i(M)$ of $M$ is the minimal codimension of a totally geodesic submanifold of $M$. In [1] we proved that $i(M)$ is bounded from below by the rank $\mathrm{rk}(M)$ of $M$, that is, $\mathrm{rk}(M) \leq i(M)$. In this paper we classify all irreducible Riemannian symmetric spaces $M$ for which the equality holds, that is, $\mathrm{rk}(M) = i(M)$. In this context we also obtain an explicit classification of all nonsemisimple maximal totally geodesic submanifolds in irreducible Riemannian symmetric spaces of noncompact type and show that they are closely related to irreducible symmetric $\mathrm{R}$-spaces. We also determine the index of some symmetric spaces and classify the irreducible Riemannian symmetric spaces of noncompact type with $i(M) \in \{4, 5, 6 \}$.

A characterization of the L2-range of the Poisson transform related to Strichartz conjecture on symmetric spaces of noncompact type

The main purpose of this paper is to solve Strichartz conjecture concerning an image characterization of the Poisson transform in the L2-theory on symmetric spaces of noncompact type. We prove that the Poisson transform provides an isomorphism between the L2-space on the boundary and a certain weighted L2-space consisting of joint eigenfunctions on symmetric spaces of noncompact type. Our approach is based on the scattering theory for the Schrödinger operator. Moreover, we give a Fourier restriction estimate and an asymptotic formula for the Poisson transform.

The distortion dimension of $$\mathbb Q$$-rank 1 lattices

Let $$X=G/K$$ be a symmetric space of noncompact type and rank $$k\ge 2$$ . We prove that horospheres in X are Lipschitz $$(k-2)$$ -connected if their centers are not contained in a proper join factor of the spherical building of X at infinity. As a consequence, the distortion dimension of an irreducible $$\mathbb {Q}$$ -rank-1 lattice $$\Gamma $$ in a linear, semisimple Lie group G of $$\mathbb R$$ -rank k is $$k-1$$ . That is, given $$m< k-1$$ , a Lipschitz m-sphere S in (a polyhedral complex quasi-isometric to) $$\Gamma $$ , and a $$(m+1)$$ -ball B in X (or G) filling S, there is a $$(m+1)$$ -ball $$B'$$ in $$\Gamma $$ filling S such that $${{\mathrm{vol}}}B'\sim {{\mathrm{vol}}}B$$ . In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension $$k-1$$ .

An order inequality characterizing invariant barycenters on symmetric cones

This paper is concerned with invariant and contractive barycenters on the Wasserstein space of probability measures on metric spaces of non-positive curvature, where the center of gravity, also called the Cartan barycenter, is the canonical barycenter on Hadamard spaces. We establish an order inequality of probability measures on partially ordered symmetric spaces of non-compact type, namely symmetric cones (self-dual homogeneous cones), characterizing the Cartan barycenter among other invariant and contractive barycenters. The derived inequality and partially ordered structures on the probability measure space lead also to significant results on (norm) inequalities including the Ando–Hiai inequality for probability measures on symmetric cones.

Hermitian symmetric space, flat bundle and holomorphicity criterion

Let K\\G be an irreducible Hermitian symmetric space of noncompact type and Γ⊂G a closed torsionfree discrete subgroup. Let X be a compact Kähler manifold and ρ:π1(X,x0)⟶Γ a homomorphism such that the adjoint action of ρ(π1(X,x0)) on the Lie algebra Lie(G) is completely reducible. A theorem of Corlette associates to ρ a harmonic map H:X⟶K\\G/Γ. We give a criterion for this harmonic map H to be holomorphic. We also give a criterion for it to be anti-holomorphic.