Noncompact Type Research Articles

Isometry Lie algebras of indefinite homogeneous spaces of finite volume

Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra equipped with a symmetric bilinear form $\langle\cdot,\cdot\rangle$. We assume that $\langle\cdot,\cdot\rangle $ is nil-invariant. This means that every nilpotent operator in the smallest algebraic Lie subalgebra of endomomorphims containing the adjoint representation of $\mathfrak{g}$ is an infinitesimal isometry for $\langle\cdot,\cdot\rangle $. Among these Lie algebras are the isometry Lie algebras of pseudo-Riemannian manifolds of finite volume. We prove a strong invariance property for nil-invariant symmetric bilinear forms, which states that the adjoint representations of the solvable radical and all simple subalgebras of non-compact type of $\mathfrak{g} $ act by infinitesimal isometries for $\langle\cdot,\cdot\rangle $. Moreover, we study properties of the kernel of $\langle\cdot,\cdot\rangle $ and the totally isotropic ideals in $\mathfrak{g} $ in relation to the index of $\langle\cdot,\cdot\rangle $. Based on this, we derive a structure theorem and a classification for the isometry algebras of indefinite homogeneous spaces of finite volume with metric index at most two. Examples show that the theory becomes significantly more complicated for index greater than two.

Negative Ricci curvature on some non-solvable Lie groups II

We construct many examples of Lie groups admitting a left-invariant metric of negative Ricci curvature. We study Lie algebras which are semidirect products $${\mathfrak {l}}= ({\mathfrak {a}} \oplus {\mathfrak {u}} ) < imes {\mathfrak {n}}$$ and we obtain examples where $${\mathfrak {u}} $$ is any semisimple compact real Lie algebra, $${\mathfrak {a}} $$ is one-dimensional and $${\mathfrak {n}}$$ is a representation of $${\mathfrak {u}} $$ which satisfies some conditions. In particular, when $${\mathfrak {u}} = {{\mathfrak {s}}}{{\mathfrak {u}}}(m)$$, $${{\mathfrak {s}}}{{\mathfrak {o}}} (m)$$ or $${{\mathfrak {s}}}{{\mathfrak {p}}} (m)$$ and $${\mathfrak {n}}$$ is a representation of $${\mathfrak {u}} $$ in some space of homogeneous polynomials, we show that these conditions are indeed satisfied. In the case $${\mathfrak {u}} = {{\mathfrak {s}}}{{\mathfrak {u}}}(2)$$ we get a more general construction where $${\mathfrak {n}}$$ can be any nilpotent Lie algebra where $${{\mathfrak {s}}}{{\mathfrak {u}}}(2)$$ acts by derivations. We also prove a general result in the case when $${\mathfrak {u}} $$ is a semisimple Lie algebra of non-compact type.

Positively ratioed representations

Let S be a closed orientable surface of genus at least 2 and let G be a semisimple real algebraic group of non-compact type. We consider a class of representations from the fundamental group of S to G called positively ratioed representations. These are Anosov representations with the additional condition that certain associated cross ratios satisfy a positivity property. Examples of such representations include Hitchin representations and maximal representations. Using geodesic currents, we show that the corresponding length functions for these positively ratioed representations are well-behaved. In particular, we prove a systolic inequality that holds for all such positively ratioed representations.

Submanifold geometry in symmetric spaces of noncompact type

In this survey article we provide an introduction to submanifold geometry in symmetric spaces of noncompact type. We focus on the construction of examples and the classification problems of homogeneous and isoparametric hypersurfaces, polar and hyperpolar actions, and homogeneous CPC submanifolds.

Intersection theory of toric 𝑏-divisors in toric varieties

We introduce toric b b -divisors on complete smooth toric varieties and a notion of integrability of such divisors. We show that under some positivity assumptions toric b b -divisors are integrable and that their degree is given as the volume of a convex set. Moreover, we show that the dimension of the space of global sections of a nef toric b b -divisor is equal to the number of lattice points in this convex set and we give a Hilbert–Samuel-type formula for its asymptotic growth. This generalizes classical results for classical toric divisors on toric varieties. Finally, we relate convex bodies associated to b b -divisors with Newton–Okounkov bodies. The main motivation for studying toric b b -divisors is that they locally encode the singularities of the invariant metric on an automorphic line bundle over a toroidal compactification of a mixed Shimura variety of non-compact type.

Scattering theory for the Laplacian on symmetric spaces of noncompact type and its application to a conjecture of Strichartz

In this paper we develop the scattering theory for the Laplacian on symmetric spaces of noncompact type. We study the asymptotic properties of the resolvent in the framework of the Agmon–Hörmander space. Our approach is based on a detailed analysis of the Helgason Fourier transform and generalized spherical functions on symmetric spaces of noncompact type. As an application of our scattering theory, we prove a conjecture by Strichartz concerning a characterization of a family of generalized eigenfunctions of the Laplacian.

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).

Finsler bordifications of symmetric and certain locally symmetric spaces

We give a geometric interpretation of the maximal Satake compactification of symmetric spaces $X=G/K$ of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable $G$-invariant Finsler metric on $X$. As an application, we establish the existence of natural bordifications, as orbifolds-with-corners, of locally symmetric spaces $X/\Gamma$ for arbitrary discrete subgroups $\Gamma< G$. These bordifications result from attaching $\Gamma$-quotients of suitable domains of proper discontinuity at infinity. We further prove that such bordifications are compactifications in the case of Anosov subgroups. We show, conversely, that Anosov subgroups are characterized by the existence of such compactifications among uniformly regular subgroups. Along the way, we give a positive answer, in the torsion free case, to a question of Ha\"issinsky and Tukia on convergence groups regarding the cocompactness of their actions on the domains of discontinuity.

A Moiré pattern on symmetric spaces of the noncompact type

We prove that if X X is a symmetric space of the noncompact type, just as adding Helgason waves which propagate in all directions will yield an elementary spherical function for X X , a Helgason wave can be produced by adding elementary spherical functions whose centers describe a horocycle in X X .

Homogeneous spin Riemannian manifolds with the simplest Dirac operator

Abstract We show the existence of nonsymmetric homogeneous spin Riemannian manifolds whose Dirac operator is like that on a Riemannian symmetric spin space. Such manifolds are exactly the homogeneous spin Riemannian manifolds (M, g) which are traceless cyclic with respect to some quotient expression M = G/K and reductive decomposition 𝔤 = 𝔨 ⊕ 𝔪. Using transversally symmetric fibrations of noncompact type, we give a list of them.

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.

Characterizations of endograph metric and Γ-convergence on fuzzy sets

This paper is devoted to the relationships and properties of the endograph metric and the Γ-convergence. The main contents can be divided into three closely related parts. Firstly, on the class of upper semi-continuous fuzzy sets with bounded α-cuts, we find that an endograph metric convergent sequence is exactly a Γ-convergent sequence satisfying the condition that the union of α-cuts of all its elements is a bounded set in Rm for each α>0. Secondly, based on investigations of level characterizations of fuzzy sets themselves, we present level characterizations (level decomposition properties) of the endograph metric and the Γ-convergence. It is worth mentioning that, using the condition and the level characterizations given above, we discover the fact: the endograph metric and the Γ-convergence are compatible on a large class of general fuzzy sets which do not have any assumptions of normality, convexity or star-shapedness. Its subsets include common particular fuzzy sets such as fuzzy numbers (compact and noncompact), fuzzy star-shaped numbers (compact and noncompact), and general fuzzy star-shaped numbers (compact and noncompact). Thirdly, on the basis of the conclusions presented above, we study various subspaces of the space of upper semi-continuous fuzzy sets with bounded α-cuts equipped with the endograph metric. We present characterizations of total boundedness, relative compactness and compactness in these fuzzy set spaces and clarify relationships among these fuzzy set spaces. It is pointed out that the fuzzy set spaces of noncompact type are exactly the completions of their compact counterparts under the endograph metric.

Higher order Dehn functions for horospheres in products of Hadamard spaces

Abstract Let X be a product of r locally compact and geodesically complete Hadamard spaces. We prove that the horospheres in X centered at regular boundary points of X are Lipschitz-(r − 2)-connected. If X has finite Assouad–Nagata dimension, then using the filling construction by R. Young in [10] this gives sharp bounds on higher order Dehn functions for such horospheres. Moreover, if Γ ⊂ Is(X) is a lattice acting cocompactly on X minus a union of disjoint horoballs, then we get a sharp bound on higher order Dehn functions for Γ. We deduce that apart from the Hilbert modular groups already considered by R. Young, every irreducible ℚ-rank one lattice acting on a product of r Riemannian symmetric spaces of the noncompact type is undistorted up to dimension r−1 and has k-th order Dehn function asymptotic to V (k+1)/k for all k ≤ r − 2.

Infinitely many solutions to the Yamabe problem on noncompact manifolds

We establish the existence of infinitely many complete metrics with constant scalar curvature on prescribed conformal classes on certain noncompact product manifolds. These include products of closed manifolds with constant positive scalar curvature and simply-connected symmetric spaces of noncompact or Euclidean type; in particular, $\mathbb S^m \times\mathbb R^d$, $m\geq2$, $d\geq1$, and $\mathbb S^m\times\mathbb H^d$, $2\leq d<m$. As a consequence, we obtain infinitely many periodic solutions to the singular Yamabe problem on $\mathbb S^m\setminus\mathbb S^k$, for all $0\leq k<(m-2)/2$, the maximal range where nonuniqueness is possible. We also show that all Bieberbach groups in $Iso(\mathbb R^d)$ are periods of bifurcating branches of solutions to the Yamabe problem on $\mathbb S^m\times\mathbb R^d$, $m\geq2$, $d\geq1$.

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.