We prove sphere packing density bounds in hyperbolic space (and more generally irreducible symmetric spaces of noncompact type), which were conjectured by Cohn and Zhao and generalize Euclidean bounds by Cohn and Elkies. We work within the Bowen–Radin framework of packing density and replace the use of the Poisson summation formula in the proof of the Euclidean bound by Cohn and Elkies with an analogous formula arising from methods used in the theory of mathematical quasicrystals.

The action of the subgroup G2 of SO(7) (resp. Spin(7) of SO(8)) on the Grassmannian space M = SO(7)/(SO(5)×SO(2)) (resp. M = SO(8)/(SO(5)×SO(3)) ) is still transitive. We prove that the spectrum (i.e. the collection of eigenvalues of its Laplace-Beltrami operator) of a symmetric metric g0 on M coincides with the spectrum of a G2-invariant (resp. Spin(7)-invariant) metric g on M only if g0 and g are isometric. As a consequence, each non-flat compact irreducible symmetric space of non-group type is spectrally unique among the family of all currently known homogeneous metrics on its underlying differentiable manifold.

Given any compact homogeneous space [Formula: see text] with [Formula: see text] simple, we consider the new space [Formula: see text], where [Formula: see text] denotes diagonal embedding, and study the existence, classification and stability of [Formula: see text]-invariant Einstein metrics on [Formula: see text], as a first step into the largely unexplored case of homogeneous spaces of compact non-simple Lie groups. We find unstable Einstein metrics on [Formula: see text] for most spaces [Formula: see text] such that their standard metric is Einstein (e.g., isotropy irreducible) and the Killing form of [Formula: see text] is a multiple of the Killing form of [Formula: see text] (e.g., [Formula: see text] simple), a class which contains [Formula: see text] families and [Formula: see text] individual examples. A complete classification is obtained in the case when [Formula: see text] is an irreducible symmetric space and [Formula: see text] is simple. We also study the behavior of the scalar curvature function on the space of all normal metrics on [Formula: see text] (none of which is Einstein), obtaining that the standard metric is a global minimum.

Abstract We construct a sequence of explicit blow-ups and blow-downs on an irreducible compact Hermitian symmetric spaces $X$ which transforms it into a projective space of the same dimension. Moreover, this resolves a birational map given by Landsberg and Manivel. Centers of the blow-ups for $X$ are constructed by loci of chains of minimal rational curves and centers of the blow-ups for the projective space are constructed from the variety of minimal rational tangents of $X$ and its higher secant varieties. The result was known in the special case where $X$ is of rank 2 and could be found in Zak’s monograph “Tangents and secants of algebraic varieties.”

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

On gap rigidity problems for compact Hermitian symmetric spaces

We study the convergence of volume-normalized Betti numbers in Benjamini-Schramm convergent sequences of non-positively curved manifolds with finite volume. In particular, we show that if $X$ is an irreducible symmetric space of noncompact type, $X \neq \mathbb H^3$, and $(M_n)$ is any Benjamini-Schramm convergent sequence of finite volume $X$-manifolds, then the normalized Betti numbers $b_k(M_n)/vol(M_n)$ converge for all $k$. As a corollary, if $X$ has higher rank and $(M_n)$ is any sequence of distinct, finite volume $X$-manifolds, the normalized Betti numbers of $M_n$ converge to the $L^2$ 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.

Totally geodesic submanifolds in exceptional symmetric spaces

We investigate the curvature operator of the second kind on Riemannian manifolds and prove several classification results. The first one asserts that a closed Riemannian manifold with three-positive curvature operator of the second kind is diffeomorphic to a spherical space form, improving a recent result of Cao–Gursky–Tran assuming two-positivity. The second one states that a closed Riemannian manifold with three-nonnegative curvature operator of the second kind is either diffeomorphic to a spherical space form, or flat, or isometric to a quotient of a compact irreducible symmetric space. This settles the nonnegativity part of Nishikawa’s conjecture under a weaker assumption.

Abstract We establish an equivalence between two approaches to quantization of irreducible symmetric spaces of compact type within the framework of quasi-coactions, one based on the Enriquez–Etingof cyclotomic Knizhnik–Zamolodchikov (KZ) equations and the other on the Letzter–Kolb coideals. This equivalence can be upgraded to that of ribbon braided quasi-coactions, and then the associated reflection operators (K-matrices) become a tangible invariant of the quantization. As an application we obtain a Kohno–Drinfeld type theorem on type $\mathrm {B}$ braid group representations defined by the monodromy of KZ-equations and by the Balagović–Kolb universal K-matrices. The cases of Hermitian and non-Hermitian symmetric spaces are significantly different. In particular, in the latter case a quasi-coaction is essentially unique, while in the former we show that there is a one-parameter family of mutually nonequivalent quasi-coactions.

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 \({\mathbb {R}}^3\)).

We give an explicit classification of maximal antipodal sets in any irreducible compact symmetric space except for spin groups and half spin groups, and some quotient symmetric spaces associated to them.

We show that for any weakly reflective submanifold of a compact isotropy irreducible Riemannian homogeneous space its inverse image under the parallel transport map is an infinite dimensional weakly reflective PF submanifold of a Hilbert space. This is an extension of the author's previous result in the case of compact irreducible Riemannian symmetric spaces. We also give a characterization of so obtained weakly reflective PF submanifolds.

We prove the linear stability with respect to the Einstein-Hilbert action of the symmetric spaces {text {SU}}(n), nge 3, and E_6/F_4. Combined with earlier results, this resolves the stability problem for irreducible symmetric spaces of compact type.

Let $G/K$ be a simply connected compact irreducible symmetric space of real rank one. For each $K$-type $\tau$ we compare the notions of $\tau$-representation equivalence with $\tau$-isospectrality. We exhibit infinitely many $K$-types $\tau$ so that, for arbitrary discrete subgroups $\Gamma$ and $\Gamma'$ of $G$, if the multiplicities of $\lambda$ in the spectra of the Laplace operators acting on sections of the induced $\tau$-vector bundles over $\Gamma\backslash G/K$ and $\Gamma'\backslash G/K$ agree for all but finitely many $\lambda$, then $\Gamma$ and $\Gamma'$ are $\tau$-representation equivalent in $G$ (i.e.\ $\dim \operatorname{Hom}_G(V_\pi, L^2(\Gamma\backslash G))=\dim \operatorname{Hom}_G(V_\pi, L^2(\Gamma'\backslash G))$ for all $\pi\in \widehat G$ satisfying $\operatorname{Hom}_K(V_\tau,V_\pi)\neq0$). In particular $\Gamma\backslash G/K$ and $\Gamma'\backslash G/K$ are $\tau$-isospectral (i.e.\ the multiplicities agree for all $\lambda$). We specially study the case of $p$-form representations, i.e. the irreducible subrepresentations $\tau$ of the representation $\tau_p$ of $K$ on the $p$-exterior power of the complexified cotangent bundle $\bigwedge^p T_{\mathbb C}^*M$. We show that for such $\tau$, in most cases $\tau$-isospectrality implies $\tau$-representation equivalence. We construct an explicit counter-example for $G/K= \operatorname{SO}(4n)/ \operatorname{SO}(4n-1)\simeq S^{4n-1}$.

For a homogeneous space [Formula: see text] of reductive type, we consider the tangential homogeneous space [Formula: see text]. In this paper, we give obstructions to the existence of compact Clifford–Klein forms for such tangential symmetric spaces and obtain new tangential symmetric spaces which do not admit compact Clifford–Klein forms. As a result, in the class of irreducible classical semisimple symmetric spaces, we have only two types of symmetric spaces which are not proved not to admit compact Clifford–Klein forms. The existence problem of compact Clifford–Klein forms for homogeneous spaces of reductive type, which was initiated by Kobayashi in 1980s, has been studied by various methods but is not completely solved yet. On the other hand, the one for tangential homogeneous spaces has been studied since 2000s and an analogous criterion was proved by Kobayashi and Yoshino. In concrete examples, further works are needed to verify Kobayashi–Yoshino’s condition by direct calculations. In this paper, some easy-to-check necessary conditions ([Formula: see text][Formula: see text]obstructions) for the existence of compact quotients in the tangential setting are given, and they are applied to the case of symmetric spaces. The conditions are related to various fields of mathematics such as associated pair of symmetric space, Calabi–Markus phenomenon, trivializability of vector bundle (parallelizability, Pontrjagin class), Hurwitz–Radon number and Pfister’s theorem (the existence problem of common zero points of polynomials of odd degree).

Let ,G/K, be an irreducible non-compact Hermitian symmetric space and let ,D, be a ,K-invariant domain in ,G/K. In this paper we characterize several classes of ,K-invariant plurisubharmonic functions on ,D, in terms of their restrictions to a slice intersecting all ,K-orbits. As applications we show that ,K-invariant plurisubharmonic functions on ,D, are necessarily continuous and we reproduce the classification of Stein ,K-invariant domains in ,G/K, obtained by Bedford and Dadok. (J Geom Anal 1:1–17, 1991).

Recently, Jablonski proved that, to a large extent, a simply connected solvable Lie group endowed with a left-invariant Ricci soliton metric can be isometrically embedded into the solvable Iwasawa group of a non-compact symmetric space. Motivated by this result, we classify codimension one subgroups of the solvable Iwasawa groups of irreducible symmetric spaces of non-compact type whose induced metrics are Ricci solitons. We also obtain the classifications of codimension one Ricci soliton subgroups of Damek-Ricci spaces and generalized Heisenberg groups.

Abstract Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of $G/K$ . For the special case of the orbital measures, $\nu _{a_{i}}$ , supported on the double cosets $Ka_{i}K$ , where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ except for the symmetric space of Cartan class $AI$ when the convolution of three orbital measures is needed (even though $\nu _{a_{1}}\ast \nu _{a_{2}}$ is absolutely continuous).

We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semi-simple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a product of quasi-isometric embeddings into irreducible symmetric spaces. We thus extend earlier rigidity results about quasi-isometric embeddings to the setting of semi-simple Lie groups. We also present some examples when the rigidity does not hold, including first examples in which every flat is mapped into multiple flats.