arXiv research

New concept of coarse medians for higher rank symmetric spaces.

This paper proves that there are no compact forms for a large class of homogeneous spaces admitting actions by higher-rank semisimple Lie groups. It builds on Zimmer's approach for studying such spaces using cocycle superrigidity. The proof involves cocycle superrigidity, measure rigidity for unipotent flows, technique…

Study on horospheres in higher rank homogeneous spaces, proving density properties.

Constructs geometries with nonvanishing curvature and essential automorphisms.

Classifies measures for Anosov subgroups in higher ranks.

We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improve…

New dHYM connections found on complex vector bundles.

We show that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid (up to an automorphism). This result is the main part in the proof of local smooth rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov…

The paper studies ergodicity of flows on subspaces, generalizing earlier work.

New tools compute index of symmetry in homogeneous fibrations.

No non-product Hessian rank 1 affine homogeneous hypersurfaces exist in dimensions 5 and above.

We give a positive answer to the Chavel's conjecture [J. Diff. Geom. 4 (1970), 13-20]: a simply connected rank one normal homogeneous space is symmetric if any pair of conjugate points are isotropic. It implies that all simply connected rank one normal homogeneous space with the property that the isotropy action is var…

Classifies foliations on specific symmetric spaces.

We give an upper bound for the rank $r$ of homogeneous (even) Clifford structures on compact manifolds of non-vanishing Euler characteristic. More precisely, we show that if $r=2^a\cdot b$ with $b$ odd, then $r\le 9$ for $a=0$, $r\le 10$ for $a=1$, $r\le 12$ for $a=2$ and $r\le 16$ for $a\ge 3$. Moreover, we describe t…

Classifies polar foliations on symmetric spaces.

We classify isometric actions of compact Lie groups on quaternionic-Kähler projective spaces with vanishing homogeneity rank. We also show that they are not in general quaternion-coisotropic.

Affine maps reveal higher rank structures in certain spaces.

Confirming a conjecture, new CAT(0) spaces of higher rank are rigid.

We study locally compact metric spaces that enjoy various forms of homogeneity with respect to Möbius self-homeomorphisms. We investigate connections between such homogeneity and the combination of isometric homogeneity with invertibility. In particular, we provide a new characterization of snowflakes of boundaries of …

Study classifies special Hessian rank 2 hypersurfaces in 4D space.

We prove that almost all geodesics on a noncompact locally symmetric space of finite volume grow with a logarithmic speed -- the higher rank generalization of a theorem of D. Sullivan (1982). More generally, under certain conditions on a sequence of subsets $A_n$ of a homogeneous space $G/Γ$ ($G$ a semisimple Lie group…

We classify noncompact homogeneous spaces which are Einstein and asymptotically harmonic. This completes the classification of Riemannian harmonic spaces in the homogeneous case: Any simply connected homogeneous harmonic space is flat, or rank-one symmetric, or a nonsymmetric Damek-Ricci space. Independently, Y. Nikola…

We prove two rigidity results for complete Riemannian three-manifolds of higher rank. Complete three-manifolds have higher spherical rank if an only if they are spherical space forms. Complete finite volume three-manifolds have higher hyperbolic rank if and only if they are hyperbolic space forms.

Characterizes higher rank model geometries using antipodal sets.

We complete our recent classification of compact inner symmetric spaces with weakly complex tangent bundle by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous spaces with positive Euler characteristic. We show that a simply connected compact equal rank …

Method constructs 2-bundles over homogeneous spaces.

The study proves conditions for CAT(0) spaces with higher rank rigidity.

New method calculates signatures of biquotients.

A Riemannian manifold is called harmonic if its volume density function expressed in polar coordinates centered at any point is radial. Flat and rank-one symmetric spaces are harmonic. The converse (the Lichnerowicz Conjecture) is true for manifolds of nonnegative scalar curvature and for some other classes of manifold…

We consider cohomogeneity one homogeneous disk bundles and adress the question when these admit a nonnegatively curved invariant metric with normal collar, i.e., such that near the boundary the metric is the product of an interval and a normal homogeneous space. If such a bundle is not (the quotient of) a trivial bundl…

Derives smooth homogeneous structures for low-rank tensors.

We show that polar actions of cohomogeneity two on simple compact Lie groups of higher rank, endowed with a biinvariant Riemannian metric, are hyperpolar. Combining this with a recent result of the second-named author, we are able to prove that polar actions induced by reductive algebraic subgroups in the isometry grou…

Study critical exponents for L^p-cohomology of higher rank Lie groups and manifolds.

The study examines conical Kähler-Einstein metrics on specific Fano manifolds and their behavior at the limit angle.

The aim of this paper is to study the spectrum of the $L^p$ Laplacian and the dynamics of the $L^p$ heat semigroup on non-compact locally symmetric spaces of higher rank. Our work here generalizes previously obtained results in the setting of locally symmetric spaces of rank one to higher rank spaces. Similarly as in t…

We generalize the higher rank rigidity theorem to a class of Finsler spaces, i.e. Berwald spaces. More precisely, we prove that a complete connected Berwald space of finite volume and bounded nonpositive flag curvature with rank at least $2$ whose universal cover is irreducible, is a locally symmetric space or a locall…

Develops sublinear Morse theory in symmetric spaces.

Study higher rank inner products and their tilings to describe tori degenerations.

Study finds only two CROSSes can have specific quaternionic structure.

The paper proves actions of lattices in higher rank groups have cost one.

In this work we study the existence of homogeneous Einstein metrics on the total space of homogeneous fibrations such that the fibers are totally geodesic manifolds. We obtain the Ricci curvature of an invariant metric with totally geodesic fibers and some necessary conditions for such a metric to be Einstein in terms …

Study eigenvalues of CROSS spaces with various metrics.

We classify compact homogeneous geometries of irreducible spherical type and rank at least 2 which admit a transitive action of a compact connected group, up to equivariant 2-coverings. We apply our classification to polar actions on compact symmetric spaces.

We give a full classification of higher order parallel surfaces in three-dimensional homogeneous spaces with four-dimensional isometry group, i.e. in the so-called Bianchi-Cartan-Vranceanu family. This gives a positive answer to a conjecture formulated in 2002. As a partial result, we prove that totally umbilical surfa…

This paper continues the work of our previous paper [8], where we generalize kth-powers of the Euclidean Dirac operator D_x to higher spin spaces in the case the target space is a degree one homogeneous polynomial space. In this paper, we reconsider the generalizations of D_x^3 and D_x^4 to higher spin spaces in the ca…

Researchers find solutions to Einstein equations in higher dimensions.

Linear inequality found for certain curved spaces.

A subset S of a Riemannian manifold N is called extrinsically homogeneous if S is an orbit of a subgroup of the isometry group of N. Thorbergsson proved the remarkable result that every complete, connected, full, irreducible isoparametric submanifold of a finite dimensional Euclidean space of rank at least 3 is extrins…