arXiv research

In this paper, we show that the simplicial volume of Q-rank one locally symmetric spaces covered by the product of R-rank one symmetric spaces is strictly positive.

Classifies foliations on specific symmetric spaces.

Right inverse found for Cartan differential in rank-1 symmetric spaces.

The paper analyzes tensor recovery from symmetric rank-one measurements using information theory.

Sharp isoperimetric inequalities for Neumann eigenvalues in symmetric spaces.

Proves conditions for Fourier transforms in rank 1 symmetric spaces.

Characterizes higher rank model geometries using antipodal sets.

Minimal submanifolds in matrix spaces proven for specific ranks.

Compact rank one symmetric spaces are rigid under certain curvature conditions.

New tools compute index of symmetry in homogeneous fibrations.

Develops sublinear Morse theory in symmetric spaces.

New concept of coarse medians for higher rank symmetric spaces.

Classifies totally geodesic submanifolds in symmetric spaces.

Volume comparison theorem for rank 1 symmetric spaces proved.

No Einstein hypersurfaces found in Damek-Ricci spaces.

Study string topology on symmetric spaces, showing non-triviality and nilpotency results.

Classifies polar foliations on symmetric spaces.

Let X be a nonsingular simply connected projective variety of dimension m, E a rank n vector bundle on X, and L a line bundle on X. Suppose that $S^2(E^{*}) \otimes L$ is an ample vector bundle and that there is a constant even rank $r \ge 2$ symmetric bundle map $E \to E^{*} \otimes L$. We prove that $m \le n-r$. We u…

Characterizes Kähler-hyperbolicity of bounded symmetric domains based on rank and genus.

In this paper, we show that there exists no equifocal submanifold with non-flat section in four irreducible simply connected symmetric spaces of compact type and rank two. Also, we show a fact for the sections of equifocal submanifolds with non-flat section in other irreducible simply connected symmetric spaces of comp…

The paper verifies a conjecture about the index of symmetric spaces.

In this paper we show that if the limit set is not small ,marked length spectrum determines geometric structure of rank one locally symmetric manifolds.

New random walk results on rank one symmetric spaces.

Affine maps reveal higher rank structures in certain spaces.

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 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…

We study the Selberg zeta and the theta function associated to bundles over even-dimensional locally symmetric spaces of rank one.

We show that codimension one dimensional Jacobian of the barycentric straightening map is uniformly bounded for most of the higher rank symmetric spaces. As a consequence, we prove that the locally finite simplicial volume of most $\mathbb Q$-rank $1$ locally symmetric spaces is positive, which has been open for many y…

Study on diagonal and separating coordinates for symmetric spaces of rank 1.

Paper proves eigenvalue inequality for Hopf-symmetric domains.

Proves symmetric spaces for certain quasi-isometric properties.

Let $X=G/K$ be a higher rank symmetric space of non-compact type, where $G$ is the connected component of the isometry group of $X$. We define the splitting rank of $X$, denoted by $\text{srk}(X)$, to be the maximal dimension of a totally geodesic submanifold $Y\subset X$ which splits off an isometric $\mathbb R$-facto…

We study the $L^p$-spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces $M=Γ\backslash X$ 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 c…

We prove that a quasiisometric map between rank one symmetric spaces is within bounded distance from a unique harmonic map. In particular, this completes the proof of the Schoen-Li-Wang conjecture.

We prove that the orbits of a polar action of a compact Lie group on a compact rank one symmetric space are tautly embedded with respect to Z_2-coefficients.

Study finds lower bounds on flat cycles in congruence covers of symmetric spaces.

We obtain Ricci flat Kähler metrics on complex symmetric spaces of rank two by using an explicit asymptotic model whose geometry at infinity is interpreted in the wonderful compactification of the symmetric space. We recover the metrics of Biquard-Gauduchon in the Hermitian case and obtain in addition several new metri…

We prove minimal entropy rigidity for complete, finite volume manifolds locally isometric to a product of rank one symmetric spaces of dimension at least 3: the locally symmetric metric uniquely minimizes (normalized) entropy among all Riemannian metrics. The corresponding theorem is true for maps into these spaces as …

New structural result classifies actions on symmetric spaces.

Paper proves geodesic ball maximizes second Robin eigenvalue in non-compact symmetric spaces.

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…

We prove an obstruction at the level of rational cohomology in small degrees to the existence of positively curved metrics with large symmetry rank. The symmetry rank bound is logarithmic in the dimension of the manifold. As an application, we provide evidence for a generalized conjecture of Hopf that says that no symm…

Study proves inequality for eigenvalues in symmetric spaces.

Let M be a geometrically finite rank one locally symmetric manifolds. We prove that the spectrum of the Laplace operator on M is finite in a small interval which is optimal.

New examples of isoparametric families on non-compact symmetric spaces.

We establish the proportionality principle between the Riemannian volume and locally finite simplicial volume for Q-rank 1 locally symmetric spaces covered by products of hyperbolic spaces, giving the first examples for manifolds whose cusp groups are not necessarily amenable. Also, we give a simple direct proof of the…

We show the existence of isometric (or Ford) fundamental regions for a large class of subgroups of the isometry group of any rank one Riemannian symmetric space of noncompact type. The proof does not use the classification of symmetric spaces. All hitherto known existence results of isometric fundamental regions and do…

We use generalised cross--ratios to prove the Ptolemaean inequality and the Theorem of Ptolemaeus in the setting of the boundary of symmetric Riemannian spaces of rank 1 and of negative curvature.