Classifies polar foliations on symmetric spaces.
problem Classifying polar foliations on symmetric spaces.
method Orbit equivalence and classification up to codimension two.
result Foliations are either hyperpolar or extensions of rank one foliations.
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…
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 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…
The paper studies mean curvature flows on specific orbits of Hermann actions.
problem Analyzing mean curvature flows on principal orbits of Hermann actions.
method Using Mathematica to illustrate and calculate the flows and orbits.
result The minimal principal orbit's position is calculated and illustrated.
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…
I show that any complex manifold that resembles a rank two compact Hermitian symmetric space (other than a quadric hypersurface) to order two at a general point must be an open subset of such a space.
We classify, up to orbit equivalence, the cohomogeneity one actions on the noncompact duals of the symmetric spaces G_2, SU_3 and the real oriented two-plane Grassmannians.
Classifies foliations on specific symmetric spaces.
problem Classifying foliations on symmetric spaces of rank one.
method Orbit equivalence classification.
result Polar homogeneous foliations classified.
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.
Right inverse found for Cartan differential in rank-1 symmetric spaces.
problem Finding a right inverse for the Cartan differential in symmetric spaces.
method Integral operator approach to the Cartan differential on exact forms.
result Extension of Gauss linking integral to rank-1 symmetric spaces.
Characterizes invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.
problem Identifying invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.
method Characterization through invariant Riemannian metrics and Killing vector fields.
result A family of invariant contact metric structures is obtained on tangent sphere bundles of compact symmetric spaces with rank greater than or equal to two.
New Calabi-Yau metrics found on complex symmetric spaces.
problem Finding Calabi-Yau metrics on complex symmetric spaces.
method Complete Calabi-Yau metrics with prescribed horospherical singular tangent cone.
result First examples of Calabi-Yau smoothings of singular tangent cones.
Sharp isoperimetric inequalities for Neumann eigenvalues in symmetric spaces.
problem Finding bounds for eigenvalues of Neumann Laplacian on domains in symmetric spaces.
method Proving sharp inequalities for eigenvalues in compact and noncompact rank-1 symmetric spaces.
result Generalization of previous results for hyperbolic space and symmetric spaces.
Characterizes higher rank model geometries using antipodal sets.
problem Identifying higher rank model geometries among Hadamard spaces.
method Using antipodal sets at infinity to characterize model geometries.
result Characterizes Riemannian symmetric spaces, Euclidean buildings, and products as higher rank model geometries.
Proves conditions for Fourier transforms in rank 1 symmetric spaces.
problem Understanding Fourier transform bounds in symmetric spaces.
method Proves sufficient and necessary conditions using Lipschitz and Fourier type integral conditions.
result Establishes bounds for Fourier transforms in rank 1 symmetric spaces with specific moduli of continuity.
We construct new homogeneous Einstein spaces with negative Ricci curvature in two ways: First, we give a method for classifying and constructing a class of rank one Einstein solvmanifolds whose derived algebras are two-step nilpotent. As an application, we describe an explicit continuous family of ten-dimensional Einst…
New tools compute index of symmetry in homogeneous fibrations.
problem Computing the index of symmetry in homogeneous fibrations.
method Developed new tools to compute the index of symmetry.
result Determined the index of symmetry of various homogeneous spaces.
No Einstein hypersurfaces found in Damek-Ricci spaces.
problem Existence of Einstein hypersurfaces in symmetric spaces.
method Analyzing properties of Damek-Ricci spaces and proving no Einstein hypersurface exists.
result No Einstein hypersurfaces in Damek-Ricci spaces.
New proof shows no equifocal submanifolds with non-flat sections in certain symmetric spaces.
problem Existence of equifocal submanifolds with non-flat sections in symmetric spaces.
method Introduced slice topology, universal covering, and topological Tits buildings to derive contradiction.
result No equifocal submanifolds with non-flat sections in certain symmetric spaces.
Classifies totally geodesic submanifolds in symmetric spaces.
problem Classifying submanifolds in symmetric spaces.
method Classification of totally geodesic submanifolds in products of rank one symmetric spaces.
result Infinitely many examples of irreducible totally geodesic submanifolds in Hermitian symmetric spaces.
Compact rank one symmetric spaces are rigid under certain curvature conditions.
problem Rigidity of compact rank one symmetric spaces under curvature constraints.
method Examined compact symmetric spaces with metric g0 of rank one, and another metric g with sectional curvature bounded by 0 to 1. result If g equals g0 outside a convex subset, then g is isometric with g0. New concept of coarse medians for higher rank symmetric spaces.
problem Understanding medians in higher rank symmetric spaces.
method Introducing coarse r-median spaces and proving their existence. result Existence of coarse higher medians on divisible and quasi-homogeneous convex domains.
Minimal submanifolds in matrix spaces proven for specific ranks.
problem Minimal submanifolds in matrix spaces.
method Proving semialgebraic sets of matrices are minimal.
result Rectangular, skew-symmetric, and symmetric matrices with prescribed eigenvalues are minimal.
The paper analyzes tensor recovery from symmetric rank-one measurements using information theory.
problem Recovering tensors with low symmetric rank from symmetric rank-one measurements.
method Covering numbers argument, Carbery-Wright inequality, orthogonal polynomials, Fano's inequality.
result Near-optimal sample complexity bounds for log-concave distributions.
Study string topology on symmetric spaces, showing non-triviality and nilpotency results.
problem Understanding the structure of string topology on symmetric spaces.
method Used cycles from Bott-Samelson and Ziller to study coproduct and product.
result Showed non-triviality and nilpotency of Chas-Sullivan product and coproduct for higher rank symmetric spaces.
Study finds only two CROSSes can have specific quaternionic structure.
problem Characterizing compact rank one symmetric spaces with quaternionic structures.
method Analyzing properties of CROSSes (compact rank one symmetric spaces).
result Only HPn and CP2 admit almost quaternionic structures. We propose a method to extend submanifolds, singular Riemannian foliations and isometric actions from a boundary component of a noncompact symmetric space to the whole space. This extension method preserves minimal submanifolds, isoparametric foliations and polar actions, among other properties. One of the several appl…
Develops sublinear Morse theory in symmetric spaces.
problem Understanding sublinear Morse properties in symmetric spaces.
method Theory of sublinearly Morse boundary and lemma in higher rank symmetric spaces.
result Proves sublinear Morse lemma in higher rank symmetric spaces.
Researchers classify hypersurfaces in specific symmetric spaces.
problem Classifying homogeneous hypersurfaces in noncompact symmetric spaces.
method Isometric congruence classification of hypersurfaces in SL(3,H)/Sp(3),SO(5,C)/SO(5), and Gr∗(2,Cn+4). result Classification of hypersurfaces up to isometric congruence.
Volume comparison theorem for rank 1 symmetric spaces proved.
problem Volume comparison for symmetric spaces of non-compact type.
method Normalized Ricci--DeTurck flow to analyze volume functional and derive monotonicity properties.
result Volume comparison theorem established for rank 1 symmetric spaces of non-compact type.
The paper verifies a conjecture about the index of symmetric spaces.
problem Calculating the minimal codimension of totally geodesic submanifolds in symmetric spaces of higher rank.
method Developed several approaches to tackle the problem, including a conjecture.
result Verified the conjecture for the index of symmetric spaces.
Affine maps reveal higher rank structures in certain spaces.
problem Characterizing spaces with higher rank structures.
method Using Hadamard spaces with geometric group actions and affine maps.
result Affine maps not dilations indicate higher rank structures.
New random walk results on rank one symmetric spaces.
problem Analyzing random walks on noncompact rank one symmetric spaces.
method Unified algebraic framework using Möbius addition and harmonic analysis of spherical functions.
result Renormalized walk converges to heat kernel on Laplace-Beltrami operator.
Proves symmetric spaces for certain quasi-isometric properties.
problem Understanding quasi-isometric spaces and their properties.
method Analyzes geometric and geometrically finite quasi-actions.
result Proves quasi-isometric spaces are symmetric or real line.
G. Pipoli and C. Sinestrari considered the mean curvature flow starting from a closed submanifold in the complex projective space. They proved that if the submanifold is of small codimension and satisfies a suitable pinching condition for the second fundamental form, then the flow has two possible behaviors: either the…
Study on diagonal and separating coordinates for symmetric spaces of rank 1.
problem Existence and nonexistence of diagonal and separating coordinates for symmetric spaces of rank 1.
method Generalization of results by Gauduchon and Moroianu, 2020, and analysis of constant sectional curvature and orthogonal separation of variables.
result Diagonal coordinates exist if and only if the symmetric space has constant sectional curvature.
The aim of this paper is to study the spectrum of the Lp Laplacian and the dynamics of the Lp 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 study the Selberg zeta and the theta function associated to bundles over even-dimensional locally symmetric spaces of rank one.
Estimates for graph embeddings into symmetric spaces derived from coarse geometry.
problem Estimating optimal volume of graph embeddings into symmetric spaces.
method Coarse geometric thick embeddings and wiring techniques.
result Optimal and lower bounds for graph embeddings in symmetric spaces of different ranks.
Study finds lower bounds on flat cycles in congruence covers of symmetric spaces.
problem Counting flat cycles in congruence covers of symmetric spaces.
method Lower bound calculation for immersed compact flat manifolds.
result Lower bounds on the contribution of flat cycles to homology.
Study proves inequality for eigenvalues in symmetric spaces.
problem Proving an inequality for Steklov eigenvalues in symmetric spaces.
method Analyzes eigenvalues on bounded domains in noncompact rank-1 symmetric spaces.
result Extends previous results to non-Euclidean spaces.
We study the Lp-spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces M=Γ\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…
New structural result classifies actions on symmetric spaces.
problem Classify cohomogeneity one actions on symmetric spaces.
method Developed new structural result and applied to specific cases.
result Classified cohomogeneity one actions on SL(n,R)/SO(n), n>1.
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 Q-rank 1 locally symmetric spaces is positive, which has been open for many y…
Paper proves geodesic ball maximizes second Robin eigenvalue in non-compact symmetric spaces.
problem Maximizing the second Robin eigenvalue in non-compact rank-1 symmetric spaces.
method Quantitative spectral inequality for the second Robin eigenvalue.
result Geodesic ball maximizes the second Robin eigenvalue among domains of the same volume.
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…
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 srk(X), to be the maximal dimension of a totally geodesic submanifold Y⊂X which splits off an isometric R-facto…