arXiv research

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

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

Volume comparison theorem for rank 1 symmetric spaces proved.

In this paper, we prove that full irreducible curvature-adapted isoparametric submanifolds of codimension greater than one in a symmetric space of non-compact type are principal orbits of Hermann actions on the symmetric spaces under certain conditions.

New spectral analysis on non-compact spaces.

Develops sublinear Morse theory in symmetric spaces.

In this paper, we assume that all isoparametric submanifolds have flat section. The main purpose of this paper is to prove that, if a full irreducible complete isoparametric submanifold of codimension greater than one in a symmetric space of non-compact type admits a reflective focal submanifold and if it of real analy…

Paper proves eigenvalue inequality for Hopf-symmetric domains.

In this paper, we obtain a Cartan type identity for curvature-adapted isoparametric hypersurfaces in symmetric spaces of compact type or non-compact type. This identity is a generalization of Cartan-D'Atri's identity for curvature-adapted(=amenable) isoparametric hypersurfaces in rank one symmetric spaces. Furthermore,…

We show that compact, locally symmetric spaces of non-compact type have positive simplicial volume. This gives a positive answer to a question that was first raised by Gromov in 1982. We provide a summary of results that are known to follow from positivity of the simplicial volume.

Paper constructs an example of a non-compact submanifold in a quaternionic Kähler symmetric space.

We study Poisson symmetric spaces of group type with Cartan subalgebra "adapted" to the Lie cobracket.

In this paper, we develop a loop group description of harmonic maps $\mathcal{F}: M \rightarrow G/K$ ``of finite uniton type", from a Riemann surface $M$ into inner symmetric spaces of compact or non-compact type. This develops work of Uhlenbeck, Segal, and Burstall-Guest to non-compact inner symmetric spaces. To be mo…

The paper establishes a duality between non-compact and compact symmetric pairs.

It is known that principal orbits of Hermann actions on a symmetric space of non-compact type are curvature-adapted isoparametric submanifolds having no focal point of non-Euclidean type on the ideal boundary of the ambient symmetric space. In this paper, we investigate the mean curvature flows for such a curvature-ada…

Paper develops a new method for harmonic maps into symmetric spaces.

New complete minimal submanifolds found in specific Riemannian spaces.

New minimal surface theory disproves a conjecture in symmetric spaces.

Lie foliations with symmetric leaves are smoothly conjugate to homogeneous ones.

Study proper actions of Lie groups on symmetric spaces, finding rigidity results and Hurwitz-Radon numbers.

Positive simplicial volume implies locally symmetric space structure.

We formulate a conjecture that arithmetic locally symmetric manifolds have simple homotopy type, and prove it for the non-compact case. More precisely, we show that, for any symmetric space S of non-compact type without Euclidean de Rham factors, there are constants a=a(S) and d=d(S) such that any non-compact arithmeti…

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…

Proves unique maps from certain spaces to others.

Classifies Ricci soliton subgroups in specific Lie groups.

Extends flat submanifold properties from hyperbolic plane to symmetric spaces.

In this paper we introduce two new methods for constructing harmonic morphisms from solvable Lie groups. The first method yields global solutions from any simply connected nilpotent Lie group and from any Riemannian symmetric space of non-compact type and rank $r\ge 3$. The second method provides us with global solutio…

Geodesic spheres in certain symmetric spaces are quantitatively stable under small perturbations.

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 classify all tight holomorphic maps between Hermitian symmetric spaces of non-compact type.

We study the barycentric straightening of simplices in irreducible symmetric spaces of non-compact type. We show that, for an n-dimensional symmetric space of rank r>1, the p-Jacobian has uniformly bounded norm, as soon as p is at least n-r+2. As a consequence, for a non-compact, connected, semisimple real Lie group G,…

In this note, we show that for any harmonic map into a non-compact symmetric space one can find naturally a "dual" harmonic map into a compact symmetric space which can be constructed from the same basic data (called "potentials" in the loop group formalism). Locally also the inverse/converse duality theorem holds.

The paper develops mathematical models for neural networks using non-compact symmetric spaces.

Classifies totally geodesic submanifolds in exceptional symmetric spaces.

In this paper we study a homological version of the higher-dimensional divergence invariants defined by Brady and Farb. We show that they are quasi-isometry invariants in the class of proper cocompact Hadamard spaces in the sense of Alexandrov and that they can moreover be used to detect the Euclidean rank of such spac…

Method extends eigenfunction construction to non-symmetric spaces.

When a discrete group admits a convex-cocompact action on a non-compact rank-one symmetric space, there is a natural lower bound for the Hausdorff dimension of the limit set, given by the Ahlfors regular conformal dimension of the boundary of the group. We show that equality is achieved precisely when the group stabili…

We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we derive an extension of Gromov's volume comparison theorem to products of negatively c…

In constant curvatures spaces, there are a lot of characterizations of geodesic balls as optimal domain for shape optimization problems. Although it is natural to expect similar characterizations in rank one symmetric spaces, very few is known in this setting. In this paper we prove that, in a non-compact rank one symm…

The paper extends rigidity results to non-compact domains and infinite energy maps.

First we investigate the evolutions of the radius function and its gradient along the volume-preserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space under certain condition for the radius function. Next, we prove that the …

In this paper we give a positive answer to the open existence problem for complex-valued harmonic morphisms from the non-compact irreducible Riemannian symmetric spaces $SL_n(R)/SO(n)$, $SU^*(2n)/Sp(n)$ and their compact duals $SU(n)/SO(n)$ and $SU(2n)/Sp(n)$. Furthermore we prove the existence of globally defined, com…

New Einstein manifolds split into symmetric and compact parts.

Constructs explicit p-harmonic functions on specific Lie groups.

Researchers create explicit p-harmonic functions on specific symmetric spaces.

We show that canonical Carnot-Caratheodory spherical and horospherical metrics, which are defined on the boundary at infinity of every rank one symmetric space of non-compact type, are visual, i.e., they are bilipschitz equivalent with universal bilipschitz constants to the inverse exponent of Gromov products based in …

We investigate periodic diffeomorphisms of non-compact aspherical manifolds (and orbifolds) and describe a class of spaces that have no homotopically trivial periodic diffeomorphisms. Prominent examples are moduli spaces of curves and aspherical locally symmetric spaces with non-vanishing Euler characteristic. In the i…

Unified mathematical framework for non-compact symmetric spaces with negative curvature.