Bounded symmetric domains are biholomorphic to tube domains over Finsler symmetric cones.
problem Characterizing biholomorphic mappings between tube domains and bounded symmetric domains.
method Analyzing properties of Finsler symmetric cones and unital JB-algebras.
result Tube domains over Finsler symmetric cones are biholomorphic to bounded symmetric domains.
Extends polydisk theorem to Hartogs domains over symmetric domains.
problem Rigidity phenomena in Riemannian manifolds.
method Extension of polydisk theorem to Hartogs domains over arbitrary symmetric domains.
result Dual of a Hartogs domain over a bounded symmetric domain admits no totally geodesic immersion into any compact Riemannian manifold.
We develop a theory of planar, origin-symmetric, convex domains that are inextensible with respect to lattice covering, that is, domains such that augmenting them in any way allows fewer domains to cover the same area. We show that origin-symmetric inextensible domains are exactly the origin-symmetric convex domains wi…
Uniformizes klt pairs using bounded symmetric domains.
problem Characterizing klt pairs uniformizable by bounded symmetric domains.
method Determines conditions for uniformization using Miyaoka-Yau-type inequalities.
result Characterizations of orbifold quotients of polydisc and classical bounded symmetric domains.
New metric defined for bounded symmetric domains.
problem Defining a new metric for bounded symmetric domains.
method Using generalized Hilbert metric and Borel embedding.
result The new metric differs from Carathéodory and Bergman metrics except for complex hyperbolic space.
Extends Polydisk Theorem to Cartan-Hartogs domains.
problem Characterizing geodesics in Cartan-Hartogs domains.
method Applying the Polydisk Theorem to Cartan-Hartogs domains.
result Cartan-Hartogs domains inherit geodesic submanifolds from bounded symmetric domains.
In this article, we study holomorphic isometric embeddings between bounded symmetric domains. In particular, we show the total geodesy of any holomorphic isometric embedding between reducible bounded symmetric domains with the same rank.
Study proves inequalities for eigenvalues of symmetric domains in space forms.
problem Eigenvalue problem for Laplacian on symmetric domains in space forms.
method Symmetry assumptions on domain, proof of inequalities for eigenvalues.
result Proves Szegö-Weinberger type inequalities for first n positive Neumann eigenvalues.
Paper proves eigenvalue inequality for Hopf-symmetric domains.
problem Eigenvalue inequality for Hopf-symmetric domains in non-compact symmetric spaces.
method Used geometric and spectral analysis on non-compact rank one symmetric spaces.
result Eigenvalue inequality for bounded Hopf-symmetric domains in non-compact symmetric spaces.
Characterizes Kähler-hyperbolicity of bounded symmetric domains based on rank and genus.
problem Understanding the Kähler-hyperbolicity of bounded symmetric domains.
method Defines Kähler-hyperbolicity length by rank and genus, and characterizes it through a special Bergman potential.
result Establishes a unique constant for Kähler-hyperbolicity based on gradient length of a Bergman potential.
We consider the Johnson-Koranyi-Hua system on symmetric Siegel domains of type two. We prove that all functions which are annihilated by the system and satisfy an H^2 integrability condition are pluriharmonic. So the situation is completely different on type two domains than on tube type domains: it was proved by Johns…
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.
New nodal domain theorems for symmetric matrices via signed graphs.
problem Establish nodal domain theorems for symmetric matrices.
method Explore signed graph structure to define nodal domains for any function.
result Improved lower bound estimates for the number of strong nodal domains.
Maps preserving Carathéodory distance between symmetric domains are rigid.
problem Rigidity of maps preserving Carathéodory distance between bounded symmetric domains.
method Large-scale geometry of Carathéodory distance, horocompactification, Gromov product.
result Maps preserving Carathéodory distance are rigid and either holomorphic or antiholomorphic.
Study on holomorphic isometries between complex domains, revealing geometric properties.
problem Characterizing holomorphic isometries between bounded symmetric domains.
method Analyzing holomorphic isometries between complex unit ball and other bounded symmetric domains, using classical results for complex-analytic subvarieties of Stein manifolds.
result Images of holomorphic isometries have specific geometric properties, including intersections with affine-linear subspaces.
Unified method to calculate Gromov norm for Kähler classes of bounded symmetric domains.
problem Calculating the Gromov norm of Kähler classes for all bounded symmetric domains.
method Combination of ideas from Domin-Toledo and Toledo, aided by the Polydisc Theorem.
result Unified and simplified calculation of Gromov norm for all bounded symmetric domains.
New domains of discontinuity found for Anosov representations.
problem Understanding Anosov representations acting on homogeneous spaces.
method Constructing open domains of discontinuity for Anosov representations acting on specific homogeneous spaces.
result Describes the largest possible open domains of discontinuity for Zariski dense Anosov representations.
This paper shows that, in dimensions two or more, there are no holomorphic isometries between Teichmüller spaces and bounded symmetric domains in their intrinsic Kobayashi metrics.
The paper finds determinantal expressions for certain symmetric space integrals.
problem Finding compact expressions for integrals on symmetric spaces.
method Expressing integrals as determinants or Pfaffians for K-invariant functions. result Determinantal expressions for specific symmetric space integrals.
We first study holomorphic isometries from the Poincaré disk into the product of the unit disk and the complex unit n-ball for n≥2. On the other hand, we observe that there exists a holomorphic isometry from the product of the unit disk and the complex unit n-ball into any irreducible bounded symmetric domain …
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 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…
Study on symmetric domains with Bergman metric properties.
problem Properties of Bergman metrics in symmetric domains.
method Combining Hermitian symmetric spaces theory, Kähler immersions, and analytic/pluripotential tools.
result Rigidity results for Bergman metrics in bounded domains.
The aim of this note is to explain a generalization to the real case of a well known result on the automorphism group of an unbounded tube type symmetric domain in a complex vector space of finite dimension.
Introduces Kähler duality between domains in complex space.
problem None explicitly stated; focuses on concept introduction.
method None explicitly stated; focuses on concept introduction.
result Introduces Kähler duality between domains in complex space.
Study modular geodesics and wedge domains in non-compactly causal symmetric spaces.
problem Understanding the geometric implementation of modular group in symmetric spaces.
method Analyzing the flow generated by Euler elements and their geometric properties.
result The wedge region W is connected and coincides with the observer domain under certain conditions.
We study the canonical complexifications of non-compact Riemannian symmetric spaces G/K by the Grauert tube construction. We determine the maximal such complexification, a domain already constructed in another context by Akhiezer and Gindikin (Math. Ann., 1990), and show that this domain is Stein. We show there is an a…
Establishes a lower bound for Kähler hyperbolicity modulus in hyperconvex domains and bounded strongly pseudoconvex domains.
problem Kähler hyperbolicity modulus for simply-connected Kähler hyperbolic manifolds
method Computes the Kähler hyperbolicity modulus for bounded symmetric domains
result Establishes a lower bound for the Kähler hyperbolicity modulus in terms of the boundary behavior of the gradient length of a plurisubharmonic function
We establish a new symmetrization procedure for the isoperimetric problem in symmetric spaces of noncompact type. This symmetrization generalizes the well known Steiner symmetrization in euclidean space. In contrast to the classical construction the symmetrized domain is obtained by solving a nonlinear elliptic equatio…
The study proves geometric inequalities for static convex domains in static rotationally symmetric spaces.
problem Proving geometric inequalities for static convex domains in static rotationally symmetric spaces.
method Locally constrained curvature flow in a static rotationally symmetric space Nn+1, proving graphical solutions and static convexity preservation. result Proves weighted geometric inequalities for static convex domains close to a slice of Nn+1. Study nearly geodesic immersions in symmetric spaces, leading to explicit fibrations.
problem Understanding geodesic immersions in higher rank symmetric spaces.
method Define and analyze nearly geodesic immersions with a bound on fundamental form.
result Explicit fibrations of domains of discontinuity for certain representations.
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.
Study reveals adversarially robust domain adaptation is harder to generalize across domains.
problem Hardness of transferring adversarial robustness across different domains.
method Analysis of adversarial Rademacher complexity over symmetric difference hypothesis space.
result Adversarial Rademacher complexity is always greater than non-adversarial, indicating intrinsic hardness.
We prove a spanning result for vector-valued Poincaré series on a bounded symmetric domain. We associate a sequence of holomorphic automorphic forms to a submanifold of the domain. When the domain is the unit ball in Cn, we provide estimates for the norms of these automorphic forms and we find asymptotics of…
An Hermitian bounded symmetric domain in a complex vector space, given in its circled realization, is endowed with two natural symplectic forms: the flat form and the hyperbolic form. In a similar way, the ambient vector space is also endowed with two natural symplectic forms: the Fubini-Study form and the flat form. I…
In the current article our primary objects of study are compact complex submanifolds of quotient manifolds of irreducible bounded symmetric domains by torsion free discrete lattices of automorphisms. We are interested in the characterization of the totally geodesic submanifolds among compact splitting complex submanifo…
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.
We combine recent developments on weakly symmetric pseudo--riemannian nilmanifolds with with geometric methods for construction of unitary representations on square integrable Dolbeault cohomology spaces. This runs parallel to construction of discrete series representations on spaces of square integrable harmonic forms…
Adapting the method of Andrews-Clutterbuck we prove an eigenvalue gap theorem for a class of non symmetric second order linear elliptic operators on a convex domain in euclidean space. The class of operators includes the Bakry-Emery laplacian with potential and any operator with second order term the laplacian whose fi…
We study general properties of holomorphic isometric embeddings of complex unit balls Bn into bounded symmetric domains of rank ≥2. In the first part, we study holomorphic isometries from (Bn,kgBn) to (Ω,gΩ) with non-minimal isometric constants k for any irreducible bounded s…
Study shows only one type of proper domain in certain spaces.
problem Classifying proper domains in Hermitian symmetric spaces.
method Analyzing Shilov boundaries and automorphism groups.
result Classification of closed proper manifolds locally modeled on Shilov boundaries.
Inspired by the work of Z. Lu and G. Tian [21] in the compact setting, in this paper we address the problem of studying the Szegö kernel of the disk bundle over a noncompact Kähler manifold. In particular we compute the Szegö kernel of the disk bundle over a Cartan-Hartogs domain based on a bounded symmetric domain. Th…
Equivalence proven between uniformizing varieties and tensors, generalizing uniformization results.
problem Characterizing complex-projective varieties with klt singularities and ample canonical divisors.
method Constructing a uniformizing variation of Hodge structure from slope zero tensors and vice versa.
result Generalization of uniformization results to singular settings, including quotients of tube domains.
The study examines special domains in S^2 supporting specific solutions to a PDE.
problem Identifying special domains in S^2 supporting positive solutions to a PDE.
method Extends moving plane method and Alexandrov reflection method to prove symmetry.
result Domains must be rotationally symmetric under specific conditions.
The paper defines Dirichlet domains for Anosov subgroups in Lie groups.
problem Finding finite-sided Dirichlet domains for Anosov subgroups in Lie groups.
method Introducing a sufficient condition for finite-sided Dirichlet domains in semisimple Lie groups for polyhedral Finsler metrics.
result Finitely generated subgroups of semisimple Lie groups can have finite-sided Dirichlet domains under certain conditions.
We study isometric maps between Teichmüller spaces and bounded symmetric domains in their intrinsic Kobayashi metric. From a complex analytic perspective, these two important classes of geometric spaces have several features in common but also exhibit many differences. The focus here is on recent results proved by the …
We give a geometric interpretation of the maximal Satake compactification of symmetric spaces X=G/K of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable G-invariant Finsler metric on X. As an application, we establish the existence of natural bordifications, as orbifold…
Study of pseudometric properties on domains in Nagano spaces.
problem Characterize pseudometrics on domains in real-type Nagano spaces.
method Analyze Kobayashi-type pseudometrics on domains, proving properties and computing specific cases.
result The pseudometric is a genuine metric under certain conditions and has specific properties in higher rank.