Extended Vaisman theorem to compact spaces with singularities.
problem Generalizing Vaisman's theorem to spaces with singularities.
method Extended Vaisman's theorem to compact complex spaces with singularities.
result Vaisman's theorem extended to compact spaces with singularities.
Classifies compact spaces by shape, finite spaces by weak homotopy.
problem Classifying compact Hausdorff spaces and finite topological spaces.
method Constructs a category that classifies spaces by shape and weak homotopy.
result Classifies compact spaces by shape, finite spaces by weak homotopy.
Compact metrics on Heisenberg manifolds have a specific condition for being relatively compact.
problem Conditions for relatively compact sets of left invariant metrics on Heisenberg manifolds.
method Necessary and sufficient condition for relatively compact sets of left invariant metrics.
result A condition for a set of left invariant metrics to be relatively compact in the moduli space.
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.
Study equigeodesics on compact homogeneous spaces using Lie algebra properties.
problem Identifying equigeodesic vectors on compact homogeneous spaces.
method Formula for equigeodesic vectors based on isotropy representation and Lie algebra structure.
result Identification of equigeodesic vectors solely through Lie algebra properties.
Proves compactness for timed-metric spaces using new distance and maps.
problem Weak convergence of space-times using timed-Hausdorff distance.
method Uses Gromov's original compactness theorem and introduces addresses.
result Establishes compactness theorem for intrinsic timed-Hausdorff convergence.
Counting geodesics on compact symmetric spaces using orbit dimensions and topological data.
problem Counting geodesics on compact symmetric spaces.
method Using orbit dimensions and topological data of the symmetric space.
result Obtained data on dimensions and connected components of focal orbits.
The paper studies spaces of non-compact real algebraic curves and their uniformisation.
problem Understanding the spaces of non-compact real algebraic curves and their uniformisation.
method Construction of spaces of non-compact real algebraic curves and description of their connected components using Fuchsian groups.
result Any connected component of the spaces of non-compact real algebraic curves is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group.
By Gromov's compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class or oriented k-dimensional Riemannian manifolds (with bound…
Paper constructs an example of a non-compact submanifold in a quaternionic Kähler symmetric space.
problem Tackles the construction of a non-compact totally complex submanifold in a quaternionic Kähler symmetric space.
method Uses an isometric action of a compact Lie group and a maximal totally geodesic sphere.
result Proves the existence of a non-compact totally complex submanifold of maximal dimension in a compact quaternionic Kähler symmetric space.
We construct a compact metric space that has any other compact metric space as a tangent, with respect to the Gromov-Hausdorff distance, at all points. Furthermore, we give examples of compact sets in the Euclidean unit cube, that have almost any other compact set of the cube as a tangent at all points or just in a den…
Computational method approximates homology groups of compact metric spaces.
problem Computing homology groups of compact metric spaces.
method Inversely sequence of finite topological spaces, inverse limit, homeomorphic copy, strong deformation retract.
result Approximates homology groups of compact metric spaces.
Study extends reflective submanifold theory to compact homogeneous spaces.
problem Characterize reflective submanifolds in compact isotropy irreducible spaces.
method Extend previous results to infinite-dimensional Hilbert spaces.
result Inverse image of reflective submanifolds is also reflective.
Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces proved.
problem Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces.
method Analyzing bounded solutions on reduced, locally irreducible compact Kähler spaces.
result Proves continuity of solutions, affirming conjectures and solving open problems.
Compact pseudo-Hermitian spaces have rigid holomorphic isometries.
problem Characterizing the rigidity of pseudo-Hermitian homogeneous spaces.
method Analysis of Tits fibration and automorphism groups of compact spaces.
result Holomorphic isometries of compact pseudo-Hermitian spaces are compact.
We prove that for a compact subgroup H of an almost connected locally compact Hausdorff group G, the following properties are mutually equivalent: (1) H is a maximal compact subgroup of G, (2) G/H is contractible, (3) G/H is homeomorphic to a Euclidean space, (4) G/H is an AE for paracompact spaces, (5) $…
Researchers classify geodesic orbit spaces for compact Lie groups of rank two.
problem Identifying geodesic orbit spaces for compact Lie groups of specific rank.
method Classification of simply connected geodesic orbit spaces where G is a compact Lie group of rank two.
result Only certain spheres and projective spaces, with metrics induced from Hopf fibrations, are geodesic orbit spaces for compact Lie groups of rank two.
Compact quotients of homogeneous spaces are studied, leading to new findings about sphere bundles.
problem Understanding compact quotients of reductive homogeneous spaces and their implications.
method Analyzing normal bundles and sphere bundles associated with these spaces, proving homotopy triviality conditions.
result Many reductive homogeneous spaces do not admit compact quotients, resolving conjectures.
We prove that an isometric action of a compact Lie group on a compact symmetric space is variationally complete if and only if it is hyperpolar.
Sharp bounds on quasimode norms on compact space forms.
problem Characterize compact manifolds using quasimode decay rates.
method Analyzes upper and lower bounds of quasimode norms on compact space forms.
result Characterizes compact manifolds of constant curvature using quasimode decay rates.
The paper studies HKKN stratifications for non-compact spaces and proves convexity properties.
problem Proving convexity properties of moment maps for non-compact subsets.
method Algebraic and analytical study of HKKN stratifications for a vector space and compact Kähler manifold, then applying to non-compact subsets.
result Convexity properties of moment maps for invariant subsets are proven.
This article discusses the existence problem of a compact quotient of a symmetric space by a properly discontinuous group with emphasis on the non-Riemannian case. Discontinuous groups are not always abundant in a homogeneous space G/H if H is non-compact. The first half of the article elucidates general machinery …
Study generalizations of chainability and compactness in metric spaces.
problem Understanding new properties of subsets in metric spaces.
method Exploring and relating new properties of chainability and compactness to existing hypertopologies.
result Relations among Hausdorff, Vietoris, and locally finite hypertopologies established.
Compact holonomy groups found in symmetric spaces.
problem Characterizing holonomy groups in symmetric spaces.
method Analyzing the holonomy group structure of locally symmetric spaces.
result Holonomy groups are compact and have finite index in the orthogonal group.
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.
On a complete, connected, locally compact, non-compact geodesic space (X,d), we assign each compact set a distance-like function. With the help of these functions, we obtain a pseudo-metric on the space of (non-empty) compact subsets of X which is less than the Hausdorff distance. The quotient metric space is close…
In math.SG/0303255, we discussed the connected components of the space of surface group representations for any compact connected semisimple Lie group and any closed compact (orientable or nonorientable) surface. In this sequel, we generalize the results in math.SG/0303255 in two directions: we consider general compact…
New compact minimal submanifolds found in Riemannian symmetric spaces.
problem Finding compact minimal submanifolds in Riemannian symmetric spaces.
method Constructing multi-dimensional families of compact minimal submanifolds via complex-valued eigenfunctions.
result New families of compact minimal submanifolds of codimension two in SU(n)/SO(n), Sp(n)/U(n), SO(2n)/U(n), and SU(2n)/Sp(n). Study shows algebraic nature of manifold submetries on compact spaces.
problem Understanding manifold submetries on compact homogeneous spaces.
method Analyzes singular Riemannian foliations and manifold submetries on compact normal homogeneous spaces.
result Establishes a one-to-one correspondence between algebras of preserved functions and manifold submetries.
The paper proves an index theorem for loop spaces of compact manifolds.
problem Defining an index theorem for loop spaces of compact manifolds.
method Formulated and proved an equivariant index theorem for non-compact manifolds with S1-actions, using a ring of formal power series. result Found an appropriate form of the index theorem for loop spaces.
Study extends Kobayashi's method to non-reductive subgroups for homogeneous spaces.
problem Existence of compact Clifford-Klein forms in homogeneous spaces.
method Extend Kobayashi's method to non-reductive subgroups and compare Cartan projections and non-compact dimensions.
result Examples of homogeneous spaces without compact Clifford-Klein forms.
New examples of isoparametric families on non-compact symmetric spaces.
problem Constructing isoparametric families with non-austere focal sets.
method New extension method from Euclidean spaces to symmetric spaces of non-compact type.
result First examples of isoparametric families on non-compact symmetric spaces.
Lifts isometries in orbit spaces for compact groups.
problem Isometries in orbit spaces of compact groups.
method Equivariant isometry of original Euclidean space.
result Simple formula for connected component of isometry group.
We study integral geometric properties of non-compact harmonic spaces.
Existence and convergence of ancient Ricci flow solutions on compact homogeneous spaces.
problem Existence and characterization of ancient solutions to the Ricci flow on compact homogeneous spaces.
method General existence theorem and Gromov-Hausdorff convergence under rescaling.
result Convergence of collapsed ancient solutions to Einstein metrics on torus fibrations.
This article studies the volume of compact quotients of reductive homogeneous spaces. Let G/H be a reductive homogeneous space and Γ a discrete subgroup of G acting properly discontinuously and cocompactly on G/H. We prove that the volume of Γ\G/H is the integral, over a certain homology class of $Γ…
The main point of this paper is that, under suitable conditions on the mean curvature and the Ricci curvature of the ambient space, we can extend Choi-Schoen's Compactness Theorem to compact embedded minimal surfaces to simple immersed compact H-surfaces in a Riemannian manifold with positive Ricci curvature (the mean …
Study shows non-compact convex hulls in certain metric spaces.
problem Compactness of convex hulls in weakly non-positive curvature spaces.
method Introduced a conical geodesic bicombing and used it to construct a counterexample.
result Existence of a metric space with a finite subset whose convex hull is not compact.
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.
We obtain a new differentiable sphere theorem for compact Lagrangian submanifolds in complex Euclidean space and complex projective space.
Proves stability of Einstein metrics on specific symmetric spaces.
problem Stability of Einstein metrics on symmetric spaces of compact type.
method Linear stability analysis of the Einstein-Hilbert action.
result Resolves stability problem for irreducible symmetric spaces of compact type.
New infinite families of flat spaces found from symmetric spaces.
problem Finding new flat homogeneous spaces.
method Starting from compact symmetric spaces, constructing infinite families of compact homogeneous spaces with invariant Bismut connections.
result Infinite families of compact homogeneous spaces with vanishing Ricci tensor.
The paper shows neural networks can approximate functions over non-compact domains with non-polynomial activation.
problem Approximating functions over non-compact domains using neural networks.
method Using single-hidden-layer feedforward neural networks with non-polynomial activation functions over non-compact subsets of Euclidean spaces.
result Neural networks can approximate functions in weighted Ck-spaces and weighted Sobolev spaces over unbounded domains. The purpose of this article is to prove existence of mass minimizing integral currents with prescribed possibly non-compact boundary in all dual Banach spaces and furthermore in certain spaces without linear structure, such as injective metric spaces and Hadamard spaces. We furthermore prove a weak∗-compactness theo…
In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which can act isometrically and locally effectively on compact Lorentzian manifolds. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special li…
Study classifies Lie group representations with non-empty boundary orbit space.
problem Classifying representations of Lie groups with non-empty boundary orbit space.
method Detailed calculations based on previous work.
result Classification of Lie group representations with non-empty boundary orbit space.
New Einstein manifolds split into symmetric and compact parts.
problem Understanding Einstein manifolds with unimodular isometry groups.
method Theory of polar actions, Lie-theoretic arguments, and maximum principles.
result Negative Einstein manifolds split into symmetric and compact parts.
A homogeneous space G/H is said to have a compact Clifford-Klein form if there exists a discrete subgroup D of G that acts properly discontinuously on G/H, such that the quotient space D\G/H is compact. When n is even, we find every closed, connected subgroup H of G = SO(2,n), such that G/H has a compact Clifford-Klein…