Study compact symplectic solvmanifolds' hard Lefschetz property.
problem Compact symplectic solvmanifolds' hard Lefschetz property.
method Analysis of compact symplectic solvmanifolds as quotients of solvable Lie groups by lattices.
result Characterization of conditions for the hard Lefschetz property.
Study compares and unifies finiteness properties of locally compact groups.
problem Understanding finiteness properties of locally compact groups.
method Comparing and unifying three families of finiteness properties: type Cn, coarse (n−1)-connectedness, and type Fn. result All three families lead to the same notion for locally compact groups.
We study compact Riemannian manifolds for which the light between any pair of points is blocked by finitely many point shades. Compact flat Riemannian manifolds are known to have this finite blocking property. We conjecture that amongst compact Riemannian manifolds this finite blocking property characterizes the flat m…
We study integral geometric properties of non-compact harmonic spaces.
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.
Thirty years after the birth of foliations in the 1950's, André Haefliger has introduced a special property satisfied by holonomy pseudogroups of foliations on compact manifolds, called compact generation. Up to now, this is the only general property known about holonomy on compact manifolds. In this article, we give a…
In this paper we establish a strong multiplicity one type property for the length-holonomy spectrum for the three dimensional compact hyperbolic spaces. We use the analytic properties of Selberg-Gangolli-Wakayama zeta functions associated to compact hyperbolic spaces.
We consider (compact or noncompact) Lorentzian manifolds whose holonomy group has compact closure. Among other results, we obtain that this property is equivalent to admitting a parallel timelike vector field. We also derive some properties of the space of all such metrics on a given manifold.
Proves regularity of extremal function on compact Kähler manifolds.
problem Regularity of extremal function on compact Kähler manifolds.
method Local property analysis and equivalence of continuity and Hölder continuity.
result Equivalence of classical notions of local L-regularity and locally Hölder continuous property. Paper proves naturally reductive property is inaudible for certain manifolds.
problem Proving naturally reductive property is inaudible for specific manifolds.
method Using a pair of non-compact 11-dimensional generalized Heisenberg groups, the paper proves the naturally reductive property is inaudible.
result The naturally reductive property is inaudible for certain manifolds.
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.
Study confirms conjecture on Hermitian manifolds with bounded mass.
problem Morse-type integrals in nef (1,1) classes on compact Hermitian manifolds with bounded mass.
method Analyzes conjecture using bounded mass property on compact Hermitian manifolds.
result Confirms Demailly-Păun and Tosatti-Weinkove's conjectures.
Study of mixed equation combining gauge theory and symplectic geometry.
problem Regularity and compactness of solutions to the mixed equation.
method Combining Uhlenbeck and Gormov compactness theorems.
result Moduli spaces of solutions to the mixed equation satisfy compactness properties.
Paper shows unique decomposition of 3-manifolds and multiplicative property of Reidemeister torsion.
problem Unique decomposition of compact 3-manifolds.
method Study of adjoint Reidemeister torsion on disk sum decompositions.
result Adjoint Reidemeister torsion has a multiplicative property on unique decompositions.
Study on compact strong HKT manifolds and their properties.
problem Characterizing the structure of compact strong HKT manifolds.
method Geometric analysis, rigidity theorems, classification, and properties of Ricci foliations.
result Compact strong HKT manifolds are Hopf fibrations over compact 4-dimensional orbifolds.
Paper proves non-compact inaudibility of symmetry and commutativity.
problem Proving inaudibility of symmetry and commutativity in non-compact settings.
method Using isospectral pairs of generalized Heisenberg groups.
result Proved inaudibility of weak symmetry and commutativity.
Geodesic completeness and flow properties of compact Brinkmann spacetimes proven.
problem Geodesic completeness and flow properties of compact Brinkmann spacetimes.
method Proof of geodesic completeness and flow properties of isotropic parallel vector fields in compact Brinkmann spaces.
result Geodesic completeness and flow properties of compact Brinkmann spacetimes proven.
In \cite{CM5}, Colding and Minicozzi describe a type of compactness property possessed by sequences of embedded minimal surfaces in $\Real^3$ with finite genus and with boundaries going to ∞. They show that any such sequence either contains a sub-sequence with uniformly bounded curvature or the sub-sequence has …
Study on compact biconservative hypersurfaces in de Sitter space.
problem Understanding compact biconservative hypersurfaces in de Sitter space.
method Investigation of hypersurfaces with constant scalar curvature under geometric constraints.
result Extension of rigidity properties of biconservative hypersurfaces.
Study decay and compact support of solutions to certain nonlinear PDEs.
problem Decay and compact support properties of positive solutions to Δpu≥Λ(u) on manifolds. method Nonlinear PDE analysis, Feller property, integral Ricci curvature conditions.
result Characterization of stochastic completeness for the p-Laplacian. The paper examines conditions for a vector field to be Killing in almost Yamabe solitons.
problem Conditions for a vector field to be Killing in almost Yamabe solitons.
method Investigation of almost Yamabe solitons on compact and non-compact manifolds.
result Sufficient conditions for the defining conformal vector field to be Killing.
Universal spaces for finite topological spaces simplify shape descriptions.
problem Describing shape properties of compact metric spaces.
method Inverse limits of finite spaces and Alexandroff extensions.
result Universal spaces simplify shape descriptions of compact metric spaces.
This article explores some properties of universal covers of compact Kahler manifolds, under the assumption of Caratheodory measure hyperbolicity. In particular, by comparing invariant volume forms, an inequality is established between the volume of canonical bundle of a compact Kahler manifolds and the Caratheodory me…
Strong formal properties for toric and homogeneous Kähler manifolds.
problem Understanding formal properties of Kähler manifolds.
method Analyzing rationally and strongly formal properties of toric and homogeneous Kähler manifolds.
result Toric and homogeneous Kähler manifolds are both rationally and strongly formal.
Study functionals on almost complex structures for Yau's Challenge.
problem Yau's Challenge on compact C-manifolds. method Variational properties of functionals on almost complex structures.
result Variational properties could be used to tackle Yau's Challenge.
Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete, conformally compact metrics. Related to this issue, an isometry extension property is proved: continuous groups of isometries at confor…
The paper studies dynamical properties in semigroups modulo ideals.
problem Analyzing shadowing, expansivity, and stability in semigroups with ideals.
method Investigates shadowing, expansivity, and stability properties in uniform transformation semigroups modulo an ideal.
result Establishes that if a semigroup exhibits shadowing and expansivity modulo an ideal, it is also topologically stable modulo that ideal.
Study presentations of groups that can be generalised over continuous open group monomorphisms.
problem Investigate presentations of groups that can be generalised over continuous open group monomorphisms.
method Systematic study of presentations with generalisation properties, focusing on right-angled Artin groups (RAAGs).
result Establish high connectivity properties for universal Salvetti-type complexes and novel examples of LC groups with prescribed compactness properties.
Harmonic functions on compact symmetric spaces exhibit strong convexity properties.
problem Understanding the convexity of harmonic functions on compact symmetric spaces.
method Analyzing the nonnegativity of the Laplacian powers of harmonic functions.
result Harmonic functions on compact symmetric spaces have nonnegative Laplacian powers, demonstrating strong convexity.
The paper classifies compact hyperbolic Coxeter polytopes and improves upper bounds.
problem Classifying compact hyperbolic Coxeter polytopes and understanding their combinatorial properties.
method Study of imes0-products of Lannér diagrams, proving superhyperbolic properties, and analyzing Lannér subdiagrams. result Improved upper bounds on the dimension of compact hyperbolic Coxeter polytopes.
Proves properties of Morse vector fields on compact manifolds.
problem Properties of gradient vector fields of Morse functions.
method Analyzes connectedness of critical points and shrinkage of flow.
result Shows connectedness of critical points through orbits and exponential shrinkage.
In the first part of this series, we defined an equivariant index without assuming the group acting or the orbit space of the action to be compact. This allowed us to generalise an index of deformed Dirac operators, defined for compact groups by Braverman. In this paper, we investigate properties and applications of th…
We prove a regularity result for the Monge--Ampère equations on compact Kaehler manifolds with degenerate rhs member.
The paper characterizes and studies compact subsets of complex projective space with specific line intersection properties.
problem Characterizing compact subsets of complex projective space with specific line intersection properties.
method Characterization and study of compact subsets of complex projective space with line intersection properties.
result Characterization of quadratic R-algebraic subsets of complex projective space.
Here we explore a variety of properties of intrinsic flat convergence. We introduce the sliced filling volume and interval sliced filling volume and explore the relationship between these notions, the tetrahedral property and the disappearance of points under intrinsic flat convergence. We prove two new Gromov-Hausdorf…
Study G-H limits of surfaces with boundary, focusing on same Euler characteristic.
problem Investigate Gromov-Hausdorff limits of compact surfaces with boundary.
method Focus on surfaces with same Euler characteristic, build on previous work on closed surfaces.
result Complete description and topological properties of limit spaces.
In this work we study the geodesic structure of the space Σ(X) of compact balls of a complete and locally compact metric length space endowed with the Hausdorff distance dH. In particular, we focus on a geometric condition (referred to as the shooting property) that enables us to give an explicit isometry between …
Study spectral and index properties of Hodge-Dirac operator on compact manifolds.
problem Investigate spectral and index-theoretic properties of Hodge-Dirac operator on compact Riemannian manifolds.
method Establish bisectoriality and H∞ functional calculus without curvature assumptions. result Prove compact Banach spectral triple and recover classical topological invariants as Lp-indices. In this paper we study numerical properties of quotients of holomorphic log-tensors.
Proves 3D Poincaré duality groups without property (T)
problem Residually finite 3D Poincaré duality groups and property (T)
method Using coboundary expansion and recent results on 3-manifold groups
result 3D Poincaré duality groups without property (T)
Study properties of balanced hyperbolic compact complex manifolds.
problem Understanding cohomology and harmonic spaces of balanced hyperbolic manifolds.
method Proved vanishing theorems and Hard Lefschetz-type theorems for balanced hyperbolic compact complex manifolds.
result Non-existence of certain L1 currents on the universal covering space of a balanced hyperbolic manifold. We are interested in global properties of systems of left-invariant differential operators on compact Lie groups: regularity properties, properties on the closedness of the range and finite dimensionality of their cohomology spaces, when acting on various function spaces e.g. smooth, analytic and Gevrey. Extending the …
We prove that for a compact subgroup H of a locally compact Hausdorff group G, the following properties are mutually equivalent: (1) G/H is a manifold, (2) G/H is finite-dimensional and locally connected, (3) G/H is locally contractible, (4) G/H is an ANE for paracompact spaces, (5) G/H is a metrizable $G…
We develop the basic topological properties of compact polygons, i.e. of compact topological Tits buildings of rank two. It is proved that the Coxeter diagram of such a building is always crystallographic, that is, compact connected n-gons exist only for n=3,4,6. We classify compact polygons which admit a transitive gr…
3-manifold groups have a property that allows them to act on quasi-trees.
problem Proving property (QT) for 3-manifold groups.
method Analyzing the geometry of 3-manifolds and using quasi-trees.
result Compact, orientable 3-manifold groups have property (QT) if they don't support Sol or Nil geometry.
This is an exposition of some recent results on ECS manifolds, by which we mean pseudo-Riemannian manifolds of dimensions greater than 3 that are neither conformally flat nor locally symmetric, and have parallel Weyl tensor. All ECS metrics are indefinite. We state two classification theorems, describing the local stru…
The paper explores low-dimensional solenoidal manifolds and their properties.
problem Characterizing and understanding solenoidal manifolds of dimensions 1, 2, and 3.
method Survey and new results about solenoidal manifolds, using theorems of A. Clark and S. Hurder.
result Topologically homogeneous, compact solenoidal manifolds are McCord solenoids and behave like laminated versions of compact manifolds.
We give an equivalent definition of compact locally conformally hyperkähler manifolds in terms of the existence of a nondegenerate complex two-form with natural properties. This is a conformal analogue of Beauville's theorem stating that a compact Kähler manifold admitting a holomorphic symplectic form is hyperkähler.