Integral volume vanishes for manifolds with circle foliations.
problem Integral foliated simplicial volume calculation.
method Regular foliation by circles analysis.
result Integral foliated simplicial volume vanishes.
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48 results for “vanishing volume”
Affine manifolds with specific properties have zero simplicial volume.
problem Understanding the simplicial volume of affine manifolds.
method Analyzing the holonomy map and using cohomological criteria.
result Affine manifolds with injective holonomy containing a pure translation have vanishing simplicial volume.
Study counts hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants up to volume .
problem Counting hyperbolic 4-manifolds with specific topological properties.
method Used volume bounds and commensurability to estimate the number of such manifolds.
result The number of hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants up to volume is asymptotically bounded by .
Estimates open sets for fibrations, leading to volume vanishing results.
problem Estimating open sets for fibrations.
method Straightforward estimate for open sets with fundamental group constraints.
result Vanishing results for simplicial volume and minimal volume entropy for certain mapping tori.
Integral foliated simplicial volume vanishes for manifolds with -action.
problem Integral foliated simplicial volume of manifolds with -action.
method Geometric construction of Yano's proof for ordinary simplicial volume combined with parametrised uniform boundary condition for .
result Integral foliated simplicial volume of aspherical manifolds with -action vanishes.
Minimal volume entropy vanishes for mapping tori over 3-manifolds.
problem Volume entropy of mapping tori over 3-manifolds.
method A variation of amenable category and minimal volume entropy of a homology class.
result Minimal volume entropy vanishes.
Study on Berwald-Weyl curvature with projective invariance and vanishing results.
problem Characterizing Berwald-Weyl curvature for spray/Finsler metrics.
method Analyzing expressions and proving vanishing conditions for curvature.
result Berwald-Weyl curvature vanishes for certain spray/Finsler metrics.
Negative curvature manifolds have vanishing bounded volume class if and only if Cheeger constant is positive.
problem Negative curvature manifolds and their volume classes.
method Integration of volume forms and isoperimetric constants.
result Vanishing of bounded volume class implies positivity of Cheeger constant and vice versa.
Graph manifolds' simplicial volume approximated by covering volumes.
problem Proving simplicial volume of graph manifolds is zero.
method Approximating simplicial volume of graph manifolds by integral simplicial volumes of their finite coverings.
result Uniform proof of rank, Betti number, and torsion homology gradients for graph manifolds.
Integral filling volume, complexity and integral simplicial volume of 3-dimensional mapping torimath.GT
Integral filling volume of mapping tori grows sublinearly with complexity.
problem Characterizing mapping classes with vanishing integral filling volume.
method Analyzing Dehn twists and mapping tori, using simplicial volume and complexity.
result Integral simplicial volume of mapping tori grows sublinearly with respect to the monodromy power.
Minimal volume entropy vanishes or is positive under certain fiber growth conditions.
problem Conditions for minimal volume entropy to be zero or positive.
method Analyzes topological conditions related to fiber growth of maps.
result Examples of finite simplicial complexes with zero simplicial volume and large minimal volume entropy.
The simplicial volume of non-R^3 contractible 3-manifolds is infinite.
problem Characterizing contractible 3-manifolds based on their simplicial volume.
method Analyzing the simplicial volume of contractible 3-manifolds and open 3-manifolds.
result The Euclidean space is the unique contractible 3-manifold with vanishing minimal volume.
Simplicial volume vanishes for 4-manifolds with open book decompositions.
problem Characterizing open book decompositions in dimension 4.
method Adapted proof by Bucher and Neofytidis.
result Simplicial volume vanishes for 4-manifolds with open book decompositions.
We study the flux homomorphism for closed forms of arbitrary degree, with special emphasis on volume forms and on symplectic forms. The volume flux group is an invariant of the underlying manifold, whose non-vanishing implies that the manifold resembles one with a circle action with homologically essential orbits.
Volume-preserving neural networks prevent gradient issues.
problem Vanishing and exploding gradients in deep neural networks.
method A new neural network architecture with volume-preserving sublayers.
result Volume-preserving neural networks maintain gradient stability.
We show that surface bundles over surfaces with base and fiber of genus at least 2 have non-vanishing simplicial volume.
Extended characterization of RAAGs with zero minimal volume entropy.
problem Characterizing RAAGs with vanishing minimal volume entropy.
method Extended characterization from geometric dimension 2 to higher dimensions.
result Extended characterization of RAAGs with zero minimal volume entropy.
New length functions on mapping class groups linked to simplicial volumes of mapping tori.
problem Understanding the relationship between mapping class groups and simplicial volumes of mapping tori.
method Introducing filling volumes as length functions and proving their properties.
result Real filling volumes equal the simplicial volume of mapping tori, while integral filling volumes are not smaller than the stable integral simplicial volume.
Non-vanishing steady Euler flows and Beltrami fields found in high dimensions.
problem Existence of non-vanishing steady Euler flows and Beltrami fields in high dimensions.
method Using open books, proved existence of non-vanishing steady solutions to the Euler equations for vector fields in odd dimensions.
result Existence of non-vanishing steady Euler flows and Beltrami fields in high dimensions.
We prove that mapping tori of 3-manifolds have zero simplicial volume.
problem The simplicial volume of mapping tori of 3-manifolds.
method Analysis of self-homeomorphisms of reducible 3-manifolds and introduction of a new technique for computing simplicial volume.
result We prove that any mapping torus of a closed 3-manifold has zero simplicial volume.
We study the bounded fundamental class in the top dimensional bounded cohomology of negatively curved manifolds with infinite volume. We prove that the bounded fundamental class of vanishes if is geometrically finite. Furthermore, when is a -rank one locally symmetric space, we show that the bou…
We provide a proof for an inequality between volume and L2-Betti numbers of aspherical manifolds for which Gromov outlined a strategy based on general ideas of Connes. The implementation of that strategy involves measured equivalence relations, Gaboriau's theory of L2-Betti numbers of R-simplicial complexes, and other …
New results on relative simplicial volume using bounded acyclicity.
problem Understanding relative simplicial volume in bounded cohomology.
method Equivariant nerve pairs, relative classifying spaces, and small relative amenable category.
result Vanishing results for -Betti numbers and mapping degrees.
The study examines conditions for minimal volume entropy of simplicial complexes.
problem Conditions for minimal volume entropy of simplicial complexes.
method Topological conditions and growth of fundamental groups.
result Examples of simplicial complexes with zero simplicial volume and large minimal volume entropy.
Study shows Bergman kernels match averages on quotient spaces, proving non-vanishing of Poincaré series.
problem Proving non-vanishing of Poincaré series on finite-volume quotients of Hermitian symmetric spaces.
method Using Bergman kernels and averaging over discrete groups, proving non-vanishing of Poincaré series.
result Large class of relative Poincaré series does not vanish on general locally symmetric spaces of finite volume.
We consider the volumes of classical supermanifolds such as the supersphere, complex projective superspace, and Stiefel and Grassmann supermanifolds, with respect to the natural metrics or symplectic structures. We show that the formulas for the volumes, upon certain universal normalization, can be obtained by an analy…
The paper proves volume stability for hyperbolic manifolds and applies it to general relativity.
problem Volume stability of hyperbolic manifolds and its implications in general relativity.
method Sharp volume-stability theorem for closed hyperbolic three-manifolds, tensorial \(C^0\)-convergence.
result Near-equality in the sharp hyperbolic volume bound forces tensorial \(C^0\)-convergence to the hyperbolic metric.
Gromov's theory of multicomplexes with applications to bounded cohomology and simplicial volumemath.GT
Gromov's theory of multicomplexes aids in bounded cohomology and simplicial volume studies.
problem Homotopy invariants of manifolds and their properties.
method Construction and study of multicomplexes, homotopy theory, completeness condition.
result Complete proofs of Gromov's Mapping and Vanishing Theorems.
Study shows complete affine manifolds have zero simplicial volume.
problem Estimating the amenable category of affine manifolds.
method Construction of manifolds with infinite amenable normal subgroups.
result Zero simplicial volume for all such manifolds.
Study on simplicial volume and Euler characteristic of aspherical manifolds.
problem Whether vanishing simplicial volume implies vanishing Euler characteristic.
method Various strategies for both affirmative and negative answers, context with other problems, and comparative analysis of additivity properties.
result Found counterexamples among aspherical spaces that are homology equivalent to manifolds but not manifolds themselves.
Degree theorems and Lipschitz simplicial volume for non-positively curved manifolds of finite volumemath.GT
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…
Study on restricted volumes on Kähler manifolds, proving conjecture under certain conditions.
problem Understanding restricted volumes on Kähler manifolds.
method Analyzing numerical restricted volumes of (1,1) classes, proving conjecture under specific conditions.
result Irreducible components of the non-Kahler locus have vanishing numerical restricted volume when the class has a Zariski decomposition.
We obtain sharp volume bound for a conic 2-sphere in terms of its Gaussian curvature bound. We also give the geometric models realizing the extremal volume. In particular, when the curvature is bounded in absolute value by , we compute the minimal volume of a conic sphere in the sense of Gromov. In order to apply th…
Study shows Betti numbers of curves and orbifolds relate to volume and genus.
problem Understanding Betti numbers of Shimura curves and 3-orbifolds.
method Analyzes asymptotic behavior of Betti numbers in relation to volume and genus.
result Gauss-Bonnet equality for Shimura curves and vanishing Betti numbers for 3-orbifolds.
The study shows simplicial volume finiteness for certain manifolds with amenable fundamental groups.
problem Determining the simplicial volume of manifolds with specific properties.
method Analyzing the fundamental group and using amenability properties.
result Simplicial volume is finite for certain manifolds with amenable fundamental groups.
We observe inequalities involving the Herzlich volume of a 4-dimensional asymptotically complex hyperbolic Einstein manifold and its Euler characteristic provided the metrics is either Kaehler or selfdual. In the selfdual case we have to assume furthermore that the Kronheimer-Mrowka invariant is non vanishing.
In this short note, exploits of constructions of -structures coupled with technology developed by Cheeger-Gromov and Paternain-Petean are seen to yield a procedure to compute minimal entropy, minimal volume, Yamabe invariant and to study collapsing with bounded sectional curvature on inequivalent smooth st…
We survey the definitions and some important properties of several asymptotic invariants of smooth manifolds, and discuss some open questions related to them. We prove that the (non-)vanishing of the minimal volume is a differentiable property, which is not invariant under homeomorphisms. We also formulate an obstructi…
Paper proves Gromov's conjecture on manifolds with certain group properties.
problem Gromov's conjecture on positive scalar curvature and simplicial volume.
method Proves conjecture under a fundamental group decay property.
result Proves Gromov's conjecture for manifolds with a weakened rapid decay property.
Morse-Bott Volume Formsmath.DG
Two Morse-Bott volume forms are diffeomorphic if their cohomology classes are equal.
problem Establishing equivalence of Morse-Bott volume forms.
method Adapting Moser's trick to Morse-Bott volume forms.
result Two Morse-Bott volume forms with the same zero set are diffeomorphic if and only if they have equal total volumes.
Study simplicial volume via foliated simplices and duality.
problem Calculate simplicial volume using foliated simplices and duality.
method Defined real singular foliated homology, constructed foliated fundamental class, and established isometric isomorphism with measurable bounded cohomology.
result Norm of foliated fundamental class equals simplicial volume of .
In this note we prove that the volume of a causal diamond associated with an inertial observer in asymptotically de Sitter 4-dimensional space-time is monotonically increasing function of cosmological time. The asymptotic value of the volume is that of in maximally symmetric de Sitter space-time. The monotonic property…
Critical metrics of volume functional on compact 4-manifolds with boundary are rigid.
problem Finding critical metrics of volume functional on compact 4-manifolds with boundary.
method Proved using critical point theory and integral curvature estimates.
result Critical metrics are isometric to geodesic balls in , or .
For a germ of a smooth map f and a subgroup G_V of any of the Mather groups G for which the source or target diffeomorphisms preserve some given volume form V in the source or in the target we study the G_V-moduli space of f that parameterizes the G_V-orbits inside the G-orbit of f. We find, for example, that this modu…
Study shows vanishing distance in fluid dynamics equations.
problem Understanding the geometric origins of fluid dynamics equations.
method Analyzing geodesic distances on diffeomorphism and symplectomorphism groups.
result Modified Constantin-Lax-Majda and surface quasi-geostrophic equations arise from metrics with vanishing geodesic distance.
On Higher Graph Manifoldsmath.GT
In this short note we introduce higher graph manifolds and use a version of the barycenter technique to characterize when they undergo volume collapse. In the case when the pure pieces are hyperbolic, we compute the exact value of the minimal volume. We verify the coarse Baum--Connes conjecture for these manifolds and …
Defines and classifies Thurston geometries and connects simplicial volume to Kodaira dimension.
problem Classifying Thurston geometries and understanding their properties.
method Introduces an axiomatic definition for the Kodaira dimension and studies its compatibility with traditional notions.
result Establishes a connection between the simplicial volume and the holomorphic Kodaira dimension, showing implications for smooth Kähler 3-folds.
Introduces trace operator for quasi-plurisubharmonic functions on Kähler manifolds.
problem Analyzing singularities of quasi-plurisubharmonic functions.
method Introduces trace operator and uses it to study singularities.
result Obtains novel extension theorems and applications to restricted volumes.