Harmonic almost complex structures on specific Lie groups and solvmanifolds identified.
problem Characterizing harmonic almost complex structures on almost abelian Lie groups and solvmanifolds.
method Adapted Gray-Hervella classification to almost abelian Lie groups, characterized harmonic structures.
result Examples of harmonic almost complex structures in different Gray-Hervella classes on compact almost abelian solvmanifolds.
Study locally conformally balanced metrics on specific Lie algebras.
problem Characterize and classify locally conformally balanced metrics on almost abelian Lie algebras.
method Characterizations and classifications based on specific properties of Lie algebras.
result Classification of six-dimensional almost abelian Lie algebras with locally conformally balanced metrics.
The paper classifies invariant structures on complex almost Abelian groups.
problem Investigating invariant geometric structures on almost Abelian Lie groups.
method Explicit formulas for Haar measures, modular function, and generator fields were derived.
result All invariant tensor fields have constant coefficients in the invariant frame.
Study on CKY forms on almost abelian Lie groups, proving parallelism and characterizing non-parallel cases.
problem Characterizing CKY forms on almost abelian Lie groups and proving parallelism.
method Analyzing almost abelian metric Lie algebras, proving parallelism for CKY forms, and classifying cases up to dimension 5.
result CKY forms are parallel on almost abelian Lie algebras, with exceptions for p=1 and p=n−1. Study of complex and Hermitian structures on specific Lie groups.
problem Classifying Lie groups with specific geometric structures.
method Analysis of left-invariant structures on almost abelian Lie groups.
result Classification of six-dimensional generalized Kähler almost abelian Lie groups.
Study pseudo-Kähler structures on almost abelian solvmanifolds.
problem Classify pseudo-Kähler structures on almost abelian Lie algebras.
method Analyzing invariant structures on solvmanifolds with specific Lie algebra properties.
result Classification of pseudo-Kähler structures on almost abelian Lie algebras.
Study balanced Hermitian structures on almost abelian Lie algebras, classifying six-dimensional cases.
problem Classify balanced Hermitian structures on almost abelian Lie algebras.
method Classify six-dimensional almost abelian Lie algebras with balanced structures, investigate flow of balanced metrics and anomaly flow.
result Prove conjecture for compact almost abelian solvmanifolds with left-invariant complex structures.
The paper constructs a new Ricci-flat metric on almost abelian Lie groups.
problem Finding Lorentzian homogeneous Ricci-flat metrics on almost abelian Lie groups.
method Constructing left-invariant metrics on almost abelian Lie groups, focusing on dimensions four or higher.
result A new Ricci-flat metric that generalizes the Petrov solution to higher dimensions.
The paper studies a flow on complex Lie groups, showing convergence to solitons.
problem The study of curvature flows on complex Lie groups.
method Positive Hermitian curvature flow on left-invariant metrics.
result The flow converges to solitons in both nilpotent and almost-abelian cases.
Unique complex structures on specific Lie algebras.
problem Existence and uniqueness of complex structures on nilpotent Lie algebras.
method Analysis of complex structures on nilpotent almost abelian Lie algebras.
result Full control over cohomology and deformations of almost abelian complex nilmanifolds.
Study of presymplectic forms on almost abelian Lie algebras, determining moduli spaces and their finiteness.
problem Determining conditions for the existence of presymplectic forms on almost abelian Lie algebras.
method Analyzing the moduli space of presymplectic forms and using matrix congruence to find canonical representatives.
result The moduli space of presymplectic forms on almost abelian Lie algebras is finite and all forms are permutations of a canonical 2-form.
Characterizes almost abelian Lie algebras with integrable complex structure
problem Classifying almost abelian Lie algebras
method Using presentations consisting of a real number, an element in a vector space, and an endomorphism
result Classifies p-Kähler, p-pluriclosed, Kähler, balanced, pluriclosed, and Gauduchon metrics Characterizes hypercomplex Lie groups and their solvmanifolds.
problem Understanding hypercomplex structures on Lie groups and their solvmanifolds.
method Characterization of almost abelian Lie groups with hypercomplex structures, analysis of Obata and Bismut connections, classification of hypercomplex Lie groups, and construction of solvmanifolds.
result Classification of hypercomplex almost abelian Lie groups in dimension 8 and properties of their solvmanifolds.
Study of abelian structures on odd-dimensional Lie algebras and their geometric properties.
problem Characterizing abelian structures on odd-dimensional Lie algebras.
method Introducing and analyzing abelian almost contact and almost 3-contact structures, and their compatibility conditions.
result Classification of 5-dimensional Sasakian Lie algebras and 7-dimensional abelian almost 3-contact Lie algebras.
This paper classifies LCSKT almost abelian Lie algebras in 6 dimensions.
problem Characterizing LCSKT structures on almost abelian Lie algebras.
method Analyzing the LCSKT condition and its compatibility with other Hermitian structures.
result Classification of LCSKT almost abelian Lie algebras in dimension 6.
The paper classifies hypercomplex Lie algebras and solvmanifolds using quaternionic Jordan form.
problem Classifying hypercomplex Lie algebras and solvmanifolds.
method Application of quaternionic Jordan form to Lie algebras and solvmanifolds.
result Infinitely many hypercomplex almost abelian solvmanifolds constructed.
In this paper we study some properties of almost abelian solvmanifolds using minimal models associated to a fibration. In particular we state a necessary and sufficient condition to formality and a method for finding symplectic strucures of this kind of solvmanifolds.
For certain manifolds, nonnegative Ricci curvature limits dimension and forces almost abelian fundamental group.
problem Bounding the dimension of manifolds with nonnegative Ricci curvature and specific fundamental group properties.
method Dimensional estimates for RCD(0,N) spaces with large Hausdorff dimension. result If dimension is less than 12, the fundamental group is almost abelian.
Characterizes almost Abelian Lie algebras with special H-structures.
problem Identifying almost Abelian Lie algebras with torsion-free H-structures. method Using linear maps and endomorphisms, characterizes the subspace of f for which the Lie algebra admits a special H-structure. result Explicitly computes the subspace of f for various linear Lie groups H. New examples show non-abelian fundamental groups for positive Ricci curvature manifolds.
problem Constructing manifolds with positive Ricci curvature and non-abelian fundamental groups.
method Constructing specific 9-dimensional manifolds with positive Ricci curvature and non-uniformly virtually abelian fundamental groups.
result Examples of manifolds with positive Ricci curvature and non-uniformly virtually abelian fundamental groups.
In this article, we determine the seven-dimensional almost Abelian Lie algebras which admit calibrated or parallel G_2-/G_2^*-structures. Along the way, we show that certain well-established curvature restrictions for calibrated and parallel G_2-structures are not valid in the G_2^* case. In more detail, we provide the…
We find explicit solutions of the Laplacian coflow of G2−structures on seven-dimensional almost-abelian Lie groups. Moreover, we construct new examples of solitons for the Laplacian coflow which are not eigenforms of the Laplacian and we exhibit a solution, which is not a soliton, having a bounded interval of existe…
We construct lattices on six dimensional not completely solvable almost abelian Lie groups, for which the Mostow condition does not hold. For the corresponding compact quotients, we compute the de Rham cohomology (which does not agree in general with the Lie algebra one) and a minimal model. We show that some of these …
Study coKähler structures on Lie algebras using Fino-Vezzoni correspondence.
problem Characterize coKähler structures on Lie algebras.
method Use Fino-Vezzoni correspondence to relate coKähler Lie algebras to Kähler Lie algebras.
result Complete the flat case for odd-dimensional Lie algebras, proving coKähler structures exist.
We study the existence of lattices in almost abelian Lie groups that admit left invariant locally conformal Kähler or locally conformal symplectic structures in order to obtain compact solvmanifolds equipped with these geometric structures. In the former case, we show that such lattices exist only in dimension 4, whi…
The note confirms a conjecture for specific Lie groups.
problem The conjecture about constant holomorphic sectional curvature in non-Kähler geometry.
method Compact quotients of Lie groups with specific properties.
result The conjecture is confirmed for almost abelian Lie algebras and those with certain abelian ideals.
Characterizes symmetric Killing tensors on specific Lie groups.
problem Understanding Killing tensors on specific Lie groups.
method Completely characterized left-invariant symmetric Killing tensors on almost abelian Lie groups.
result All such tensors are decomposable into polynomial expressions of Killing vector fields and metric.
Paper confirms conjecture for specific Lie algebras.
problem Fino-Vezzoni conjecture on Lie algebras with abelian ideals of codimension two.
method Analyzes unimodular Lie algebras with abelian ideals of codimension two.
result Confirms the Fino-Vezzoni conjecture for this specific class of Lie algebras.
New representation theory for surface groups to SO0(2,3).
problem Understanding representations of surface groups into special orthogonal groups.
method Non-maximal Anosov representations via Higgs bundles and non-Abelian Hodge correspondence.
result Generalization of Filip's result on weight 3 variation of Hodge structures.
Classifies complex symplectic structures on Lie algebras with large abelian ideals.
problem Classifying complex symplectic structures on Lie algebras with large abelian ideals.
method Two constructions of complex symplectic structures on Lie algebras with large abelian ideals, considering compact quotients of Lie groups.
result Complete classification of complex symplectic structures on almost abelian Lie algebras.
We prove that a fundamental group of codimension one nonnegative Ricci curvature C2-foliation of a closed Riemannian manifold is finitely generated and almost abelian, i.e. it contains abelian subgroup of finite index. In particular, we confirm the Milnor conjecture for manifolds which are leaves of codimension one non…
Locally conformal SKT structures are introduced and studied on Lie groups and their compact quotients.
problem Existence and classification of LCSKT structures on Lie groups and their quotients.
method Introducing LCSKT structures and studying their properties on Lie groups and their quotients.
result Existence of non-trivial LCSKT structures on 6-dimensional nilpotent Lie algebras and almost abelian Lie algebras.
We prove the closure for the sequential weak Lp-topology of the class of vectorfields on B3 having integer flux through almost every sphere. We show how this problem is connected to the study of the minimization problem for the Yang-Mills functional in dimension higher than critical, in the abelian case.
The paper constructs toric vector bundles using spectral networks and non-abelianization.
problem Understanding how holomorphic vector bundles arise from spectral networks and non-abelianization.
method Constructing toric vector bundles on complete toric surfaces via spectral networks and non-abelianization.
result The moduli space of rank 2 toric vector bundles over toric surfaces admits an A-type X-cluster structure. Abstract classifies Lie algebras with complex or symplectic structures.
problem Classifying Lie algebras with specific structures.
method Analyzing Jordan normal form and restrictions on matrix A. result Classification reduces to nilpotent case, with specific structure implications.
We first extend Cheeger-Colding Almost Splitting Theorem to smooth metric measure spaces. Arguments utilizing this extension of the Almost Splitting Theorem show that if a smooth metric measure space has almost nonnegative Bakry-Emery Ricci curvature and a lower bound on volume, then its fundamental group is almost abe…
Given a non compact semisimple Lie group G we describe all homogeneous spaces G/L carrying an invariant almost Kähler structure (ω,J). When L is abelian and G is of classical type, we classify all such spaces which are Chern-Einstein, i.e. which satisfy ρ=λω for some λ∈R, where ρ is the Ricci…
Prove long-time existence of pluriclosed flow on certain fibrations
problem Long-time existence of pluriclosed flow on fibrations
method General theorem on holomorphic submersions
result Long-time existence of pluriclosed flow on nilmanifolds, almost-abelian solvmanifolds, and certain complex surfaces
Study characterizes G2-structures on specific Lie groups and identifies harmonic conditions.
problem Characterizing and identifying harmonic G2-structures on almost Abelian Lie groups. method Analyzing left-invariant G2-structures, characterizing torsion forms, and using algebraic conditions. result Established algebraic conditions for harmonic G2-structures and identified admissible torsion classes. We study the spinor flow on homogeneous spin manifolds. After providing the general setup we discuss the homogeneous spinor flow in dimension 3 and on almost abelian Lie groups in detail. As a further example the flag manifold in dimension 6 is treated.
We describe an action of the concordance group of knots in the three-sphere on concordances of knots in arbitrary 3-manifolds. As an application we define the notion of almost-concordance between knots. After some basic results, we prove the existence of non-trivial almost-concordance classes in all non-abelian 3-manif…
Classifies two-step solvable Lie groups with SKT structures.
problem Classifying Lie groups with SKT structures.
method Shear construction and analysis of SKT shear data on Abelian Lie algebras.
result Large part of the classification for two-step solvable SKT algebras of dimension six.
We apply the general Ansatz in geometric flows on homogeneous spaces proposed by Jorge Lauret for the Laplacian co-flow of invariant G2-structures on a Lie group, finding an explicit soliton on a particular almost Abelian 7-manifold.
Moduli spaces of Abelian and quadratic differentials are stratified by multiplicities of zeroes; connected components of the strata correspond to ergodic components of the Teichmuller geodesic flow. It is known that the strata are not necessarily connected; the connected components were recently classified by M. Kontse…
We develop a Chern character map for twisted equivariant non-abelian cohomology.
problem Understanding non-abelian cohomology theories and their applications.
method General construction of the Chern character map for twisted equivariant non-abelian cohomology.
result Illustrated the construction by computing the equivariant Sullivan model of Cohomotopy.
The article finds non-Abelian DT-instantons on non-Kähler manifolds.
problem Finding solutions to DT-instanton equations on non-Kähler manifolds.
method Constructing examples of DT-instantons for homogeneous almost Hermitian structures on the manifold of full flags in C^3.
result Explicit classification and phenomena of reducibility and disappearance of DT-instantons.
Study second-Chern-Einstein metrics on 4D manifolds, finding Killing vector fields and examples.
problem Investigate second-Chern-Einstein metrics on 4D almost-Hermitian manifolds.
method Analyze compact and unimodular almost-abelian Lie algebras, use Killing vector fields and parallel non-zero Lee forms.
result Describe 4D compact second-Chern-Einstein locally conformally symplectic manifolds and classify unimodular almost-abelian Lie algebras with second-Chern-Einstein metrics.
We study the cohomology properties of the singular foliation $\F$ determined by an action Φ:G×M→M where the abelian Lie group G preserves a riemannian metric on the compact manifold M. More precisely, we prove that the basic intersection cohomology $\lau{\IH}{*}{\per{p}}{\mf}$ is finite dimensiona…