We show how certain diffeomorphism-invariant functionals on differential forms in dimensions 6,7 and 8 generate in a natural way special geometrical structures in these dimensions: metrics of holonomy G2 and Spin(7), metrics with weak holonomy SU(3) and G2, and a new and unexplored example in dimension 8. The general f…
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48 results for “G2' invariant metric”
We develop a powerful new analytic method to construct complete non-compact G2-manifolds, i.e. Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction starts with a complete non-compact asymptotically conical Calabi-Yau 3-fold B and a circle bundle M over B satisfying…
The article classifies G2-structures with conformally flat metrics.
problem Identifying G2-structures with specific geometric properties.
method Classifying closed G2-structures with conformally flat metrics.
result Any closed G2-structure with conformally flat metric is locally equivalent to one of three explicit examples.
Bryant and Salamon gave a construction of metrics of G2 holonomy on the total space of the bundle of anti-self-dual (ASD) 2-forms over a 4-dimensional self-dual Einstein manifold. We generalise it by considering the total space of an SO(3) bundle (with fibers R^3) over a 4-dimensional base, with a connection on this bu…
The paper constructs a family of SKT metrics on the exceptional Lie group G2.
problem Constructing SKT metrics on the exceptional Lie group G2.
method Left-invariant integrable almost complex structure and construction of 7-parameter family of metrics.
result A 3-parameter family of left-invariant SKT metrics on G2.
The Bergman metric on symmetrized bidisc has negative curvature properties.
problem Holomorphic curvature properties of the Bergman metric on symmetrized bidisc.
method Analysis of holomorphic sectional and bisectional curvatures.
result Negative pinched holomorphic sectional curvature and non-positive holomorphic bisectional curvature.
Found a new compact G2-structure on a 7-manifold.
problem Identifying compact G2-structures on manifolds.
method Lattice in solvable Lie group, Laplacian coflow analysis.
result Obtained a 4-parameter subfamily of expanding solitons.
We overview the properties of non-infinitesimal deformations of G2-structures on seven-manifolds, and in particular, focus on deformations that lie in the seven-dimensional representation of G2 and are thus defined by a vector. We then consider deformations from G2-structures with the torsion class having one-dimension…
We find G2-manifolds with specific asymptotic properties.
problem Existence and structure of G2-manifolds with ALC asymptotics.
method Robust Fredholm theory for ALC spaces, proving existence and rigidity results.
result Existence of a G2-analogue of the Atiyah-Hitchin metric and good moduli theory for ALC G2-holonomy metrics.
We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) or G2 invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For virtual knots and graphs these invariants provide new graphical information that allo…
We give an answer to a question posed recently by R.Bryant, namely we show that a compact 7-dimensional manifold equipped with a G2-structure with closed fundamental form is Einstein if and only if the Riemannian holonomy of the induced metric is contained in G2. This could be considered to be a G2 analogue of the Gold…
Study G2-instantons on specific Lie groups, finding conditions and structures.
problem Characterize G2-instantons on 2-step nilpotent Lie groups.
method Analyze connections arising from characteristic connections, use Lie group structure and torsion.
result Establish necessary and sufficient conditions for G2-instantons, define naturally reductive structures.
Among closed G2-structures there are two very distinguished classes: Laplacian solitons and Extremally Ricci-pinched G2-structures. We study the existence problem and explore possible interplays between these concepts in the context of left-invariant G2-structures on solvable Lie groups. Also, some Ricci pinching prope…
The study shows that certain nearly G2 and nearly Kähler conifolds cannot be resolved by gluing asymptotically conical G2 and Calabi-Yau manifolds.
problem The inability to resolve nearly G2 and nearly Kähler conifolds by gluing asymptotically conical G2 and Calabi-Yau manifolds.
method Topological analysis of asymptotically conical G2 and Calabi-Yau manifolds to show conditions under which resolutions are impossible.
result For certain rates of the metric, the G2 4-form and Kähler form cannot be simultaneously exact, leading to non-existence of resolutions.
In this paper most of the classes of G2-structures with Einstein induced metric of negative, null or positive scalar curvature are realized. This is carried out by means of warped G2-structures with fiber an Einstein SU(3) manifold. The torsion forms of any warped G2-structure are explicitly described in terms of the t…
Study on -type flag manifolds, focusing on invariant metrics and Ricci flow.
problem Characterizing and analyzing metrics on -type flag manifolds.
method Investigation of invariant metrics, analysis of g.o. metrics, and Ricci flow techniques.
result Characterization of metrics invariant under maximal compact subgroups.
Researchers compute the ν-invariant for specific G2-structures on nilmanifolds.
problem Detecting connected components of G2-structure moduli spaces.
method Defined and computed the ν-invariant using Mathai-Quillen currents, harmonic spinors, and η-invariants.
result Determined the parity of harmonic spinor dimensions and deduced ν vanishing on invariant spinors.
A complete classification of left-invariant closed G2-structures on Lie groups which are extremally Ricci pinched, up to equivalence and scaling, is obtained. There are five of them, they are defined on five different completely solvable Lie groups and the G2-structure is exact in all cases except one, given by the onl…
Study nearly parallel G2-structures with torus symmetry using multi-moment maps.
problem Characterize and construct nearly parallel G2-structures with torus symmetry.
method Use multi-moment map techniques and analyze the geometry of the base spaces.
result Locally, the construction may produce examples with four-torus symmetry.
We use a G2-structure on a 7-dimensional Riemannian manifold with a fixed metric to define an octonion bundle with a fiberwise non-associative product. We then define a metric-compatible octonion covariant derivative on this bundle that is compatible with the octonion product. The torsion of the G2-structure is then sh…
Let M be a 7-manifold with a G2-structure defined by φ\inΩ^{3}_{+}(M). We prove that φ is conformal-Killing with respect to the associated metric g(φ) if and only if the G2-structure is nearly parallel. Let M be an 8-manifold with a Spin7-structure defined by ψ\inΩ^{4}_{+}(M). We prove that ψ is conformal-Killing with …
We construct propose an anzatz for Spin(7) metrics as an R-bundle over closed G2 structures. These G2 structures are R3 bundles over 4-dimensional compact quaternion Kahler spaces. The inspiration for the anzatz metric comes from the Bryant-Salamon construction of G2 holonomy metrics and from the fact that the twistor …
Article explores G2-structures with quadratic conditions, finding new ERP and complete solitons.
problem Investigating closed G2-structures satisfying quadratic conditions.
method Analyzing a second-order PDE system involving parameter λ, producing new examples of ERP and complete solitons.
result First examples of ERP G2-structures, including complete inhomogeneous ERP G2-structure.
We give the first examples of closed Laplacian solitons which are shrinking, and in particular produce closed Laplacian flow solutions with a finite-time singularity. Extremally Ricci pinched G2-structures (introduced by Bryant) which are steady Laplacian solitons have also been found. All the examples are left-invaria…
Researchers compute the ν-invariant for specific G2-structures on Aloff-Wallach spaces.
problem Computing the ν-invariant for G2-structures on Aloff-Wallach spaces.
method Using Goette's formulas for η-invariants of homogeneous spaces, derived an explicit expression for ν.
result For the nearly-parallel G2-structures φ± on Nk,l, the ν-invariant is ν(φ±) = ±41.
Researchers found G2-structures with zero torsion on specific Lie groups.
problem Finding G2-structures with divergence-free torsion.
method Isometric flow to evolve G2-structures with divergence-free torsion as critical points.
result Three families of G2-structures on solvable Lie groups have zero torsion.
Geodesic concavity and hypersymplectic structures in -structures space.
problem Analyzing the geodesic concavity and hypersymplectic structures in the space of closed -structures.
method Utilising the geodesic constructed in the previous article, we show geodesic concavity and decrease in length of Laplacian flow.
result Hitchin's volume functional is geodesically concave and the Laplacian flow decreases the length.
Derives an integral formula for G2-structures.
problem Calculating properties of G2-structures.
method Applies an integral formula for G-structures to G2.
result Derives an integral formula relating curvatures and quadratic invariants.
The problem of recovering the asymptotics of a short range perturbation of the Euclidean metric on R^n from fixed energy scattering data is studied. It is shown that if two such metrics, g1, g2, have scattering data at some fixed energy which are equal up to smoothing, then there exists a diffeomorphism ψ`fixing infini…
Promotes Poisson deformations to hyperkähler structures.
problem Deforming hyperkähler cone metrics.
method Twistor methods for universal Poisson deformations.
result Produces incomplete hyperkähler metrics with applications.
Study 3-Sasakian and G2 structures on manifolds.
problem Investigate algebra of forms on 3-Sasakian manifolds.
method Examine cohomogeneity-one actions on spheres and Berger's space.
result Identify nearly-parallel G2 forms.
New isometric solutions found for heterotic G2-system.
problem Solving the heterotic G2-system with non-abelian gauge groups.
method An ansatz to vary G2-structure and gauge data while keeping metric fixed.
result Distinct isometric solutions on the same manifold with different gauge groups.
We study isometric Lie group actions on the compact exceptional groups E6, E7, E8, F4 and G2 endowed with a biinvariant metric. We classify polar actions on these groups. We determine all isometric actions of cohomogeneity less than three on E6, E7, F4 and all isometric actions of cohomogeneity less than 20 on E8. More…
G2-manifolds with a cohomogeneity-one action of a compact Lie group G are studied. For G simple, all solutions with holonomy G2 and weak holonomy G2 are classified. The holonomy G2 solutions are necessarily Ricci-flat and there is a one-parameter family with SU(3)-symmetry. The weak holonomy G2 solutions are Einstein o…
We present an analytic construction of complete non-compact 8-dimensional Ricci-flat manifolds with holonomy Spin(7). The construction relies on the study of the adiabatic limit of metrics with holonomy Spin(7) on principal Seifert circle bundles over asymptotically conical G2 orbifolds. The metrics we produce have an …
Study heterotic G2-system on 2-step nilmanifolds with torus bundles.
problem Investigate G2-structures on 7D nilmanifolds with torus bundles.
method Prove coclosure of G2-structures, discuss existence of solutions for all isomorphism classes.
result Existence of solutions for various 7D nilpotent Lie algebras with constant dilaton.
The heterotic G2 system is solved on Calabi-Yau 3-orbifolds.
problem Solving the heterotic G2 system on Calabi-Yau 3-orbifolds.
method Adjusting circle bundles and powers of α' to obtain cocalibrated G2 structures.
result Non-trivial solutions to the heterotic G2 system are found.
The study examines the stability of Einstein metrics on Sasaki Einstein and nearly parallel G2 manifolds.
problem Linear instability of Einstein metrics on Sasaki Einstein and nearly parallel G2 manifolds.
method Analysis of the second and third Betti numbers for Sasaki Einstein and nearly parallel G2 manifolds.
result Positive second and third Betti numbers lead to linear instability for the respective manifolds.
We study the natural structure on the moduli space of deformations of compact coassociative submanifolds. We show that a G2-manifold with a T^4-action of isomorphisms such that the orbits are coassociative tori is locally equivalent to a minimal 3-manifold in R^{3,3} = H^2(T^4,R) with positive induced metric. By studyi…
Study on dimensions of G2-structures on nilmanifolds, proving non-abelian automorphism groups.
problem Dimensions and automorphisms of G2-structures on nilmanifolds.
method Computational analysis of moduli spaces of G2-structures on 7-dimensional nilmanifolds.
result Coclosed G2-structures have non-abelian automorphism groups, unlike closed ones.
Study coclosed G2-structures on SU(2)²-invariant manifolds.
problem Existence and classification of coclosed G2-structures on specific manifolds.
method Analysis of half-flat SU(3)-structures and boundary conditions.
result Existence of coclosed G2-structures on R⁴ × S³, no such structures on S⁴ × S³.
Classifies 7D manifolds with specific G2-structures and automorphisms.
problem Classifying 7D manifolds with closed G2-structures and transitive automorphisms.
method Complete classification through detailed analysis of manifold properties and group actions.
result The center of the automorphism group is one-dimensional and the manifold is a product of a flat factor and a homogeneous six-manifold with a specific SU(3)-structure.
Completes classification of G2-structures on specific nilpotent Lie groups.
problem Classifying seven-dimensional nilpotent Lie groups with purely coclosed G2-structures.
method Analyzing nilpotent Lie groups of various steps and dimensions.
result Classification of indecomposable 5- and 6-step nilpotent Lie groups.
We introduce the notion of a subregular subalgebra, which we believe is useful for classification of subalgebras of Lie algebras. We use it to construct a non-regular invariant generalized complex structure on a Lie group. As an illustration of the study of invariant generalized complex structures, we compute them all …
This paper is a review of current developments in the study of moduli spaces of G2 manifolds. G2 manifolds are 7-dimensional manifolds with the exceptional holonomy group G2. Although they are odd-dimensional, in many ways they can be considered as an analogue of Calabi-Yau manifolds in 7 dimensions. They play an impor…
Geometric flows study nearly parallel G2-structures on 3-Sasakian 7-manifolds.
problem Analyzing geometric flows of G2-structures on 3-Sasakian manifolds.
method Study of Laplacian flow and Laplacian coflow of G2-structures on 3-Sasakian manifolds.
result Distinct behavior of flows, notably regarding stability of nearly parallel G2-structures.
New collapsing mechanism for G2-manifolds discovered.
problem Understanding collapsing behavior of G2-manifolds.
method Adiabatic description involving weighted maximal submanifold equation; formal power series solutions.
result Existence of formal power series solutions and heuristic discussion of compactification.
Study on Hermitian curvature flow on special linear groups, disproving a conjecture and finding non-algebraic solitons.
problem Disproving a conjecture about the stability of canonical metrics on special linear groups.
method Investigation of invariant solutions to the Positive Hermitian Curvature Flow on complex Lie groups.
result Discovered non-algebraic solitons on special linear groups, contradicting Ustinovskiy's conjecture.