The study examines asymptotic properties of G2-monopoles on nonparabolic G2-manifolds.
problem Investigating the asymptotic behavior of G2-monopoles on nonparabolic G2-manifolds.
method Proof of finite mass for monopoles with bounded curvature on nonparabolic G2-manifolds; sharp decay estimates and convergence to pseudo-Hermitian--Yang--Mills connection on asymptotically conical G2-manifolds.
result Sharp decay estimates and convergence to pseudo-Hermitian--Yang--Mills connection on asymptotically conical G2-manifolds.
This paper studies normalized Ricci flow on a nonparabolic surface, whose scalar curvature is asymptotically -1 in an integral sense. By a method initiated by R. Hamilton, the flow is shown to converge to a metric of constant scalar curvature -1. A relative estimate of Green's function is proved as a tool.
Derives formulas from Green function Hessian assumption.
problem Deriving formulas from Green function Hessian assumption.
method Assumption on Hessian of Green function leads to monotonicity formulas.
result Explicit examples of manifolds satisfying assumption.
We extend classical Euclidean stability theorems corresponding to the nonrelativistic Hamiltonians of ions with one electron to the setting of non parabolic Riemannian 3-manifolds.
We prove that polyharmonic maps of arbitrary order from complete nonparabolic Riemannian manifolds to arbitrary Riemannian manifolds must be harmonic if certain smallness and integrability conditions hold.
Let M be a G2-manifold. We consider an almost CR-structure on the sphere bundle of unit tangent vectors on M, called the CR twistor space. This CR-structure is integrable if and only if M is a holonomy G2 manifold. We interpret G2-instanton bundles as CR-holomorphic bundles on its twistor space.
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…
We study the duality between M-theory on compact holonomy G2-manifolds and the heterotic string on Calabi-Yau three-folds. The duality is studied for K3-fibered G2-manifolds, called twisted connected sums, which lend themselves to an application of fiber-wise M-theory/Heterotic Duality. For a large class of such G2-man…
In G2 manifolds, 3-dimensional associative submanifolds (instantons) play a role similar to J-holomorphic curves in symplectic geometry. In [21], instantons in G2 manifolds were constructed from regular J-holomorphic curves in coassociative submanifolds. In this exposition paper, after reviewing the background of G2 ge…
We calculate explicitly the Betti numbers of a class of barely G2 manifolds - that is, G2 manifolds that are realised as a product of a Calabi-Yau manifold and a circle, modulo an involution. The particular class which we consider are those spaces where the Calabi-Yau manifolds are complete intersections of hypersurfac…
3-dimensional Harvey Lawson submanifolds were introduced in an earlier paper by Akbulut-Salur, as examples of Lagrangian-type manifolds inside G2 manifold. In this paper, we first show that the space of deformations of a smooth, compact, orientable Harvey-Lawson submanifold HL in a G2 manifold M can be identified with …
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.
A knot space in a manifold M is a space of oriented immersions from a circle S^1 to M up to Diff(S^1). Brylinski has shown that a knot space of a Riemannian threefold is formally Kahler. We prove that a space of knots in a holonomy G2 manifold is formally Kahler.
We show that K3 surfaces with non-symplectic automorphisms of prime order can be used to construct new compact irreducible G2-manifolds. This technique was carried out in detail by Kovalev and Lee for non-symplectic involutions. We use Chen-Ruan orbifold cohomology to determine the Hodge diamonds of certain complex thr…
Non-formal G2 manifold found with holonomy.
problem Existence of non-formal G2 manifolds.
method Construction method of compact torsion-free G2 manifolds.
result Found a compact, simply connected G2 manifold that is non-formal.
In an earlier paper we showed that the space of deformations of a smooth, compact, orientable Harvey-Lawson submanifold HL in a G2 manifold M can be identified with the direct sum of the space of smooth functions and closed 2-forms on HL. In that paper, we also introduced a new class of Lagrangian-type 4-dimensional su…
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.
This article studies the geometry of moduli spaces of G2-manifolds, associative cycles, coassociative cycles and deformed Donaldson-Thomas bundles. We introduce natural symmetric cubic tensors and differential forms on these moduli spaces. They correspond to Yukawa couplings and correlation functions in M-theory. We ex…
Study on deformation theory of nearly G2 manifolds with obstructions.
problem Deformation theory of nearly G2 manifolds with obstructions.
method Study of real Killing spinors and cohomology of nearly G2 manifolds.
result Infinitesimal deformations of nearly G2 structures are obstructed in general.
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…
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…
The paper defines capacities for minimal graphs over manifolds and proves the half-space property.
problem Characterizing minimal graphs and their properties over manifolds.
method Defining capacities using relative volume, studying solutions of bounded variation, and analyzing boundary behavior.
result Proves the half-space property for M-parabolic manifolds. Study of interactions between functions on manifolds via submersions.
problem Understanding interactions between convex, subharmonic, and pluri-subharmonic functions on manifolds.
method Application of pluri-potential theory and analysis of Kähler and G2 manifolds.
result Previous results on Lagrangian fibrations can be viewed as applications of this framework.
This paper has been withdrawn by the author, due to errors in Groebner basis calculations in the cases of five and six dimensional groups.
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.
Investigates nearly Kähler and parallel G2 manifolds using Hitchin functionals.
problem Stability analysis of nearly Kähler and parallel G2 manifolds.
method Gradient flow of Hitchin functionals, spectral decomposition of Hessians, Hitchin index.
result Hitchin index provides a lower bound for the Einstein co-index.
We consider immersions of a Riemann surface into a manifold with G2-holonomy and give criteria for them to be conformal and harmonic, in terms of an associated Gauss map.
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 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.
Study on p-Green functions on specific manifolds, proving monotonicity.
problem Monotonicity of p-Green functions on certain 3D manifolds. method Sharp monotonicity formula for p-Green functions along level sets. result Established monotonicity for 1<p<3 on specific manifolds. Study deformations of G2-instantons on nearly G2 manifolds.
problem Deformations of G2-instantons on nearly G2 manifolds.
method Formulated in terms of spinors and Dirac operators, proved isomorphism of infinitesimal deformations to kernel of an elliptic operator.
result Proved abelian instantons are rigid and described the deformation space of the canonical connection on specific nearly G2 manifolds.
Study infinitesimal deformations of Killing spinors on nearly parallel G2-manifolds.
problem Infinitesimal deformations of Killing spinors on nearly parallel G2-manifolds.
method Examined using the correspondence between nearly parallel G2-structures and Killing spinors.
result Identified that the space of Rarita-Schwinger fields coincides with a subspace of the eigenspace of the Laplacian.
Method constructs rigid associative submanifolds in twisted G2-manifolds.
problem Constructing rigid associative submanifolds in twisted G2-manifolds.
method Introducing a gluing theorem for asymptotically cylindrical associative submanifolds in ACyl G2-manifolds.
result Yields many new topological types of rigid associative submanifolds.
The article develops deformation theory for ACyl associative submanifolds in ACyl G2-manifolds.
problem Deformation theory of ACyl associative submanifolds in ACyl G2-manifolds.
method Study of moduli spaces with fixed and varying asymptotic data, computing virtual dimensions.
result The moduli space of ACyl associative submanifolds embeds as a Lagrangian submanifold in the moduli space of holomorphic curves.
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…
The paper proves a Minkowski inequality on specific Riemannian manifolds.
problem Establishing a Minkowski inequality on manifolds with nonnegative Ricci curvature.
method Analyzing Riemannian manifolds with nonnegative Ricci curvature and Euclidean Volume Growth.
result Validated an optimal Minkowski inequality for certain subsets.
We give a sufficient condition for an associative submanifold in a G2-manifold to appear as the bubbling locus of a sequence of G2-instantons, related to the existence of a Fueter section of a bundle of ASD instanton moduli spaces over said submanifold.
The paper extends the Hopf differential concept to associative submanifolds in G2-manifolds.
problem Understanding the geometry of associative submanifolds in G2-manifolds.
method Analogy with CMC surfaces in R^3 and use of spinor theory.
result Every non-totally-geodesic associative 3-fold in R^7, T^7, and S^7 admits non-vanishing harmonic twisted spinors.
Proves conjecture about special Lagrangians in G2-manifolds.
problem Existence of special Lagrangians in G2-manifolds.
method Solves real Monge-Ampère equation with singular right-hand side.
result Smoothness and asymptotic properties of special Lagrangians proved.
Twistor theory applied to special holonomy manifolds.
problem Understanding special holonomy manifolds.
method Application of twistor theory to G2 and Spin(7) manifolds. result Natural Riemannian twistorial structures on G2 and Spin(7) manifolds. The paper examines properties of deformed Donaldson-Thomas connections on G2-manifolds.
problem Properties of deformed Donaldson-Thomas connections on G2-manifolds.
method Analyzes the existence, description, and flow of dDT connections.
result Observations in dDT connections mirror those in other geometric problems.
Constructs new coassociative fibrations for G2 manifolds.
problem Tackles the construction of new coassociative fibrations for G2 manifolds.
method Constructs fibrations by coassociative 4-folds, relates to hypersymplectic geometry and Donaldson's work.
result Shows natural generalizations of known coassociative fibrations.
Study constructs associative submanifolds in G2-manifolds from orbifolds.
problem Constructing associative submanifolds in G2-manifolds from orbifolds. method Using Joyce's generalised Kummer construction.
result The volume of associative submanifolds tends to zero as they approach orbifolds.
Building on ideas from [DT98; DS11; Wal17; Hay17], we outline a proposal for constructing Floer homology groups associated with a G2-manifold. These groups are generated by associative submanifolds and solutions of the ADHM Seiberg-Witten equations. The construction is motivated by the analysis of various transitions w…
A concrete model for a 7-dimensional gauge theory under special holonomy is proposed, within the paradigm outlined by Donaldson and Thomas, over the asymptotically cylindrical G2-manifolds provided by Kovalev's noncompact version of the Calabi conjecture. One obtains a solution to the G2-instanton equation from the …
Researchers solve porous medium equation on noncompact manifolds with Ricci curvature.
problem Solving porous medium equation on noncompact manifolds with nonnegative Ricci curvature.
method Constructing a space X of functions larger than L1, in which the Green function on M appears as a weight, to solve the PME.
result The porous medium equation admits a solution in the weak dual sense for certain initial data.
We study deformations of associative submanifolds Y3⊂M7 of a G2 manifold M7. We show that the deformation space can be perturbed to be smooth, and it can be made compact and zero dimensional by constraining it with an additional equation. This allows us to associate local invariants to associative subm…
Study geometry on G2-manifolds, focusing on Spin(7) and G2.
problem Classify harmonic morphisms from G2-manifolds. method Analyze geometry of G-manifolds, classify harmonic morphisms. result Classify harmonic morphisms with one-dimensional fibres from G2-manifolds.