Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

We provide a significant extension of the twisted connected sum construction of G_2-manifolds, i.e. Riemannian 7-manifolds with holonomy group G_2, first developed by Kovalev; along the way we address some foundational questions at the heart of the twisted connected sum construction. Some of the main contributions of the paper are: 
(i) We correct, clarify and extend several aspects of the K3 "matching problem" that occurs as a key step in the twisted connected sum construction. 
(ii) We show that the large class of asymptotically cylindrical Calabi-Yau 3-folds built from semi-Fano 3-folds (a subclass of weak Fano 3-folds) can be used as components in the twisted connected sum construction. 
(iii) We construct many new topological types of compact G_2-manifolds by applying the twisted connected sum to asymptotically Calabi-Yau 3-folds of semi-Fano type studied in arXiv:1206.2277. 
(iv) We obtain much more precise topological information about twisted connected sum G_2-manifolds; one application is the determination for the first time of the diffeomorphism type of many compact G_2-manifolds. 
(v) We describe "geometric transitions" between G_2-metrics on different 7-manifolds mimicking "flopping" behaviour among semi-Fano 3-folds and "conifold transitions" between Fano and semi-Fano 3-folds. 
(vi) We construct many G_2-manifolds that contain rigid compact associative 3-folds. 
(vii) We prove that many smooth 2-connected 7-manifolds can be realised as twisted connected sums in numerous ways; by varying the building blocks matched we can vary the number of rigid associative 3-folds constructed therein. This leads to speculation that the moduli space of G_2-metrics on a given 7-manifold may consist of many different connected components.

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G_2-manifolds and associative submanifolds via semi-Fano 3-folds

We construct many new topological types of compact G 2 -manifolds, that is, Riemannian 7 -manifolds with holonomy group G 2 . To achieve this we extend the twisted connected sum construction first developed by Kovalev and apply it to the large class of asymptotically cylindrical Calabi–Yau 3 -folds built from semi-Fano 3 -folds constructed previously by the authors. In many cases we determine the diffeomorphism type of the underlying smooth 7 -manifolds completely; we find that many 2 -connected 7 -manifolds can be realized as twisted connected sums in a variety of ways, raising questions about the global structure of the moduli space of G 2 -metrics. Many of the G 2 -manifolds we construct contain compact rigid associative 3 -folds, which play an important role in the higher-dimensional enumerative geometry (gauge theory/calibrated submanifolds) approach to defining deformation invariants of G 2 -metrics. By varying the semi-Fanos used to build different G 2 -metrics on the same 7 -manifold we can change the number of rigid associative 3 -folds we produce.

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G2-manifolds and associative submanifolds via semi-Fano 3-folds

We conjecture the existence of four independent gradings in colored HOMFLYPT homology, and make qualitative predictions of various interesting structures and symmetries in the colored homology of arbitrary knots. We propose an explicit conjectural description for the rectangular colored homology of torus knots, and identify the new gradings in this context. While some of these structures have a natural interpretation in the physical realization of knot homologies based on counting supersymmetric configurations (BPS states, instantons, and vortices), others are completely new. They suggest new geometric and physical realizations of colored HOMFLYPT homology as the Hochschild homology of the category of branes in a Landau–Ginzburg B-model or, equivalently, in the mirror A-model. Supergroups and supermanifolds are surprisingly ubiquitous in all aspects of this work.

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Quadruply-graded colored homology of knots

We develop foundational theory for the Laplacian flow for closed G2 structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor Rm and torsion tensor T along the flow, i.e. that a bound on

 $$\Lambda(x,t)=\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}$$
 will imply bounds on all covariant derivatives of Rm and T. (2). We show that $${\Lambda(x,t)}$$
 will blow up at a finite-time singularity, so the flow will exist as long as $${\Lambda(x,t)}$$
 remains bounded. (3). We give a new proof of forward uniqueness and prove backward uniqueness of the flow, and give some applications. (4). We prove a compactness theorem for the flow and use it to strengthen our long time existence result from (2) to show that the flow will exist as long as the velocity of the flow remains bounded. (5). Finally, we study soliton solutions of the Laplacian flow.

/pdf/laplacian-flow-for-closed-g2-structures-shi-type-estimates-163bue10z8.pdf

Laplacian flow for closed G2 structures: Shi-type estimates, uniqueness and compactness

We consider some infinitesmal and global deformations of G_2 structures on 7-manifolds. We discover a canonical way to deform a G_2 structure by a vector field in which the associated metric gets "twisted" in some way by the vector cross product. We present a system of partial differential equations for an unknown vector field whose solution would yield a manifold with holonomy G_2. Similarly we consider analogous constructions for Spin(7) structures on 8-manifolds. Some of the results carry over directly, while others do not because of the increased non-linearity of the Spin(7) case.

/pdf/deformations-of-g-2-and-spin-7-structures-on-manifolds-36aopmkl0m.pdf

Deformations of G_2 and Spin(7) Structures on Manifolds

This is a foundational paper on flows of G_2 Structures. We use local coordinates to describe the four torsion forms of a G_2 Structure and derive the evolution equations for a general flow of a G_2 Structure on a 7-manifold. Specifically, we compute the evolution of the metric, the dual 4-form, and the four independent torsion forms. In the process we obtain a simple new proof of a theorem of Fernandez-Gray. 
As an application of our evolution equations, we derive an analogue of the second Bianchi identity in G_2-geometry which appears to be new, at least in this form. We use this result to derive explicit formulas for the Ricci tensor and part of the Riemann curvature tensor in terms of the torsion. These in turn lead to new proofs of several known results in G_2 geometry.

Flows of G_2-structures, I

We consider the Laplacian "co-flow" of $G_2$-structures: $\frac{d}{dt} \psi = - \Delta_d \psi$ where $\psi$ is the dual 4-form of a $G_2$-structure $\phi$ and $\Delta_d$ is the Hodge Laplacian on forms. This flow preserves the condition of the $G_2$-structure being coclosed ($d\psi =0$). We study this flow for two explicit examples of coclosed $G_2$-structures with symmetry. These are given by warped products of an interval or a circle with a compact 6-manifold $N$ which is taken to be either a nearly Kahler manifold or a Calabi-Yau manifold. In both cases, we derive the flow equations and also the equations for soliton solutions. In the Calabi-Yau case, we find all the soliton solutions explicitly. In the nearly Kahler case, we find several special soliton solutions, and reduce the general problem to a single \emph{third order} highly nonlinear ordinary differential equation.

Soliton solutions for the Laplacian coflow of some $G_2$-structures with symmetry

We consider the Laplacian “co-flow” of G 2 -structures: ∂ ∂ t ψ = − Δ d ψ where ψ is the dual 4-form of a G 2 -structure φ and Δ d is the Hodge Laplacian on forms. Assuming short-time existence and uniqueness, this flow preserves the condition of the G 2 -structure being coclosed ( d ψ = 0 ). We study this flow for two explicit examples of coclosed G 2 -structures with symmetry. These are given by warped products of an interval or a circle with a compact 6-manifold N which is taken to be either a nearly Kahler manifold or a Calabi–Yau manifold. In both cases, we derive the flow equations and also the equations for soliton solutions. In the Calabi–Yau case, we find all the soliton solutions explicitly. In the nearly Kahler case, we find several special soliton solutions, and reduce the general problem to a single third order highly nonlinear ordinary differential equation.

Soliton solutions for the Laplacian co-flow of some G2-structures with symmetry

We collect together various facts about G_2 and Spin(7) geometry which are likely well known but which do not seem to have appeared explicitly in the literature before. These notes should be useful to graduate students and new researchers in G_2 and Spin(7) geometry.

Spiro Karigiannis

Papers

Deformations of G_2 and Spin(7) Structures on Manifolds

Flows of G_2-structures, I

Soliton solutions for the Laplacian coflow of some $G_2$-structures with symmetry

Soliton solutions for the Laplacian co-flow of some G2-structures with symmetry

Some Notes on G_2 and Spin(7) Geometry