Schedule for: 19w5145 - $G_2$ Geometry and Related Topics

The notion of a holomorphic bundle and Hermitian-Yang-Mills connections is one that proved to be very fruitful in complex geometry. There are some natural generalizations of these notions in almost complex geometry such as those of pseudo-holomorphic and pseudo-Hermitian-Yang-Mills connections. In this talk, I will focus on a system of partial differential equations, the DT-instanton equations, whose solutions give a further generalization of the notion of a Hermitian-Yang-Mills connection in the setting of real 6 dimensional almost Hermitian manifolds. (Joint work with Gavin Ball)

I shall discuss recent work with Andrew Swann that proposes a framework for studying exceptional holonomy manifolds with torus symmetry, much similar to the study of hyperkaehler manifolds with a tri-Hamiltonian action of a torus. The talk is mainly based on the preprints arXiv:1803.06646 and 1810.12962.

In this talk we explain how to describe pure types of $\mathrm{Spin}(7)$ structures in terms of spinors and focus on the construction of balanced examples. An $8$-dimensional Riemannian manifold admitting a $\mathrm{Spin}(7)$ structure determined by a $4$-form $\Omega$ is spin and the structure can also be described in terms of a spinor $\eta$. Balanced $\mathrm{Spin}(7)$ structures are a pure class and are characterized by the equation $(\ast d\Omega)\wedge \Omega=0$ or, equivalently, by the condition that $\eta$ is harmonic, that is, $D \eta=0$ where $D$ is the Dirac operator. For our purposes, the description of balanced structures in terms of spinors turns out to be much simpler. Our examples are products $(N\times T,g+g_k)$, where $(N,g)$ is a $k$-dimensional nilmanifold endowed with a left-invariant metric, $(T,g_k)$ is an $(8-k)$-dimensional flat torus, and $k=5,6$. Under these assumptions, the presence of a left-invariant balanced $\mathrm{Spin}(7)$ structure on the product is equivalent to the fact that $(N,g)$ admits a left-invariant non-zero harmonic spinor. For this reason we search left-invariant metrics on $N$ that admit left-invariant harmonic spinors. The results of our investigation are a list of $5$ and $6$-dimensional nilmanifolds that verify this condition, and the description of the set of left-invariant metrics with left-invariant harmonic spinors in the particular case $k=5$.

The most natural maps into nearly Kaehler manifolds are pseudoholomorphic curves. When taking the cone of a nearly Kaehler manifold one gets a torsion-free $G_2$ manifold, and the cone of a pseudoholomorphic curve will be an associative submanifold. Most of the work on this topic has been done for specific examples of ambient manifolds since compared to pseudoholomorphic curves in symplectic manifolds little is known about them in the general setting. After reviewing the relevant background I will show how holomorphic data can be used to construct integer invariants and examples of these curves, as done by Bryant for $S^6$ and by Xu for $\mathbb C \mathbb P^3$. I will then show how the latter construction can be combined via the Eells-Salamon correspondence with a generalisation of a formula of Friedrich to compute the Euler number of a conformal and harmonic map into $S^4$. In the end, I will present open problems I am working on such as finding new examples of pseudoholomorphic curves.

It is known that higher codimensional mean curvature flow appears naturally in the $G_2$-Laplacian flow of coassociative fibrations. With this motivation, we consider entire higher codimensional mean curvature flow in $\mathbb R^{n,m}$ of $n$-dimensional spacelike manifolds and prove a long time existence theorem starting from arbitrary spacelike initial data. We will see that the key to the proof is to demonstrate local spacelike gradient estimates, and to get around difficulties with cutoff functions in $\mathbb R^{n,m}$. This yields new long time existence results for the $G_2$-Laplacian flow.

In this talk we will discuss the deformation theory for $G_2$ instantons on $7$-dimensional manifolds with nearly parallel $G_2$-structure. We prove that the space of infinitesimal deformations of a $G_2$ instanton on such manifolds can be identified with the eigenspaces of a $1$-parameter family of Dirac operators and as a result prove that the abelian instantons are rigid, yielding a different proof of a recent result of Ball-Oliveira. This identification along with some other properties of these instantons can then be further used to gather more information about the deformation space in the non abelian case. This talk is based on a work in progress and we aim in future to determine the deformation space of some particular instantons on some known examples of nearly $G_2$ manifolds.

$G_2$-structures defined by a closed positive 3-form constitute the starting point in known and potentially effective methods to obtain holonomy $G_2$ metrics on seven-dimensional manifolds. Currently, no general results guaranteeing the existence of such structures on compact 7-manifolds are known. Moreover, the construction of new explicit examples requires substantial efforts. In the first part of this talk, I will discuss the properties of the automorphism group of a compact 7-manifold endowed with a closed $G_2$-structure, showing how they impose strong constraints on the construction of examples with high degree of symmetry (e.g. homogeneous, cohomogeneity one). Then, I will focus on the non-compact case. Here, motivated by known results on symplectic Lie groups, I will discuss some structure theorems for Lie groups admitting left invariant closed $G_2$-structures. This is based on joint works with F. Podestà (Firenze), A. Fino (Torino), and M. Fernández (Bilbao).

I will talk about a special class of closed $G_2$-structures, those satisfying the `quadratic' condition. This is a second order PDE system first written down by Bryant that can be interpreted as a condition on the Ricci curvature of the induced metric, that includes the extremally Ricci-pinched (ERP) condition as a special case. I will talk about various constructions of quadratic closed $G_2$-structures, including the first examples of ERP closed $G_2$-structures that are not locally homogeneous and the first examples of quadratic closed $G_2$-structures that are not ERP. I will discuss the relationship with the Laplace flow, and give new examples of Laplace solitons.

All known examples of instantons on (non-trivial) asymptotically conical $G_2$ manifolds asymptote to nearly K\"ahler instantons which live on the link of the cone. I will use this observation to develop a framework for studying the moduli space of such a $G_2$ instanton, showing that the expected dimension of this space is the index of a twisted Dirac operator on a weighted Sobolev space. I will focus on the example of $\mathbb R^7$ where the link is the homogeneous space $G_2/\mathrm{SU}(3)$ and show how one can use representation theoretic methods to calculate the expected dimension of the moduli space.

The Laplacian flow is an evolution equation of closed $G_2$-structures arising as the gradient flow of the so-called Hitchin volume functional. In this talk, we shall consider the flow of those $G_2$ structures admitting $S^1$ symmetry and derive explicitly the evolution equations of the $\mathrm{SU}(3)$-structure on the quotient manifold together with a connection 1-form. We describe these equations in a couple of examples and mention some partial results of ongoing work.

One of the most important ideas in the study of differential equations is the use of symmetries to cut down the number of variables. As manifolds with exceptional holonomy cannot be homogeneous the most symmetric case are group actions with cohomogeneity one, i.e. where a generic orbit has codimension one. In this case the PDE system is reduced to an ODE system. I will give an overview of recent progress in the construction of cohomogeneity one metrics with holonomy $G_2$ and Spin(7). All complete examples have a asymptotically locally conical (ALC) or asymptotically conical (AC) geometry at infinity.

We will discuss analogues of toric geometry in the $G_2$ and nearly Kaehler settings using a multisymplectic generalization of the moment map called the multi-moment map. We will then present recent work on complete toric nearly Kaehler manifolds, demonstrating some information about their global structure and giving evidence to the conjecture that the homogeneous nearly Kaehler structure on the product of two three spheres is the only example.

The first compact examples of $G_2$-manifolds were constructed by D. Joyce by desingularising $G_2$-orbifolds with singularities modeled on flat $T^3 \times (\mathbb C^2/{\pm 1})$. T. Walpuski constructed $G_2$-instantons on these compact $G_2$-manifolds. D. Joyce and S. Karigiannis generalised the original construction of $G_2$-manifolds by resolving orbifold singularities modeled on $L \times (\mathbb C^2/{\pm 1})$, where $L$ is an associative submanifold in a $G_2$-manifold. The obvious question is how the original instanton construction can be adapted to the new setting. In the talk I will explain (1) the construction by Joyce and Karigiannis of compact $G_2$-manifolds, (2) the construction by Walpuski of $G_2$-instantons on the original construction, and (3) the construction that I hope will produce $G_2$-instantons on the Joyce-Karigiannis $G_2$-manifolds. I will explain possible sources of examples using desingularisations of $G_2$-orbifolds of the form $T^3 \times (\text{K3-surface/involution})$.

In a hyper-Kaehler 4-manifold, holomorphic curves are stable minimal surfaces. One may wonder whether those are all the stable minimal surfaces. Micallef gave an affirmative answer in many cases. However, this cannot be true in general. The minimal sphere in the Atiyah--Hitchin manifold is a counter-example. In this talk, we will first recall the hyper-Kaehler geometry of Atiyah--Hitchin manifold. We will then explain that the minimal sphere is quite rigid in various senses. This is based on a joint work with Mu-Tao Wang.

We will talk about a flow of isometric $G_2$ structures. We consider the negative gradient flow of the energy functional restricted to the class of $G_2$ structures inducing a given Riemannian metric. We will discuss various an- alytic aspects of the flow including global and local derivative estimates, a compactness theorem and a local monotonicity formula for the solutions. We also study the evolution equation of the torsion and show that under a modification of the gauge and of the relevant connection, it satisfies a nice reaction-diffusion equation. After defining an entropy functional we will prove that low entropy initial data lead to solutions that exist for all time and converge smoothly to a $G_2$ structure with divergence free torsion. We will also discuss finite time singularities and show that at the singular time the flow converges to a smooth $G_2$ structure outside a closed set of finite 5- dimensional Hausdorff measure. Finally, we will prove that if the singularity is Type I then a sequence of blow-ups of a solution has a subsequence which converges to a shrinking soliton of the flow. This is a joint work with Pana- giotis Gianniotis (University of Athens) and Spiro Karigiannis (University of Waterloo).

Seven-manifolds with a $G_2$-structure possess two distinguished classes of submanifolds: associative 3-folds and coassociative 4-folds. When the $G_2$-structure is torsion-free, such submanifolds are minimal (in fact, calibrated) and exist locally. We are led to ask: For which classes of $G_2$-structures is it the case that associative 3-folds (respectively, coassociative 4-folds) are always minimal submanifolds? We will answer this by deriving a simple formula for the mean curvature, in the process uncovering new obstructions to the local existence of coassociatives. Time permitting, we will discuss the analogous results for special Lagrangian 3-folds (respectively Cayley 4-folds) in 6-manifolds with $\mathrm{SU}(3)$-structures (respectively 8-manifolds with $\mathrm{Spin}(7)$-structures). This is joint work with Gavin Ball.