Circle and Torus Actions in Exceptional Holonomy

The work in this thesis is an investigation of the geometric structures arising on S 1 and T 2 quotients of manifolds endowed with G2 and Spin(7)-structures. This was motivated by the work of Apostolov and Salamon who studied the circle reduction of G2 manifolds and showed that imposing that the quotient is Kähler leads to a rich geometry. We shall consider the following problems: 1. The S 1 quotient of Spin(7)-structures 2. The Kähler reduction of Spin(7) manifolds with T 2 actions 3. The S 1 -invariant G2 Laplacian flow 4. The SU(2) 2 ×U(1)-invariant G2 Laplacian flow on S 3 ×R 4 Our key results include expressions relating the intrinsic torsion of S 1 -invariant Spin(7)-structures to that of the quotient G2-structures, a new expression for the Ricci curvature of Spin(7)-structures only in terms of the intrinsic torsion, infinitely many new examples of (incomplete) Spin(7) metrics arising as T 2 bundles over Kähler manifolds with trivial canonical bundle, the first example of an inhomogeneous shrinking gradient G2 Laplacian soliton and a local classification of closed SU(2) 2 ×U(1)-invariant G2-structures on S 3 ×R 4

Explicit abelian instantons on S1S^1-invariant K\"ahler Einstein 66-manifolds

We consider a dimensional reduction of the (deformed) Hermitian Yang-Mills condition on $S^1$-invariant K\"ahler Einstein $6$-manifolds. This allows us to reformulate the (deformed) Hermitian Yang-Mills equations in terms of data on the quotient K\"ahler $4$-manifold. In particular, we apply this construction to the canonical bundle of $\mathbb{C}\mathbb{P}^2$ endowed with the Calabi ansatz metric to find explicit abelian $SU(3)$ instantons and we show that these are determined by the spectrum of $\mathbb{C}\mathbb{P}^2$. We also find $1$-parameter families of explicit deformed Hermitian Yang-Mills connections. As a by-product of our investigation we find a coordinate expression for its holomorphic volume form which leads us to construct a special Lagrangian foliation of $\mathcal{O}_{\mathbb{C}\mathbb{P}^2}(-3)$.Comment: v2: 31 pages, added results (the content of section 8.

S1S^1 invariant Laplacian flow

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.Non UBCUnreviewedAuthor affiliation: University College LondonGraduat