Special holonomy: geometric flow and boundary value problems

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A fundamental notion in my research area is a Riemannian metric: this allows us to describe how the curvature of a geometric object varies from point to point. For example, on the surface of a round sphere like a football the metric is the same at every point, whereas on a rugby ball the metric is much more curved near the ends than the middle. The football and the rugby ball are the same basic geometric shape (i.e. a sphere) with different metrics, because you can imagine squashing and stretching to transform one to the other. Intuitively the round metric is, in some sense, the "best" metric on the sphere. One of the biggest problems in geometry is to find "optimal" metrics and has been studied for more than a hundred years, yet continues to be at the forefront of modern research. The quest for optimal metrics has led to pioneering research in mathematics and to the development of major new techniques.An important piece of data associated with a Riemannian metric is its holonomy group. A natural class of optimal metrics are those with so-called special holonomy groups. Two particular examples of geometric objects with metrics with special holonomy are called hyperkaehler and G_2 manifolds (whose dimension has to be a multiple of four or be seven, respectively). Some of the greatest problems in the field are to find examples of these metrics and to determine necessary and sufficient conditions on a given geometric object which ensure the existence of such a metric. The aim of the proposed project is to shed light on these problems using two completely different approaches.To tackle the problem for G_2 manifolds we intend to follow an elegant approach using a geometric flow. Geometric flow techniques have been employed to prove celebrated results in geometry and topology but are also used in engineering applications, for example Mean Curvature Flow is used as a robust means to remove noise from empirical data such as occurs when obtaining brain images from various scanners. The flow allows us to start with a simpler metric and evolve it so that it approaches the G_2 holonomy metric, in a similar way to how heat dissipates from a heat source. The general equation is very complicated, so we consider the simpler situation where the seven-dimensional objects have symmetries.The other half of the project, for hyperkaehler and G_2 manifolds, is to consider the boundary value problem. Such boundary value problems arise throughout geometry and analysis, but also occur naturally in physical applications such as modelling bending beams in engineering and interactions between molecules and cells in biology. These problems are typically substantially more challenging than so-called initial value problems, like geometric flows, so we again simplify the problem, now by considering perturbations of a known solution. In this way we aim to identify which deformations of the boundary metric can be extended to define special holonomy metrics and thus hopefully find new examples of such metrics.