Special holonomy: geometric flow and boundary value problems

我研究领域的一个基本概念是黎曼度规：它允许我们描述几何物体的曲率如何在点与点之间变化。例如，在像足球这样的圆球表面上，度规在每个点上都是相同的，而在橄榄球上，度规在接近末端的地方比在中间的地方弯曲得多。足球和橄榄球是相同的基本几何形状（即球体），具有不同的度量，因为你可以想象挤压和拉伸以将一个转换为另一个。直观地说，在某种意义上，圆度规是球体上的“最佳”度规。几何中最大的问题之一是寻找“最优”度量，这个问题已经研究了一百多年，但仍然处于现代研究的前沿。对最佳度量的追求导致了数学领域的开创性研究和主要新技术的发展。与黎曼度规相关的一个重要数据是它的完整群。最优度量的一个自然类别是那些具有所谓的特殊完整群的度量。具有特殊完整度的几何对象的两个特殊例子被称为超凯勒流形和G_2流形（其维数必须分别是4的倍数或7）。该领域的一些最大问题是找到这些度量的例子，并确定给定几何对象上确保这种度量存在的充分必要条件。拟议项目的目的是用两种完全不同的方法来阐明这些问题。为了解决G_2流形的问题，我们打算采用一种使用几何流的优雅方法。几何流技术已被用于证明几何和拓扑的著名结果，但也用于工程应用，例如，平均曲率流被用作从经验数据中去除噪声的稳健手段，例如从各种扫描仪获取大脑图像时发生的噪声。流动允许我们从一个更简单的度规开始，发展它，使它接近G_2完整度规，以类似于热量如何从热源消散的方式。一般方程非常复杂，所以我们考虑更简单的情况，即七维物体具有对称性。项目的另一半，对于hyperkaehler和G_2流形，是考虑边值问题。这样的边值问题出现在几何和分析中，但也自然地出现在物理应用中，如工程中的弯曲梁建模和生物学中分子和细胞之间的相互作用。这些问题通常比所谓的初值问题更具挑战性，比如几何流，所以我们再次简化问题，现在通过考虑已知解的扰动。通过这种方式，我们的目标是确定哪些边界度量的变形可以扩展到定义特殊的完整度量，从而有希望找到这样的度量的新例子。

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.