Publication of the week: Dr Stuart Hall
9 June 2014
Hall, S. & T. Murphy, “On the spectrum of the Page and the Chen-LeBrun-Weber metrics”, Annals of Global Analysis and Geometry 46.1, 87-101 (2014). DOI: 10.1007/s10455-014-9412-6
The spectrum of an operator on a space tells one how functions on the space break up into fundamental pieces. This is completely analogous to decomposing a note as a series of fundamental tones (often called harmonics). As the geometry of a shape changes, the fundamental tones also change (think of twanging an elastic band stretched between your fingers).
This paper is about the fundamental tones associated to two 4-dimensional shapes (both having the geometry of a solution of Einstein’s equations). The first is called the Page metric and was discovered by physicist Don Page in 1978, the second is called the Chen-LeBrun-Weber metric and was discovered by mathematicians Xiuxiong Chen, Claude LeBrun and Brian Weber in 2007.
The main thrust of the article is to estimate the first fundamental tone which contains a lot of information about the space. For example, it can determine whether the geometry is a stable solution of Einstein’s equations. As the equations describing the metrics are extremely complicated, the authors have to translate the integrals that appear in such estimates into ones that can be easily computed. This is done by a technique called integration-by-parts, a staple of any A-level student’s toolbox!
The idea for this article occured to the authors whilst they were working together in Buckingham. The visit of Dr Thomas Murphy was funded by a Dennison research grant.
A preprint of the paper is available at: http://arxiv.org/abs/1206.5489
Stuart Hall is a research lecturer in mathematics based in the Department for Applied Computing at Buckingham.