Publication of the week: Dr Stuart Hall

Hall, S. & T. Murphy, “Bounding Lambda_2 for some Kähler-Einstein metrics with large symmetry groups”, Annals of Global Analysis and Geometry46.2 (2014), 145-158.

The Einstein equations govern how the high-dimensional spaces used in modern theories of space-time (known as manifolds) curve.  One central problem is whether a geometric invariant called the spectrum of a manifold can characterise particular solutions of the Einstein equations (often paraphrased to, “can one hear the shape of space-time?”).

This paper continues the study of the spectrum (set of fundamental tones) of manifolds that carry solutions of the Einstein equations. In this paper the authors focus on the second eigenvalue “lambda 2” and obtain estimates for it in terms of the topological data of the underlying manifold.  They calculate this bound for a whole variety of spaces of varying dimension.  These calculations seem to suggest that the topologically simplest manifold (complex projective space) has the highest value of lambda_2. Proving whether or not this is true would be an interesting future project.

This research was supported by a Dennison research grant from The University of Buckingham.

A preprint of the paper is available at: http://arxiv.org/abs/1308.1312