Published on March 12th, 2014 | by Emily Corbett0
Einstein metrics on domains in a solid torus
By Benjamin Babao, University of Queensland
This student took part in the 2012/13 AMSI Vacation Research Scholarship program. For more information on this years program please click here
Problems in geometry have always been of extreme interest to mathematicians. Thousands of years ago, Pythagoras discovered a special relationship between the lengths of the sides of a Right angled triangle. Now, Pythagoras’ theorem is one of the first things taught in high school mathematics classes around the world.
More recently, in 2002-2003, Russian mathematician Gregori Perelman solved the Poincare Conjecture, one of the most difficult problems in the history of modern mathematics. The problem was of such complexity, that it was granted the title of a Millennium Prize problem, one of only seven that carry a $1 million reward for the finding of a correct solution. The deep theory developed to solve the problem has had resounding applications in mathematics, and its solution is critical in the understanding of our geometrical world.
In essence, Perelman was able to show that ‘n-dimensional surfaces’ (manifolds) with a specific set of properties admitted ‘nice’ geometries. In fact, some argue that the fundamental task in differential geometry is to find manifolds with these ‘nice’ geometries, called Einstein metrics.
Central to Perelman’s proof was the concept of Ricci Curvature. As its name suggests, Ricci curvature is a property of a manifold that measures how curved the manifold is. The reason it is important is because one needs to understand Ricci curvature in order to understand the Einstein Equation – the equation that must be solved in order to find these magical Einstein metrics which gives manifolds their nice geometries.
In my project over summer, I looked at the problem of solving the Einstein equation on an altered solid torus, or more informally, a doughnut with a wormhole through the middle. It was my task to see if we could actually find these Einstein metrics on this specific manifold, or if there were any conditions that needed to be satisfied in order to guarantee a solution to the Einstein equation. This was an interesting problem as the ‘worm-eaten doughnut’ is an example of a manifold with boundary, and little is yet known about solutions to the Einstein equation on these objects.
After a lot of hard work, and learning lots of new material, I was actually able to prove a result about the existence of Einstein metrics on an altered torus. Essentially what my result says, is that the boundaries of the worm-eaten torus must behave in a sufficiently ‘nice’ way in order to guarantee the existence of these Einstein metrics.
Throughout my honours year in 2013, I look to extend the work I have done over summer to a more general class of manifolds.
I would like to thank my supervisor, Artem Pulemotov, for his patience and help. Thanks must also go to AMSI and CSIRO for putting on the Big Day In in Sydney. It was a great couple of days!