Constant curvature – the special metrics of Kähler manifolds

Kähler metrics are geometrical tools that are key to many areas of Mathematics. Dr Tamás Darvas and his team at the University of Maryland use Kähler metrics to tackle one of the most pressing questions in differential geometry – how to impose constant scalar curvature on an existing geometrical object. This work has applications across a wide range of mathematical theory.
From early childhood, we are unwittingly introduced to the theoretical topic of ‘geometry’. We play with simple shapes like circles and squares, in order to slowly digest much deeper concepts that, without thinking, we will use in our later life: ‘geometry’ makes us aware of the world around us and even an everyday object such as a football can shine a light on complex mathematical problems.
Imagine a soccer ball (or British football) and an American football ball. The former is ‘evenly curved’ – no matter what point of the ball you look at, the curvature is the same – but the latter shows some differences in its curvature. For example, if you compared the shape near the two pointed ends with the flatter middle sections, the curvature would be very different. At the same time, the American football ball could be moulded into a soccer ball by pushing it at the more pointed ends.

Objects like a soccer ball, that have a constant curvature (the same at every point), are considered ideal shapes in geometry

Both the soccer ball and the American football ball are geometric shapes that are examples of Kähler manifolds. This simple image of one geometric object (the football ball) being moulded into another (the symmetrical shape of a soccer ball) gives us a glimpse into one of the most important problems in differential geometry. Put simply, a manifold is a set of points that, locally, seems to form a flat space where the distance between two points can be computed or understood as usual (termed a Euclidean space). These manifolds can have different shapes and, most importantly, they can be curved like the balls described above.
Kähler manifolds
Kähler manifolds are a special class of geometric objects that find application in a wide spectrum of topics, spanning from pure Mathematics to Theoretical Physics. They are manifolds on which the distance between any two points can be measured using a Kähler metric, a specialised mathematical gadget that naturally appears in applications. The wide relevance of these geometric objects is one of the reasons mathematicians are motivated to look for formal descriptions of their structure. Notably, to this end, it is fundamental to find an opportune Kähler metric, with special curvature properties.
Using the Kähler metric we can compute the curvature at every point on the manifold. According to Einstein’s Theory of General Relativity, we live in a hyperbolic universe, i.e., a manifold with a negative curvature. To really understand the meaning of the term curvature, picture a spherical object, like a simple ball, or like our planet. Spherical objects are not flat, as they have positive curvature. However, when we walk between two locations, for example, from our home to a restaurant, we are not able to perceive the curvature of the Earth – to us, it appears that the spherical object is in fact flat. This occurs precisely because locally, on a very small scale, manifolds appear flat.
Objects like the soccer ball, that have a constant curvature (the same at every point), are considered ideal shapes. Partly, they are more aesthetically pleasing, being symmetrical, but there is a deeper reason for this: geometric objects with constant curvature are also useful. For example, the Einstein universe, emerging from his ground-breaking model of gravity (General Relativity), can be described by a 4-dimensional manifold, with an evenly distributed curvature (named Ricci curvature).

As we said before, we could change the shape of the American football ball into that of a soccer ball to give it constant curvature. This transformation can be easily visualised by thinking of what we do every time we want to relax sitting in a bean-bag: we modify its shape into a more comfortable position. We shape a bean-bag for the same reason mathematicians transform geometrical objects to provide them with constant curvature: in both cases to achieve more comfort or ease of use. Abstractly, the American football ball and the soccer ball represent the same Kähler manifold with two different Kähler metrics. We can mould one into the other, because we can ‘deform’ the Kähler metric of the American football ball into the Kähler metric of the soccer ball.
Very few Kähler manifolds admit Kähler metrics with constant curvature. On the other hand, a very large class seems to admit metrics of constant scalar curvature, which is a weaker condition that suffices for many natural applications.

Prof Darvas and his team aim to find Kähler manifolds that can be transformed to have constant scalar curvature

And this is where the cutting-edge work of Prof Darvas and his research group comes in. Notably, the results of their investigations are making steps towards identifying special Kähler metrics for the Kähler manifolds we have just discussed. In other words, they aim to find Kähler manifolds that can be transformed to have constant scalar curvature – the football balls that are able to be moulded into soccer balls.
Evening out curvature
The team’s strategy is based on a ‘variational’ method, where a quantity defined as ‘K-energy’ plays a key role. The latter was introduced about 30 years ago by the mathematician T. Mabuchi, and allows us to make evaluations on the metrics that can be adopted in a Kähler manifold. In particular, if a metric minimises the K-energy, then it must have constant scalar curvature, fulfilling the aim of mathematicians to create more comfortable manifolds. The idea of minimising a quantity defined, in general, as energy is quite common in other domains. For instance, in Statistical Physics, the area of Theoretical Physics adopts the theory of probability for describing physical systems like gases. Here, minimising a quantity called Free energy leads a system towards equilibrium, where thermodynamic variables become constant (over time intervals), like our curvature in a Kähler manifold.

In their work, Prof Darvas and his colleagues conjectured that it is possible to find a minimum of the K-energy “if and only if the K-energy is ‘proper’”: this refers to the point when its global behaviour shows a certain rate of growth. Remarkably, Prof Darvas and his team have already taken the first steps in proving this theory, and so further exciting results are expected in the near future.

What, do you envision, is the most important application, of your research?
With the results we have already obtained and also with the ones that we will hopefully obtain in the future, mathematicians and physicists will gain a better understanding in the basic properties of manifolds that have special curvature properties.
Do you think that some applications might also involve other domains, like information theory or computer science, where geometry has a relevant role?
The most promising applications are in theoretical physics where manifolds with special curvature play an important role.

Kähler geometry lies at the intersection of algebraic geometry, differential geometry and complex analysis

The minimisation of the K-energy is reminiscent of a kind of free energy in statistical physics. Do you see any particular connection between the two quantities?
This is a very intriguing parallel. Though it is unlikely that a direct connection exists, we have learned a great deal from the methods applied in statistical physics in our variational study of the K-energy.
Considering the other branches of mathematics, in which of them are your results expected to provide a relevant contribution in the short-term period?
Kähler geometry lies at the intersection of algebraic geometry, differential geometry and complex analysis. Thus, any advance made in our project will automatically push the boundaries of our understanding in all of these subjects. To my mind, this is one of the most attractive features of my research.
Since the K-energy might constitute a quantity to measure ‘how constant’ the scalar curvature is, and this can be related to some forms of art, do you see some connection between this measure and the world of human perception?
Shapes with constant curvature do play a prominent role in the arts. The most striking example that comes to mind is M.C. Esher’s work that heavily used hyperbolic geometry. Given that most of our manifolds that we now study are at least four dimensional, very few tools are available to visualise them, hence I see little chance for them being used in visual arts. However, our work may help physicists with their perception of the universe.

Research Objectives
Dr Darvas focuses his research on complex differential geometry. He is particularly interested in special metrics and complex Monge-Ampère equations arising in Kähler geometry, and more recently in the special Lagrangians of Calabi-Yau manifolds.
Funding
National Science Foundation (NSF)
Bio
Tamás Darvas graduated with an MSc in Mathematics from Babeş-Bolyai University and completed his PhD in Mathematics at Purdue University in 2014. He is currently a Postdoctoral Research Associate at University of Maryland.
Contact
Dr Tamás Darvas
Department of Mathematics
William E Kirwan Hall
4176 Campus Drive
University of Maryland
MD 20742 – 4015
USA
E: [email protected]
T: +1 765 464 9843
W: www2.math.umd.edu/~tdarvas/