The shape of spaces with curvature bounds - a research case study by Daniele Semola

Curvature is a way of measuring the distortion of a space from being flat, and it is ubiquitous in Science. Ricci Curvature, in particular, appears in Einstein’s equations of General Relativity. It controls the heat diffusion in general ambient spaces and it plays a fundamental role in Hamilton-Perelman’s solution of the Poincaré conjecture and of Thurston’s geometrisation conjecture.

The modern notion finds its roots in the works of Gauss and Riemann in the mid-nineteenth century, where “space” meant “smooth differentiable manifold”, locally modelled by the Euclidean geometry through atlases and charts. However, smoothness is an unsatisfactory condition from various points of view: it is unstable, therefore it can be lost under limits, and it is not a priori available in several important situations. 

These observations motivate the interest in synthetic notions of curvature with no regularity assumptions.

The analogy with convexity in real analysis can be useful in understanding the meaning of the word “synthetic” in this field. Real functions with two derivatives are convex if their second derivative is non negative. However, convexity does not require smoothness at all if formulated with the usual interpolation inequality f(tx+(1-t)y)≤(1-t)f(x)+tf(y). The second point of view is synthetic, in opposition to the first one, which is analytic.

As for convexity, the condition “space with non negative Ricci curvature” seems to require two derivatives (of the metric tensor) to be formulated. This regularity assumption would make it unsuitable for the analysis of discontinuous matter distributions in General Relativity or of the heat diffusions in spaces with edges and corners, for instance.

However, there are at least two “synthetic” alternative perspectives on this condition, that do not require such regularity.

The first one was already implicit in the works of Hawking and Penrose about General Relativity in the 1960s: a space has non negative Ricci curvature if whenever we look at a light source from a very far position and we try to determine its volume, we overestimate it.

The second one is more recent, based on Fluid Mechanics and due to McCann and Otto-Villani, at the end of the 1990s. Imagine that a perfect gas moves along the least action path between two prescribed distributions. If the space has non negative Ricci curvature, the entropy of the gas at any intermediate time lies above the line joining the entropies of the gas at the initial and final times. In none of these points of view is regularity of the ambient required and this makes them suitable for extension to more general settings.

After the introduction of synthetic notions of spaces with Ricci Curvature bounds it is natural to try to understand their shape. Do they look like smooth manifolds or are singular pathological behaviours admitted?

In a joint work with Elia Bruè (Institue for Advanced Study, Princeton) three years ago we proved that spaces with synthetic lower Ricci curvature bounds have at least a well defined dimension that cannot jump between different regions. The proof employs the fluid mechanics perspective: a jump of dimension is shown to force a perfect gas to have a behaviour incompatible with the lower Ricci curvature bound.

Indeed the proof of this regularity property has opened up a series of recent developments and questions. Is there a notion of boundary for these spaces, as happens for classical manifolds? And, since we know that they have a regular and a singular part, how can we control their geometry near to regular points and how degenerate can they be near to singular ones? Answering these questions is likely to impact on our future understanding of the shape of spaces with Ricci Curvature bounds.

Daniele Semola is a Postdoctoral Research Associate in Oxford Mathematics.