Mathematicians at USTC have resolved a long‑standing problem in complex geometry, offering new insights into the fabric of higher‑dimensional space and its role in fundamental physics.

What is the underlying structure of higher dimensional space, and how does it behave at distances approaching infinity? In tackling this question, Yu Li and Bing Wang of the University of Science and Technology of China (USTC) have paved the way for advances in string theory and other areas of theoretical physics.

Li and Wang’s field of complex differential geometry involves the study of manifolds. They are mathematical spaces of multiple dimensions that are curved, even though they appear flat locally — somewhat like the curvature of the Earth.

The term ‘complex’ refers to the use of imaginary numbers, which are multiples of the square root of -1. Though these numbers defy everyday intuition, they unlock powerful tools for solving equations that have no real-number solutions.

“Every equation that might fail to have real solutions becomes solvable in the complex world, giving a more elegant and unified framework,” explains Li. “We were drawn to complex differential geometry because it sits at the crossroads of several branches of mathematics. It connects deep and elegant ideas across these fields and allows us to see how they come beautifully together.”

Illustration of the Ricci flow smoothing complex geometric space, researchers at USTC are uncovering the elegant structure of higher-dimensional manifolds.

Within the field, the two mathematicians have managed to map out much of the geometry of a special class of manifolds known as ‘Kähler Ricci shrinkers’, which are distinguished by their behavior under a tool called the Ricci flow.

Their work builds on the achievements of Russian mathematician Grigori Perelman whose three-dimensional theory demonstrated how geometric spaces can evolve smoothly over time and helped to solve a century-old problem about the shape of three-dimensional space.

“The Ricci flow is an equation that gradually smooths out the irregularities of a shape — much as flowing water polishes a rough stone into something round,” says Wang. “When we evolve a geometric space under the Ricci flow, it can sometimes develop ‘singularities’, which are points where the flow breaks down. Kähler Ricci shrinkers are special geometric shapes that model these singularities.”

Li and Wang showed any Kähler Ricci shrinker surface has bounded sectional curvature, thereby achieving a complete classification of all Kähler Ricci shrinkers. In simple terms, this means that even if the space stretches out infinitely, its shape remains controlled and never becomes wildly distorted.

“We were struck by how ideas from Perelman’s three-dimensional theory reappear in our four-dimensional setting, revealing deep connections across dimensions,” says Li. “Another surprise was the simplicity of the final picture at infinity, a beautiful result reflecting the elegance of complex geometry itself.”

Going forward, the two mathematicians plan to continue exploring the Ricci flow in four dimensions, which could have major implications for the study of our universe’s space-time. A long-term goal is to use it to tackle the classification of all four-dimensional spaces.

Reference:

1. Li, Y., & Wang, B. arXiv preprint arXiv:2301.09784. (2023).