Ricci flow

Ricci flow is an evolution equation for a Riemannian metric in which the metric changes according to its Ricci curvature. It smooths geometric irregularities and can develop singularities that reveal a manifold’s structure.

Ricci curvature: A tensor measuring how volumes of small geodesic balls differ from those in Euclidean space. It supplies the curvature term that dictates the metric’s rate of change.
Richard S. Hamilton: An American mathematician who introduced Ricci flow and developed its foundational theory. Hamilton defined the flow in 1982 and established its role in geometric analysis.
Riemannian manifold: A smooth manifold equipped with a Riemannian metric that defines lengths and angles. Ricci flow evolves metrics on this underlying geometric space.
Differential geometry: The study of smooth spaces using calculus, including their metrics, curvature, and geodesics. Ricci flow is a central technique for studying and classifying curved spaces.
Ricci flow singularity: A finite-time or asymptotic breakdown in a Ricci flow, typically marked by unbounded curvature. Understanding singularity formation determines when surgery or a different analysis is needed.
Riemannian metric: A smoothly varying inner product on tangent spaces that defines lengths, angles, and volumes on a manifold. The evolving metric is the geometric quantity Ricci flow changes.
Thurston geometrization conjecture: A conjecture that every compact three-manifold decomposes into pieces carrying one of eight standard geometries. Hamilton developed Ricci flow as a possible route to proving geometrization.
Riemann curvature tensor: A tensor encoding the curvature of a connection, including how parallel transport changes around loops. Ricci curvature is obtained by contracting this fuller description of curvature.
Geometric analysis: The use of analytic methods, especially differential equations, to study geometric structures. The flow’s estimates and long-term behavior are core problems in this field.
Type I singularity: A Ricci-flow singularity whose maximum curvature grows no faster than inversely with remaining time. Type I blow-up has controlled rescalings that often lead to shrinking soliton models.

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