Geometric analysis

Builds on

Required prerequisites, all the way down: 50

You will learn to

• Derive the minimal surface equation as the Euler–Lagrange equation of area and describe solutions of the Plateau problem.
• Explain the Ricci flow equation, its behaviour on round spheres and why singularities and surgery arise.
• Survey the Yamabe problem, harmonic maps and mean curvature flow, distinguishing established results from open problems.

Definition

• A minimal surface is a surface whose mean curvature vanishes everywhere; equivalently, a critical point of area.
• Ricci flow is the evolution equation ∂g/∂t = −2 Ric(g) for a Riemannian metric g.
• A harmonic map between Riemannian manifolds is a critical point of the energy ∫ |df|².

Theorem

• Douglas and Radó: every rectifiable Jordan curve in ℝ³ bounds a disc-type minimal surface (in general a branched immersion; Osserman and others ruled out interior branch points for area minimisers).
• Hamilton: a closed 3-manifold with positive Ricci curvature admits a metric of constant positive curvature.
• Perelman: Ricci flow with surgery proves the geometrisation conjecture.

Common misconceptions

• Thinking a minimal surface minimises area among all surfaces with the same boundary; minimal means stationary, not necessarily minimising.
• Believing Ricci flow always smooths a metric to a round one; singularities can form in finite time.

Optional discoveries

• Low-dimensional topology: 3- and 4-manifolds Ricci flow with surgery proved Thurston's geometrisation conjecture, and with it the Poincaré conjecture.
• The geometry of spacetime: special and general relativity Schoen and Yau proved the positive mass theorem of general relativity with minimal surfaces.
• ← Riemannian geometry
• ← Partial differential equations: modern theory
• ← The geometry of spacetime: special and general relativity

Learning pathways

• Analysis, PDE and dynamics

Further reading

• Bennett Chow, Peng Lu and Lei Ni, Hamilton's Ricci Flow (2006)
• Tobias Holck Colding and William P. Minicozzi II, A Course in Minimal Surfaces (2011)