Geometric Analysis

The field got its name and its confidence in the 1970s, largely through Shing-Tung Yau and his collaborators. Eugenio Calabi had conjectured in the 1950s that certain complex manifolds could be given metrics with any prescribed Ricci curvature. In 1976 Yau proved it by solving the complex Monge–Ampère equation,

$$\det\!\left( g_{i\bar{j}} + \partial_i \partial_{\bar{j}} \varphi \right) = e^{F} \det\!\left( g_{i\bar{j}} \right),$$

a fully nonlinear equation whose solution is the new metric. The manifolds this produced, now called Calabi–Yau manifolds, would within a decade become the extra dimensions of string theory.

Solving such equations means controlling how solutions can fail. In 1981 Jonathan Sacks and Karen Uhlenbeck showed what happens when a sequence of approximate minimal spheres does not converge. The energy concentrates at a few points and splits off as small spheres, or "bubbles." Once the bubbles are accounted for, nothing else can go wrong. Tracking bubbles became the standard way to take limits of geometric equations, from harmonic maps to the gauge theories Simon Donaldson used to probe dimension four.

In 1979 Richard Schoen and Yau proved the positive mass theorem. In general relativity, the total mass of an isolated system, measured from far away, cannot be negative. Physicists expected this but could not prove it. Schoen and Yau reduced it to a question about minimal surfaces in curved three-dimensional space and solved that. Edward Witten gave an independent proof in 1981 with methods from quantum field theory.

This is the field's recurring pattern. A statement that sounds physical is really a statement about curvature, and the right equation proves it. Hamilton's Ricci flow, which grew up alongside this field, followed the same philosophy all the way to Perelman's proof of the Poincaré conjecture in geometric topology.

A soap film is a physical minimisation problem that turns into a differential equation. Surface tension $\gamma$ pulls the film to reduce its area, and the pressure difference across a curved film is given by the Young–Laplace law,

$$\Delta P = 2\gamma H ,$$

where $H$ is the mean curvature, the average of the two principal curvatures. That is for a single surface. A soap film has two, which doubles the jump. A soap bubble holds higher pressure inside, so it curves with $H > 0$ and becomes a sphere. A film spanning a wire loop has the same air pressure on both sides, so $\Delta P = 0$, which forces $H = 0$ everywhere. The film bends one way in one direction and equally the other way in the perpendicular direction, like a saddle.

The same equation comes from calculus. Push a surface slightly in the normal direction by an amount $\varphi$. The area changes, to first order, by

$$\frac{d}{dt}\text{Area} = -\int H\,\varphi\,dA$$

(up to a convention factor). A surface of least area cannot decrease under any small push, so $H$ must vanish. Minimal surfaces are exactly the critical points of area.

Dip two parallel rings into soapy water and pull them apart, and the film between them forms a catenoid, the surface made by spinning the curve $y = a\cosh(x/a)$. Pull the rings too far apart and the catenoid snaps into two flat discs: beyond a critical distance, no catenoid spans the rings at all. Deciding when solutions exist and when they break down is the central work of geometric analysis.

Minimal surfaces returned to the centre after 2012. The methods Fernando Codá Marques and André Neves revived to prove the Willmore conjecture could find minimal surfaces that are not area-minimising: saddle points of area rather than minima. Within six years they, together with Kei Irie, and then Antoine Song had proved Yau's 1982 conjecture that every closed three-manifold contains infinitely many of them.

The open questions now sit at the two edges of the field: pure spectral geometry, where Yau's eigenvalue conjecture still stands, and general relativity, where the full Penrose inequality waits for the right flow.