Yamabe problem

The Yamabe problem is the question of whether every smooth Riemannian metric on a compact manifold can be conformally deformed to a metric of constant scalar curvature. Proposed by Hidehiko Yamabe in 1960, the problem was believed solved by Yamabe's own paper, but a critical error in the proof was discovered by Neil Trudinger in 1968. The complete solution, achieved through the independent work of Richard Schoen and others in 1984, is one of the landmark achievements of geometric analysis.

The problem reduces to finding a positive solution to a nonlinear elliptic partial differential equation — the Yamabe equation — on the manifold. The difficulty lies in the critical Sobolev exponent, where standard variational techniques fail due to the lack of compactness. Schoen's resolution introduced the positive mass theorem from general relativity as a tool for ruling out the non-compactness: the geometry of the manifold at small scales is governed by the asymptotic behavior of solutions on Euclidean space, and the positive mass theorem guarantees that this asymptotic geometry carries positive mass, which provides the necessary a priori estimates.

The Yamabe problem exemplifies a pattern common in geometric analysis: a purely geometric question (constant curvature) is answered by importing physical intuition (mass-energy positivity) into an analytic framework (elliptic PDE). The result is not merely a theorem but a demonstration that the boundaries between geometry, physics, and analysis are administrative conveniences, not mathematical necessities.