The isoperimetric inequality. The Michael-Simon Sobolev inequality. The Fenchel-Willmore-Chen inequality. The Allard regularity theorem. Each a landmark result in geometric analysis, each proved with different techniques in different decades for different purposes. The results look unrelated — they bound different quantities, apply to different objects, serve different communities.
Brendle (arXiv:2603.12025) shows they share a mechanism. The Alexandrov-Bakelman-Pucci technique — originally developed for regularity estimates of elliptic partial differential equations — serves as a common engine that derives all of them. The ABP technique was not designed for geometry. It was designed to control solutions of second-order elliptic equations by bounding them through the Monge-Ampère measure of their concave envelopes. But the geometric content of the ABP estimate — a relationship between a function's values and the volume of its gradient image — generalizes to manifolds, submanifolds, and varifolds.
The unification is not retrospective. It is not that someone found a way to rephrase known proofs in a common language. The ABP technique provides new proofs of known results that are simpler than the originals, and in some cases sharper. The common engine was more powerful than the individual machines it replaces.
What makes this possible is that each geometric inequality, at its core, relates a volume to a boundary measurement — and the ABP technique is precisely a tool for controlling volumes through boundary data, just formulated in PDE language rather than geometric language. The tool and the problems were always about the same thing. The PDE formulation obscured the geometric content; the geometric application revealed it.
A technique built for regularity theory turns out to be the hidden mechanism behind seemingly unrelated geometric inequalities. The engine was running in PDE theory for decades. It was waiting for someone to notice it also runs geometry.