  The maximum principle is one of the most important tools in the analysis of
geometric partial differential equations. Traditionally, the maximum principle
is applied to a scalar function defined on a manifold, but in recent years more
sophisticated versions have emerged. One particularly interesting direction
involves applying the maximum principle to functions that depend on a pair of
points. This technique is particularly effective in the study of problems
involving embedded surfaces.
  In this survey, we first describe some foundational results on curve
shortening flow and mean curvature flow. We then describe Huisken's work on the
curve shortening flow where the method of two-point functions was introduced.
Finally, we discuss several recent applications of that technique. These
include sharp estimates for mean curvature flow, as well as the proof of
Lawson's 1970 conjecture concerning minimal tori in S^3.
