An evolution of minimal surfaces with Plateau condition (Q1879313)

This is an extensive study of the heat flow associated to the minimal surface equation for a Plateau boundary condition. This article represents a generalization to Riemannian geometry of a method previously found by the same authors for the Euclidean case. The authors prove that the natural \(L^2\)-gradient flow in the current setting is equivalent to a variational inequality (called the weak heat flow) for a certain class of surfaces. They also state and prove some very important global existence and uniqueness theorems for surfaces whose initial Dirichlet energy is below a certain level (depending on \(N\) and the Plateau boundary condition).