The regularity theory for the area functional (in geometric measure theory)

Abstract

The aim of this article is to give a rather extensive, and yet nontechnical, account of the birth of the regularity theory for generalized minimal surfaces, of its various ramifications along the decades, of the most recent developments, and of some of the remaining challenges.

Rectifiability; a survey

“…Currents and varifolds are very well presented also in [Sim83], [LY02] and [KP08], and more informally in [Mor88]. De Lellis's survey [Del21] enlightens the background and recent developments for this and next two chapters.…”

“…The currents modulo p exhibit many interesting new phenomena. They are extensively discussed in [Del21]. Simple illustrative examples are presented in [Mor88,Section 11.1].…”

“…Quasiminimizers. In addition to currents and varifolds there are several other ways to model minimal surfaces and related objects, see [Dav19] and [Del21]. Quasiminimizers provide a very natural and general setting for many variational problems.…”

“…Mass minimizing currents and stationary varifolds. De Lellis's survey [Del21] gives an excellent up-to-date view of this wide topic. Currents give a very convenient setting for the Plateau problem for orientable surfaces.…”

“…To overcome this Almgren developed a theory of multivalued functions minimizing Dirichlet integral. Much of Almgren's work has been simplified and extended by De Lellis and Spadaro, see the surveys [Del16a], [Del21] and the references given there.…”