Frank Morgan

The most important lemma of regularity for an m-dimensional area-minimizing surface S in Rⁿ is the monotonicity of the mass ratio about a point p of S, i.e., the ratio of the area inside an n-ball about p of radius r to the volume of an m-ball of radius r [FF, Thm. 9.13 and proof], [M, 9.3]. Here in Lahore, Pakistan, at the Abdus Salam School of Mathematical Sciences, Rein Zeinstra told me that such monotonicity generalizes a similar result by Lelong [L] for complex analytic varieties (which are area minimizing). Similarly, the compactness theorem of geometric measure theory generalizes a later such result for complex analytic varieties. See a Math Review by Stoll.

[FF] Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. 72 (1960), 458-520.

[L] Pierre Lelong, Propriétés métriques des variétés analytiques complexes définies par une équation, Ann. Sci. École Norm. Sup. 67 (1950), 393–419.

[M] Frank Morgan, Geometric Measure Theory: a Beginner’s Guide, Academic Press, 4th ed., 2009.