nLab
geometric measure theory

Geometric measure theory studies various measures of subsets of Euclidean spaces (and possibly of some geometric generalizations) and their geometric properties. Especially, one studies rectifiability of subsets of some lower dimensionality, to define notions like area, arc length etc. and to study distributions and currents on such spaces. Very central questions and motivations belong to the variational problems including the study of minimal surfaces.

• wikipedia geometric measure theory
• Jürgen Jost, The geometric calculus of variations: a short survey and a list of open problems, Exposition. Math. 6 (1988), no. 2, 111–143, MR89h:58036
• Herbert Federer, Geometric measure theory, Springer 1969(especially appendices to Russian transl.)
• Frederick J., Jr. Almgren, Almgren’s big regularity paper (book form of a 1970s preprint)
• F. J. Almgren, Jr., Geometric measure theory and elliptic variational problems, Proc. ICM Nice 1970, vol. 2, djvu:350 K, pdf:649 K
• eom: T.C.O’Neil (2001), Geometric measure theory
• H. Federer, W. H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 1960 458–520, MR123260, doi
• Stephen H. Schanuel, What is the length of a potato? An introduction to geometric measure theory, in: Categories in continuum physics (Buffalo, N.Y., 1982), 118–126, Lecture Notes in Math. 1174, Springer 1986, MR842922,doi