Rectifiability of flat chains

Brian White

Review from Zentralblatt MATH:

One of the profound results in geometric measure theory is that an integral flat chain having finite mass is necessarily rectifiable. Previously, several proofs of this result have been given and it has been extended to flat chains with coefficients in a finite group (the author provides references to the appropriate papers). As the coefficient group for flat chains one can consider any normed abelian group that forms a complete metric space. A natural problem is to characterize the coefficient groups for which the finite mass flat chains are rectifiable. The author gives a beautifully simple answer: those groups that contain no continuous path of finite length.

The method by which the author proves the above result is itself of interest and significance. The main tool for the rectifiablity proof presented in this paper is the theorem that a finite mass flat chain over any coefficient group is rectifiable if and only if almost all of its $0$-dimensional slices are rectifiable. Needed for that slicing result is a deformation theorem proved elsewhere by the author [Acta Math. 183, No. 2, 255-271 (1999)].

Reviewed by H.Parks

Keywords: flat chains; rectifiability; coefficient group

Classification (MSC2000): 49Q15 49Q20

Full text of the article:

• PDF file (182 kilobytes)