Federer Geometric Measure Theory Pdf May 2026

From the introduction to Chapter 4 on currents:

“A k‑dimensional current in an open subset U of ℝⁿ is a continuous linear functional on the space of smooth k‑forms with compact support in U. The boundary of a k‑current is defined by duality with the exterior derivative. The mass of a current is the supremum of its values on forms of pointwise norm ≤ 1.”

Then follows 50 pages of dense estimates, culminating in the deformation theorem. federer geometric measure theory pdf

While Federer is the "source code" of GMT, the PDF is often used alongside "interface" books that explain Federer’s dense concepts. These include:

There is a running joke in the mathematical community: Federer’s book is the best reference on the subject, provided you already know the subject. From the introduction to Chapter 4 on currents:

The text is infamous for several reasons:

Despite this, it remains the definitive reference. There is no other book that covers the breadth of material—particularly regarding currents, varifolds, and the structure of sets—quite like Federer does. “A k‑dimensional current in an open subset U

Federer applies the machinery to prove the existence of area-minimizing surfaces. He introduces varifolds (a measure-theoretic notion of a surface that allows for multiplicities and tangencies) and proves regularity theorems (showing that minimizers are smooth except on a small singular set).

A legitimate PDF of Federer’s book (copyright permitting, e.g., from a university library subscription or personal scan) contains the following structure:

If you succeed in obtaining a legitimate copy (via your university), you will notice something strange. The PDF is not "modern." It is a direct scan of the 1969 print.

Federer does not assume you know set theory. He starts with ordinal numbers, cardinal numbers, and the Zorn’s Lemma. He then builds vector spaces, topological spaces, and the basics of measure theory (outer measures, Carathéodory’s criterion) from scratch.