The aim of this paper is to provide a simple, complete and self-contained refer-
ence for Almgren’s theory of Dir-minimizing Q-valued functions, so to make it an
easy step for the understanding of the remaining parts of the Big regularity paper
[Alm00]. We propose simpler and shorter proofs of the central results on Q-valued
functions contained there, suggesting new points of view on many of them. In ad-
dition, parallel to Almgren’s theory, we elaborate an intrinsic one which reaches
his main results avoiding the extrinsic mappings ξ and ρ (see Section 2.1 and
compare with 1.2 of [Alm00]). This “metric” point of view is clearly an original
contribution of this paper. The second new contribution is Theorem 0.12 where we
improve Almgren’s estimate of the singular set in the planar case, relying heavily
on computations of White [Whi83] and Chang [Cha88].
Simplified and intrinsic proofs of parts of Almgren’s big regularity paper have
already been established in [Gob06a] and [Gob06b]. In fact our proof of the
Lipschitz extension property for Q-valued functions is essentially the one given in
[Gob06a] (see Section 1.2). Just to compare this simplified approach to Almgren’s,
note that the existence of the retraction ρ is actually an easy corollary of the ex-
istence of ξ and of the Lipschitz extension theorem. In Almgren’s paper, instead,
the Lipschitz extension theorem is a corollary of the existence of ρ, which is con-
structed explicitly (see 1.3 in [Alm00]) . However, even where our proofs differ
most from his, we have been clearly influenced by his ideas and we cannot exclude
the existence of hints to our strategies in [Alm00] or in his other papers [Alm83]
and [Alm86]: the amount of material is very large and we have not explored it in
all the details.
Almgren asserts that some of the proofs in the first chapters of [Alm00] are
more involved than apparently needed because of applications contained in the
other chapters, where he proves his celebrated partial regularity theorem for area-
minimizing currents. We instead avoid any complication which looked unnecessary
for the theory of Dir-minimizing Q-functions. For instance, we do not show the
existence of Almgren’s improved Lipschitz retraction
(see 1.3 of [Alm00]), since
it is not needed in the theory of Dir-minimizing Q-valued functions. This retraction
is instead used in the approximation of area-minimizing currents (see Chapter 3 of
[Alm00]) and will be addressed in the forthcoming paper [DLS].
In our opinion the portion of Almgren’s Big regularity paper regarding the the-
ory of Q-valued functions is simply a combination of clean ideas from the theory of
elliptic partial differential equations with elementary observations of combinatorial
nature, the latter being much less complicated than what they look at a first sight.
In addition our new “metric” point of view reduces further the combinatorial part,
at the expense of introducing other arguments of more analytic flavor.