Area‐Minimizing Currents mod 2Q: Linear Regularity Theory

Abstract

We establish a theory of Q‐valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currents mod(p) when p = 2Q, and to establish a first general partial regularity theorem for every p in any dimension and codimension . © 2020 Wiley Periodicals LLC.

An existence theorem for Brakke flow with fixed boundary conditions

“…for every φ as above , which shows the other statement N i=1 ∇χ E i (t) ≤ 2 V t in (8). Since the claim of ( 9) is interior in nature, the proof is identical to the case without boundary as in [20,Theorem 3.5(6)].…”

“…(7) says that (t) has the fixed boundary ∂ 0 . In general, the reduced boundary of the partition and V t may not match, but the latter is bounded from below by the former as in (8). By (10), the Lebesgue measure of each E i (t) changes continuously in time, so that arbitrary sudden loss of measure of V t is not allowed.…”

“…Conversely, let x ∈ ∂ 0 , and suppose for a contradiction that x / ∈ clos (spt V t ), so that there is a radius r > 0 with the property that B r (x) ∩ spt V t = ∅. Then, Theorem 2.3 (8)…”

“…It is worth noticing that analogous generic nonuniqueness holds true also for Plateau's problem: in that context, different definitions of the key words surfaces, area, spanning in its formulation lead to solutions with dramatically different regularity properties, thus making each model a better or worse predictor of the geometric complexity of physical soap films; see e.g. the survey papers [6,15] and the references therein, as well as the more recent works [7][8][9][22][23][24]27]. It is then interesting and natural to investigate different formulations for Brakke flow as well.…”

The regularity theory for the area functional (in geometric measure theory)

“…In the papers [41,42] the author, Jonas Hirsch, Andrea Marchese, and Salvatore Stuvard developed a theory to bound the dimension of flat singular points of a general area-minimizing current Σ mod p (i.e. in any dimension and codimension), which implies that the Hausdorff dimension of the set of flat singular points of Σ is at most m − 1.…”