Regularity of area minimizing currents mod p
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GAFA Geometric And Functional Analysis
REGULARITY OF AREA MINIMIZING CURRENTS MOD p Camillo De Lellis, Jonas Hirsch, Andrea Marchese and Salvatore Stuvard
Abstract. We establish a first general partial regularity theorem for area minimizing currents mod(p), for every p, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an m-dimensional area minimizing current mod(p) cannot be larger than m − 1. Additionally, we show that, when p is odd, the interior singular set is (m − 1)-rectifiable with locally finite (m − 1)-dimensional measure.
1 Introduction 1.1 Overview and main results. In this paper we consider currents mod(p) (where p ≥ 2 is a fixed positive integer), for which we follow the definitions and the terminology of [Fed69]. In particular, given an open subset Ω ⊂ Rm+n , we will let Rm (Ω) and Fm (Ω) denote the spaces of m-dimensional integer rectifiable currents and m-dimensional integral flat chains in Ω, respectively. If C ⊂ Rm+n is a closed set (or a relatively closed set in Ω), then Rm (C) (resp. Fm (C)) denotes the space of currents T ∈ Rm (Rm+n ) (resp. T ∈ Fm (Rm+n )) with compact support spt(T ) contained in C. Currents modulo p in C are defined introducing an appropriate family of pseudo-distances on Fm (C): if S, T ∈ Fm (C) and K ⊂ C is compact, then p (T − S) := inf M(R) + M(Z) : R ∈ Rm (K) , Z ∈ Rm+1 (K) FK such that T − S = R + ∂Z + pP for some P ∈ Fm (K) . Two flat currents in C are then congruent modulo p if there is a compact set K ⊂ C p (T − S) = 0. The corresponding congruence class of a fixed flat chain such that FK T will be denoted by [T ], whereas if T and S are congruent we will write T = S mod(p) . Keywords and phrases: Minimal surfaces, Area minimizing currents mod(p), Regularity theory, Multiple valued functions, Blow-up analysis, Center manifold Mathematics Subject Classification: 49Q15, 49Q05, 49N60, 35B65, 35J47
C. DE LELLIS ET AL.
GAFA
p p The symbols Rm (C) and Fm (C) will denote the quotient groups obtained from Rm (C) and Fm (C) via the above equivalence relation. The boundary operator ∂ has the obvious property that, if T = S mod(p), then ∂T = ∂S mod(p). This allows to define an appropriate notion of boundary mod(p) as ∂ p [T ] := [∂T ]. Correspondingly, we can define cycles and boundaries mod(p) in C:
• a current T ∈ Fm (C) is a cycle mod(p) if ∂T = 0 mod(p), namely if ∂ p [T ] = 0; • a current T ∈ Fm (C) is a boundary mod(p) if ∃S ∈ Fm+1 (C) such that T = ∂S mod(p), namely [T ] = ∂ p [S]. Note that every boundary mod(p) is a cycle mod(p). In what follows, the closed set C will always be sufficiently smooth, more precisely a complete submanifold Σ of Rm+n without boundary and of class C 1 . Remark 1.1. Note that the congruence classes [T ] depend on the set C, and thus our notation is not precise in this regard. In particular, when two currents are congruent modulo p in Σ ⊂ Rm+n , then they are obviously congruent in Rm+n , but the opposite implication is generally false, see also the discussion in [MS18, Remark
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