Regularity Theory for 2-Dimensional Almost Minimal Currents II: Branched Center Manifold

Abstract

Abstract. We construct a branched center manifold in a neighborhood of a singular point of a 2-dimensional integral current which is almost minimizing in a suitable sense. Our construction is the first half of an argument which shows the discreteness of the singular set for the following three classes of 2-dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of 3-dimensional area minimizing cones.This paper is the third in a series of works aimed at establishing an optimal regularity theory for 2-dimensional integral currents which are almost minimizing in a suitable sense. Building upon the monumental work of Almgren [1], Chang in [4] established that 2-dimensional area minimizing currents in Riemannian manifolds are classical minimal surfaces, namely they are regular (in the interior) except for a discrete set of branching singularities. The argument of Chang is however not entirely complete since a key starting point of his analysis, the existence of the so-called "branched center manifold", is only sketched in the appendix of [4] and requires the understanding (and a suitable modification) of the most involved portion of the monograph [1].An alternative proof of Chang's theorem has been found by Rivière and Tian in [16] for the special case of J-holomorphic curves. Later on the approach of Rivière and Tian has been generalized by Bellettini and Rivière in [3] to handle a case which is not covered by [4], namely that of special Legendrian cycles in S 5 (see also [2] for a further generalization). Meanwhile the first and second author revisited Almgren's theory giving a much shorter version of his program for proving that area minimizing currents are regular up to a set of Hausdorff codimension 2, cf. [6,8,7,9,10]. In this note and its companion papers [11,12] we build upon the latter works in order to give a complete regularity theory which includes both the theorems of Chang and Bellettini-Rivière as special cases. In order to be more precise, we introduce the following terminology (cf. [13, Definition 0.3 ]).Definition 0.1. Let Σ ⊂ R m+n be a C 2 submanifold and U ⊂ R m+n an open set.(a) An m-dimensional integral current T with finite mass and spt(T ) ⊂ Σ ∩ U is area minimizing in Σ ∩ U if M(T + ∂S) ≥ M(T ) for any m + 1-dimensional integral current S with spt(S) ⊂⊂ Σ ∩ U . (b) A semicalibration (in Σ) is a C 1 m-form ω on Σ such that ω x c ≤ 1 at every x ∈ Σ, where · c denotes the comass norm on Λ m T x Σ. An m-dimensional integral current T with spt(T ) ⊂ Σ is semicalibrated by ω if ω x ( T ) = 1 for T -a.e. x. In what follows, given an integer rectifiable current T , we denote by Reg(T ) the subset of spt(T ) \ spt(∂T ) consisting of those points x for which there is a neighborhood U such that T U is a (constant multiple of) a C 2 submanifold. Correspondingly, Sing(T ) is the set spt(T ) \ (spt(∂T ) ∪ Reg(T )). Observe that Reg(T ) is relatively open in spt(T ) \ spt(∂T ) and thus Sing(T ) is relatively closed. The main result of this and the works ...

An upper Minkowski bound for the interior singular set of area minimizing currents

“…In contrast to the codimension one case, the fine structure of the interior singular set is still unknown in general. In the case m " 2, Chang [11] and De Lellis, Spadaro & Spolaor [8][9][10] established a structure theorem for the singular set: all singularities are isolated. 1 The latter series of references in fact generalize this result to the wider class of semicalibrated currents and spherical cross sections of area minimizing cones.…”

“…The results of this article are an initial step towards better understanding the problem of how to determine the structure of the singular set. 2 Before we state 1 Chang's result relies heavily on the existence of a branched center manifold; the rigorous proof of the existence of such an object is due to the authors in [9]. 2 In particular, 'flat' singular points in S m pT qzS m´1 pT q.…”

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

“…the appendix of [20]), invoking suitable modifications of Almgren's statements (it must be noted that the construction of the center manifold occupies more than half of Almgren's monograph [11]). Based on the works [47,48,49,50,51] the author, Spadaro, and Luca Spolaor gave a complete independent proof of the existence of a branched center manifold in [53]. We also developed a suitable more general counterpart of Chang's theory in the papers [52,54,55], proving in particular the same regularity result for spherical cross-sections of area-minimizing 3-dimensional cones and for semicalibrated 2-dimensional currents (previous theorems in [12,13] proved some cases of particular interest, based on the works of Rivière and Tian, see [99,100,101]).…”

An inequality for length and volume in the complex projective plane

“…One possible tool in investigating geometric inequalities is the existence of minimizing surfaces in homology classes. Such existence is guaranteed by Geometric Measure Theory; see Chang ([4], 1988), De Lellis et al [6,7,8]. It is in the nature of the techniques used that the geometry of the minimizer is rather inexplicit.…”