On the boundary behavior of mass-minimizing integral currents

Abstract

Let Σ be a smooth Riemannian manifold, Γ ⊂ Σ a smooth closed oriented submanifold of codimension higher than 2 and T an integral area-minimizing current in Σ which bounds Γ. We prove that the set of regular points of T at the boundary is dense in Γ. Prior to our theorem the existence of any regular point was not known, except for some special choice of Σ and Γ. As a corollary of our theorem• we answer to a question of Almgren showing that, if Γ is connected, then T has at least one point p of multiplicity 1 2 , namely there is a neighborhood of the point p where T is a classical submanifold with boundary Γ;• we generalize Almgren's connectivity theorem showing that the support of T is always connected if Γ is connected; • we conclude a structural result on T when Γ consists of more than one connected component, generalizing a previous theorem proved by Hardt and Simon when Σ = R m+1 and T is m-dimensional. ContentsChapter 1. Introduction Chapter 2. Corollaries, open problems and plan of the paper 2.1. Indecomposable components of T 2.2. Almgren's question and proof of Theorem 1.9 2.3. Proof of Theorem 1.8 2.4. Plan of the proof of Theorem 1.6 2.5. Open problems Chapter 3. Stratification and reduction to collapsed points 3.1. First variation and monotonicity formula 3.2. Stratification 3.3. Proof of Theorem 3.8 3.4. Proof of Theorem 3.2, of Corollary 3.11 and of Corollary 3.12 Chapter 4. Regularity for Q − 1 2 Dir-minimizers 4.1. Preliminaries and proof of Theorem 4.2 4.1.1. Interpolation lemma 4.1.2. A simple measure theoretical lemma 4.1.3. Proof of Theorem 4.8: Compactness 4.1.4. Proof of Theorem 4.8: Minimality 4.2. The main frequency function estimate 4.2.1. H and D 4.2.2. Lower bound on H 4.2.3. Outer variations 4.2.4. Inner variations 4.2.5. A good function d 4.2.6. Proof of Theorem 4.15 4.3. Further consequences of the frequency function estimate 4.4. Blowup: proof of Theorem 4.5 with ϕ ≡ 0 4.5. Proof of Theorem 4.5: general case Chapter 5. First Lipschitz approximation and harmonic blow-up 5.1. Proof of Theorem 5.5 5.1.1. Artificial sheet and "bad set" 5.1.2. Lipschitz estimate 9.4. First variations 9.4.1. Outer variation 9.4.2. Inner variation 9.5. Key identities 9.6. Estimates on the error terms 9.6.1. Families of subregions 9.6.2. Lower and upper bounds in the subregions 9.6.3. Estimates on the error terms 9.7. Proof of Theorem 9.3 Chapter 10. Final blow-up argument 10.1. Asymptotics for D(r) 10.2. Vanishing of the average 10.3. Minimality and convergence in energy Bibliography Index CHAPTER 1

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

“…While Theorem 13.2 might look very far from optimal, it turns out that a naive counterpart of the bound of the dimension of the interior singular set is in fact false. In [34] we prove also the following.…”

“…[34]. In [34] the author, Guido De Philippis, Hirsch, and Annalisa Massaccesi were able to develop a suitable "Almgren-type" regularity theory for boundary points, building on a previous important step of Hirsch [72]. In particular we proved the following Theorem 13.2.…”

“…It can be easily shown that, if Γ = γ 1 + γ 2 , then Σ = D 1 + D 2 is the unique area-minimizing integral current bounded by Γ. Σ can be described as the sum of the corona D 2 \ D 1 , counted with multiplicity 1, and the disk D 1 , counted with multiplicity 2. A point p ∈ γ 1 is what can be naturally called a "2-sided" boundary point and note that its density is 3 2 (for a more rigorous definition, cf [34]). The regularity theory at such points is rather subtle and (in codimension 1) it was handled in the famous work [69] by Hardt and Simon.…”

“…And as in the case of Theorem 10.1 the most problematic issue is that unfortunately the existence of a flat tangent cone at the boundary does not guarantee regularity: flat boundary singular points exist as soon as the codimension is larger than 1, cf. [34]. In [34] the author, Guido De Philippis, Hirsch, and Annalisa Massaccesi were able to develop a suitable "Almgren-type" regularity theory for boundary points, building on a previous important step of Hirsch [72].…”

Rectifiability of the free boundary for varifolds

“…In section 5 we first derive a monotonicity identity for distorted balls and next we prove Theorem 1. 4.…”