Multiple Valued Functions and Integral Currents

We prove several results on Almgren’s multiple valued functions and their links to integral currents. In particular, we give a simple proof of the fact that a Lipschitz multiple valued map naturally defines an integer rectifiable current; we derive explicit formulae for the boundary, the mass and the first variations along certain specific vectorfields; and exploit this connection to derive a delicate reparametrization property for multiple valued functions. These results play a crucial role in our new proof of the partial regularity of area minimizing currents [5, 6, 7]. 0. Introduction It is known since the pioneering work of Federer and Fleming [10] that one can naturally associate an integer rectifiable current to the graph of a Lipschitz function in the Euclidean space, integrating forms over the corresponding submanifold, endowed with its natural orientation. It is then possible to derive formulae for the boundary of the current, its mass and its first variations along smooth vector-fields. Moreover, all these formulae have important Taylor expansions when the current is sufficiently flat. In this paper we provide elementary proofs for the corresponding facts in the case of Almgren’s multiple valued functions (see [4] for the relevant definitions). The connection between multiple valued functions and integral currents is crucial in the analysis of the regularity of area minimizing currents for two reasons. On the one hand, it provides the necessary tools for the approximation of currents with graphs of multiple valued function. This is a fundamental idea for the study of the regularity of minimizing currents in the classical “single-vaued” case, and it also plays a fundamental role in the proof of Almgren’s partial regularity result (cf. [1, 5]). In this perspective, explicit expressions for the mass and the first variations are necessary to derive the right estimates on the main geometric quantities involved in the regularity theory (cf. [5, 6, 7]). On the other hand, the connection can be exploited to infer interesting conclusions about the multiple valued functions themselves. This point of view has been taken fruitfully in many problems for the case of classical functions (see, for instance, [11, 12] and the references therein), and has been recently exploited in the multiple valued setting in [3, 14]. The prototypical example of interest here is the following: let f : R ⊃ Ω→ R be a Lipschitz map and Gr(f) its graph. If the Lipschitz constant of f is small and we change coordinates in R with an orthogonal transformation close to the identity, then the set Gr(f) is the graph of a Lipschitz function f̃ over some domain Ω̃ also in the new system of coordinates. In fact it is easy to see that there exist suitable maps Ψ and Φ such that f̃(x) = Ψ ( x, f(Φ(x)) ) . In the multiple valued 1 2 CAMILLO DE LELLIS AND EMANUELE SPADARO case, it remains still true that Gr(f) is the graph of a new Lipschitz map f̃ in the new system of coordinates, but we are not aware of any elementary proof of such statement, which has to be much more subtle because simple relations as the one above cannot hold. It turns out that the structure of Gr(f) as integral current gives a simple approach to this and similar issues. Several natural estimates can then be proved for f̃ , although more involved and much harder. The last section of the paper is dedicated to these questions; more careful estimates obtained in the same vein will also be given in [6], where they play a crucial role. Most of the conclusions of this paper are already established, or have a counterpart, in Almgren’s monograph [1], but we are not always able to point out precise references to statements therein. However, also when this is possible, our proofs have an independent interest and are in our opinion much simpler. More precisely, the material of Sections 1 and 2 is covered by [1, Sections 1.5-1.7], where Almgren deals with general flat chains. This is more than what is needed in [5, 6, 7], and for this reason we have chosen to treat only the case of integer rectifiable currents. Our approach is anyway simpler and, instead of relying, as Almgren does, on the intersection theory of flat chains, we use rather elementary tools. For the theorems of Section 3 we cannot point out precise references, but Taylor expansions for the area functional are ubiquitous in [1, Chapters 3 and 4]. The theorems of Section 4 do not appear in [1], as Almgren seems to consider only some particular classes of deformations (the “squeeze” and “squash”, see [1, Chapter 5]), while we derive fairly general formulas. Finally, it is very likely that the conclusions of Section 5 appear in some form in the construction of the center manifold of [1, Chapter 4], but we cannot follow the intricate arguments and notation of that chapter. In any case, our approach to “reparametrizions” of multiple valued maps seems more flexible and powerful, capable of further applications, because, as it was first realized in [4], we can use tools from metric analysis and metric geometry developed in the last 20 years. Acknowledgments The research of Camillo De Lellis has been supported by the ERC grant agreement RAM (Regularity for Area Minimizing currents), ERC 306247. The authors are warmly thankful to Bill Allard for several enlightening conversations and his constant enthusiastic encouragement; and very grateful to Luca Spolaor and Matteo Focardi for carefully reading a preliminary version of the paper and for their very useful comments. Camillo De Lellis is also very thankful to the University of Princeton, where he has spent most of his sabbatical completing this and the papers [5, 6, 7]. 1. Q-valued push-forwards We use the notation 〈, 〉 for: the euclidean scalar product, the naturally induced inner products on p-vectors and p-covectors and the duality pairing of p-vectors and p-covectors; we instead restrict the use of the symbol · to matrix products. Given a C m-dimensional submanifold Σ ⊂ R , a function f : Σ→ R and a vector field X tangent to Σ, we denote by DXf the derivative of f along X, that is DXf(p) = (f ◦ γ)′(0) whenever γ is a smooth curve on Σ with γ(0) = p and γ′(0) = X(p). When k = 1, we denote by ∇f the vector field tangent to Σ such that 〈∇f,X〉 = DXf for every tangent vector field X. For general MULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 3 k, Df |x : TxΣ → R will be the linear operator such that Df |x ·X(x) = DXf(x) for any tangent vector field X. We write Df for the map x 7→ Df |x and sometimes we will also use the notation Df(x) in place of Df |x. Having fixed an orthonormal base e1, . . . em on TxΣ and letting (f1, . . . , fk) be the components of f , we can write ∇fi = ∑m j=1 aijej and |Df | for the usual Hilbert-Schmidt norm: |Df | = m ∑