MULTIPLE VALUED JACOBI FIELDS SALVATORESTUVARD Abstract. We develop a multivalued theory for the stability operator of (a constant mul- tiple of) a minimally immersed submanifold Σ of a Riemannian manifold M. We define themultiplevaluedcounterpartoftheclassical Jacobi fieldsastheminimizersofthesecond variation functional defined on a Sobolev space of multiple valued sections of the normal 7 bundle of Σ in M, and we study existence and regularity of such minimizers. Finally, we 1 provethatanyQ-valuedJacobifieldcanbewrittenasthesuperpositionofQclassicalJacobi 0 fields everywhere except for a relatively closed singular set having codimension at least two 2 in the domain. n Keywords: Almgren’sQ-valuedfunctions;secondvariation;stabilityoperator;Jacobifields; a existence and regularity. J AMS subject classification (2010): 49Q20, 35J57, 54E40, 53A10. 0 3 ] P A . h t a m [ Contents 1 v 0. Introduction 2 3 0.1. Main results 3 5 7 0.2. Aknowledgements 6 8 1. Notation and preliminaries 6 0 1.1. The geometric setting 6 . 1 1.2. Multiple valued functions 7 0 2. Q-valued second variation of the area functional 15 7 1 3. Jacobi Q-fields 23 : 4. Existence of Jacobi Q-fields 26 v i 4.1. Q-valued Luckhaus Lemma and the extension theorem 26 X 4.2. The compactness theorem 34 r a 5. Hölder regularity of Jacobi Q-fields 37 6. Blow-up analysis 41 6.1. First variations 41 6.2. The frequency function 46 6.3. Some fundamental consequences of the almost monotonicity of the frequency function 52 6.4. The top-multiplicity singular stratum. Blow-up 58 6.5. The closing argument: proof of Theorem 0.7 67 References 70 1 2 SALVATORESTUVARD 0. Introduction Given an m-dimensional area minimizing integer rectifiable current T in Rm+n and any point x spt(T) spt(∂T), it is a by now well known consequence of the monotonicity of the ∈ \ function r kTk(Br(x)) (cf. [All72, Section 5]) that for any sequence of radii r ∞ with 7→ ωmrm { j}j=1 rj ↓0 there exists a subsequence rj′ such that the corresponding blow-ups Tx,rj′ := (ηx,rj′)♯T (where η (y) := y−x) converge to a (locally) area minimizing m-dimensional current C x,r r which is invariant with respect to homotheties about the origin: such a limit current is called a tangent cone to T at x. If x is a regular point, and thus spt(T) is a classical m-dimensional minimal submanifold in a neighborhood of x, then the cone C is certainly unique, and in fact C = Q π , where π = T (spt(T)) is the tangent space to spt(T) at x and Q = Θ( T ,x) x k k is the m-dimensional density of the measure T at x. On the other hand, singularities k k do occur for area minimizing currents of arbitrary codimension as soon as the dimension J K of the current is m 2: indeed, by the regularity theory developed by F. Almgren in his ≥ monumental Big Regularity Paper [Alm00] and recently revisited by C. De Lellis and E. Spadaro in [DLS14, DLS16a, DLS16b], we know that area minimizing m-currents in Rm+n may exhibit a singular set of Hausdorff dimension at most m 2, and that this result is sharp − whenn 2([Fed65]). Now, ifxhappenstobesingular, thennotonlywehavenoinformation ≥ about the limit cone, but in fact it is still an open question whether in general such a limit cone is unique (that is, independent of the approximating sequence) or not. The problem of uniqueness of tangent cones at the singular points of area minimizing currents of general dimension and codimension stands still today as one of the most celebrated of the unsolved problems in Geometric Measure Theory (cf. [ope86, Problem 5.2]), and only a few partial answers corresponding to a limited number of particular cases are available in the literature. In [Whi83], B. White showed such uniqueness for two-dimensional area minimizing currents in any codimension, building on a characterization of two-dimensional area minimizing cones proved earlier on by F. Morgan in [Mor82]. In general dimension, W. Allard and F. Almgren [AA81] were able to prove that uniqueness holds under some additional requirements on the limit cone. Specifically, they have the following theorem, which is valid in the larger class of stationary integral varifolds. Theorem 0.1 ([AA81]). Let V be an m-dimensional stationary integral varifold in Rm+n, and let x spt(V) be an isolated singular point. Assume that there exists a tangent cone C ∈ to V at x satisfying the following hypotheses: (H1) C is the cone over an (m 1)-dimensional minimal submanifold Σ of Sm+n−1, and − thus C has an isolated singularity at 0 and Θ( C ,x) = 1 for every x spt(C) 0 ; k k ∈ \{ } (H2) all normal Jacobi fields N of Σ in Sm+n−1 are integrable, that is for every normal Jacobi field N there exists a one-parameter family of minimal submanifolds of Sm+n−1 having velocity N at Σ. Then, C is the unique tangent cone to V at x. Furthermore, the blow-up sequence V x,r converges to C as r 0 with rate rµ for some µ > 0. ↓ The hypotheses (H1) and (H2) are however quite restrictive. Allard and Almgren were able to show that (H2) holds in case Σ is the product of two lower dimensional standard spheres (of appropriate radii to ensure minimality), since in this case all normal Jacobi fields of Σ in Sm+n−1 arise from isometric motions of Sm+n−1. It seems however rather unlikely that the condition can hold for any general Σ admitting normal Jacobi fields other than those generated by rigid motions of the sphere. In [Sim83a], L. Simon was able to prove Theorem MULTIPLE VALUED JACOBI FIELDS 3 0.1 droppingthe hypothesis (H2), with a quite different approach with respect to [AA81] and purely PDE-based techniques. Not much has been done, instead, in the direction of removing the hypothesis (H1): to our knowledge, indeed, the only result concerning the case when a tangent cone C has more than one isolated singularity at the origin is contained in L.Simon’s work [Sim94], where the author proves uniqueness of tangent cones to any codimension one area minimizing m-current T whenever one limit cone C is of the form C = C R, with C a 0 0 × strictly stable, strictly minimizing (m 1)-dimensional coneinRm withanisolated singularity − at theorigin, andunderadditionalassumptionson theJacobi fieldsof C andon thespectrum of the Jacobi normal operator of C . 0 However, all the results discussed above do not cover the cases when a tangent cone has higher multiplicity: it is remarkable that uniqueness is still open even under the strong assumption that all tangent cones to an area minimizing m-current T (m > 2) at an interior singular point x are of the form C = Q π , where π is the rectifiable current associated with an oriented m-dimensional linear subspace of Rm+n. The purpose of this work is to present a multivalued theory of the Jacobi normal operator: J K J K we believe that such a theory may facilitate the understanding of the qualitative behaviour of the area functional near a minimal submanifold with multiplicity, and eventually lead to a generalization of Theorem 0.1(and neighbouringresults)torelevant cases whenthecondition that Θ( C ,x) = 1 for every x spt(C) 0 fails to hold. k k ∈ \{ } In our investigation, we will make use of tools and techniques coming from the theory of multiple valued functions minimizing the Dirichlet energy, developed by Almgren in [Alm00] and revisited by De Lellis and Spadaro in [DLS11]. A quick tutorial on the theory of multiple valued functions is contained in 1.2, in order to ease the reading of the remaining part § of the paper. As a byproduct, the theory of multiple valued Jacobi fields will show that the regularity theory for Dir-minimizing Q-valued functions is robust enough to allow one to produce analogous regularity results for minimizers of functionals defined on Sobolev spaces of Q-valued functions other than the Dirichlet energy (see also [DLFS11] for a discussion about general integral functionals defined on spaces of multiple valued functions and their semi-continuity properties, and [Hir16] for a regularity theory for multiple valued energy minimizing maps with values into a Riemannian manifold). 0.1. Main results. Letusfirstrecallwhatisclassically meantbyJacobioperatorandJacobi fields. Let Σ be an n-dimensional compact oriented submanifold (with or without boundary) of an m-dimensional Riemannian manifold RM, and assume that Σ is stationary with M ⊂ respect to the n-dimensional area functional. Then, a one-parameter family of normal varia- tions of Σ in can be defined by setting Σ := F (Σ), where F is the flow generated by a t t t M smooth cross-section N of the normalbundle Σ of Σ in which has compact supportin Σ. N M Itis known thatthe second variation formulacorrespondingto sucha family of variations can be expressed in terms of an elliptic differential operator defined on the space Γ( Σ) of the L N cross-sections of the normal bundle. This operator, usually called the Jacobi normal operator, is given by = ∆⊥ A R, where ∆⊥ is the Laplacian on Σ, and A and R are linear L − Σ − − Σ N transformations of Σ defined in terms of the second fundamental form of the immersion N ι: Σ and of apartial Ricci tensor of theambient manifold , respectively. Thenotions → M M of Morse index, stability and Jacobi fields, central in the analysis of the properties of the class of minimal submanifolds of a given Riemannian manifold, are all defined by means of the Jacobi normal operator and its spectral properties (see Section 2 for the precise definitions and for a discussion about the most relevant literature related to the topic). In particular, 4 SALVATORESTUVARD Jacobi fields are defined as those sections N Γ( Σ) lying in the kernel of the operator , ∈ N L and thus solving the system of partial differential equations (N) = 0. L In this work, we consider instead multivalued normal variations in the following sense. Let Σ and be as above, and consider, for a fixed integer Q > 1, a Lipschitz multiple valued vector fiMeld N: Σ (RM) vanishing at ∂Σ and having the form N = Q Nl , where → AQ l=1 Nl(x) is tangent to and orthogonal to Σ at every point x Σ and for every l = 1,...,Q. M ∈ P The flow of such a multiple valued vector field generates a one-parameter famJily KΣt of n- dimensional integer rectifiable currents in such that Σ = Q Σ and ∂Σ = Q ∂Σ for 0 t M every t. The second variation d2 J K J K δ2 Σ (N):= M(Σ ) dt2 t t=0 is a well-defined functional on the space Γ1,2( Σ) of Q(cid:12)(cid:12)-valued W1,2 sections of the normal J K Q N bundle Σ of Σ in . Such a Jacobi functional is denoted Jac, and it is given by N M Q Jac(N,Σ) := ⊥Nl 2 A Nl 2 (Nl,Nl) d n, (0.1) ˆ |∇ | −| · | −R H ΣXl=1(cid:16) (cid:17) where ⊥ is the projection of the Levi-Civita connection of onto Σ, A Nl is the ∇ M N | · | Hilbert-Schmidt norm of the projection of the second fundamental form of the embedding Σ ֒ onto Nl and (Nl,Nl) is a partial Ricci tensor of the ambient manifold in the → M R M direction of Nl. Unlike the classical case, it is not possible to characterize the stationary maps of the Jac functional as the solutions of a certain Euler-Lagrange equation, and no PDE techniques are allowed to study their regularity. Therefore, the theory of multiple valued Jacobi fields must be completely variational. Hence, we give the following definition. Definition 0.2. Let Ω Σ ֒ be a Lipschitz open set. A map N Γ1,2( Ω) is said ⊂ → M ∈ Q N to be a Jac-minimizer, or a Jacobi Q-field in Ω, if it minimizes the Jacobi functional among all Q-valued W1,2 sections of the normal bundle of Ω in having the same trace at the M boundary, that is 1,2 Jac(N,Ω) Jac(u,Ω) for all u Γ ( Ω) such that u = N . (0.2) ≤ ∈ Q N |∂Ω |∂Ω We are now ready to state the main theorems of this paper. They develop the theory of Jacobi Q-fields along three main directions, concerning existence, regularity and estimate of the singular set. Theorem 0.3 (Conditional existence). Let Ω be an open and connected subset of Σ ֒ → M with C2,β boundary. Assume that the following strict stability condition is satisfied: the only Q-valued Jacobi field N in Ω such that N = Q 0 is the null field N Q 0 . Then, for ∂Ω 0 | ≡ any g Γ1,2( Ω) such that g W1,2(∂Ω; (RM)) there is a Jacobi Q-field N such that ∈ Q N |∂Ω ∈ AQ N = g . J K J K ∂Ω ∂Ω | | Remark 0.4. Notice that the above result strongly resembles the classical “Fredholm alter- native” condition for solving linear elliptic boundary value problems: the solvability of the 1,2 minimum problem for the Jac functional in Γ ( Ω) for any given boundary datum g as Q N in the statement is guaranteed whenever Ω does not admit any non-trivial Jacobi Q-field vanishing at the boundary. MULTIPLE VALUED JACOBI FIELDS 5 Theorem 0.5 (Regularity). Let Ω Σ be an open subset, with Σ ֒ as above. There ⊂ → M 1,2 exists a universal constant α = α(n,Q) (0,1) such that if N Γ ( Ω) is Jac-minimizing ∈ ∈ Q N then N C0,α(Ω; (RM)). ∈ loc AQ The statement of the next theorem requires the definition of regular and singular points of a Jacobi Q-field. Definition 0.6 (Regular and singular set). Let N Γ1,2( Ω) be Jac-minimizing. A point ∈ Q N p Ω is regular for N (and we write p reg(N)) if there exists a neighborhood B of p in Ω ∈ ∈ and Q classical Jacobi fields Nl: B RM such that → Q N(x) = Nl(x) x B ∀ ∈ l=1 X and either Nl Nl′ or Nl(x) = Nl′(x) foJr all xK B, for any l,l′ 1,...,Q . The singular ≡ 6 ∈ ∈ { } set of N is defined by sing(N) := Ω reg(N). \ Theorem 0.7 (Estimate of the singular set). Let N be a Q-valued Jacobi field in Ω Σn. ⊂ Then, the singular set sing(N) is relatively closed in Ω. Furthermore, if n = 2, then sing(N) is at most countable; if n 3, then the Hausdorff dimension dim sing(N) does not exceed H ≥ n 2. − Theorems 0.3, 0.5 and 0.7 have a counterpart in Almgren’s theory of Dir-minimizing mul- tiple valued functions (cf. Theorems 1.24 and 1.26 below). The existence result for Jacobi Q-field is naturally more difficult than its Dirichlet counterpart, because in general the space of Q-valued W1,2 sections of Σ with bounded Jacobi energy is not weakly compact. There- N fore, the proof of Theorem 0.3 requires the development of an extension result (cf. Corollary 4.3) for multiple valued Sobolev functions defined on the boundary of an open subset of Σ to a tubular neighborhood, which eventually allows one to exploit the strict stability condition in order to gain the desired compactness. In turn, such an extension theorem is obtained as a corollary of a multivalued version of the celebrated Luckhaus’ Lemma, cf. Proposition 4.1. The proof of Theorem 0.5 is obtained from the Hölder regularity of Dir-minimizing Q- valued functions by means of a perturbation argument. Finally, the estimate of the Hausdorff dimension of the singular set of a Jac-minimizer, Theorem 0.7, relies on its Dirichlet coun- terpart once that we have shown that the blow-ups of a Jacobi Q-field at a collapsed point are non-trivial homogeneous Dir-minimizing functions, see Theorem 6.21. In turn, the proof of the Blow-up Theorem is based on a delicate asymptotic analysis of an Almgren’s type frequency function, which is shown to be almost monotone and bounded at every collapsed point. This is done by providing fairly general first variation integral identities satisfied by the Jac-minimizers. This note is organized as follows. In Section 1 we fix the terminology and notation that will be used throughout the paper and we recap the main results of the theory of multiple valued functions. Section 2 contains the derivation of the second variation formula generated by a Q-valued section of Σ which leads to the definition of the Jac functional. In section 3 N we investigate the first elementary properties of the Jac functional, we show that it is lower semi-continuous with respect to W1,2 weak convergence (cf. Proposition 3.3) and we study the strict stability condition mentioned in the statement of Theorem 0.3 (cf. Lemma 3.4). Section 4 contains the proof of Theorem 0.3. The proof of Theorem 0.5 (and actually of a 6 SALVATORESTUVARD quantitative version of it including an estimate of the α-Hölder seminorm, cf. Theorem 5.1) is contained in Section 5, whereas Section 6 is dedicated to the proof of Theorem 0.7. 0.2. Aknowledgements. Theauthor is warmly thankfulto Camillo De Lellis for suggesting him to study this problem, and for his precious guidance and support; and to Guido De Philippis, Francesco Ghiraldin and Luca Spolaor for several useful discussions. The research of S.S. has been supported by the ERC grant agreement RAM (Regularity for Area Minimizing currents), ERC 306247. 1. Notation and preliminaries 1.1. The geometric setting. We start immediately specifying the geometric environment and fixing the notation that will be used throughout the paper. We will denote by a M closed (i.e. compact with empty boundary) m-dimensional Riemannian manifold, and by Σ a compact oriented minimal submanifold (possibly with boundary) of the ambient manifold of dimension n = m k. We will assume C3,β regularity for both and Σ, for some M − M β (0,1). ∈ Without loss of generality, we will regard as an isometrically embedded submanifold of M some Euclidean space RM, with K := M m being the codimension of in RM. − M The Euclidean scalar product in RM is denoted , . The metric on and Σ is induced h· ·i M by the flat metric in RM: therefore, the same symbol will also denote the scalar product between tangent vectors to or to Σ. M The tangent space to at a point z will be denoted T . The maps pM: RM T z z z M M → M and pM⊥: RM T⊥ denote orthogonal projections of RM onto the tangent space to z z → M M at z and its orthogonal complement in RM respectively. If x Σ, the tangent space T x ∈ M can be decomposed into the direct sum T =T Σ T⊥Σ, x x x M ⊕ where T⊥Σ is the orthogonal complement of T Σ in T . At each point x Σ, we define x x x M ∈ orthogonal projections p : T T Σ and p⊥: T T⊥Σ. x x x x x x M→ M→ This decomposition at the level of the tangent spaces induces an orthogonal decomposition at the level of the tangent bundle, namely = Σ Σ, TM T ⊕N where Σ denotes the normal bundle of Σ in . N M If f: Σ Rq is a C1 map and ξ is a vector field tangent to Σ, the symbol D f will denote ξ → the directional derivative of f along ξ, that is d D f(x):= (f γ) ξ dt ◦ t=0 whenever γ = γ(t) is a C1 curve on Σ with γ(0) = x an(cid:12)(cid:12)d γ˙(0) = ξ(x). The differential of f at x Σ will be denoted Df : we recall that this is the linear operator Df : T Σ Rq such x x x ∈ | | → that Df ξ(x) = D f(x) for any tangent vector field ξ. The notation Df(x) will sometimes x ξ | · be used in place of Df . Moreover, the derivative along ξ of a scalar function f: Σ R will x | → be sometimes simply denoted by ξ(f). The symbol , instead, will identify the Levi-Civita connection on . If ξ and X are ∇ M tangent vector fields to Σ, then for every x Σ we have ∈ X(x) = p X(x)+p⊥ X(x) =: ΣX(x)+A (ξ(x),X(x)), ξ x ξ x ξ ξ x ∇ ·∇ ·∇ ∇ MULTIPLE VALUED JACOBI FIELDS 7 where Σ is the Levi-Civita connection on Σ and A is the 2-covariant tensor with values in ∇ Σ defined by A (X,Y) := p⊥ Y for any x Σ, for any X,Y T Σ. A is called the x x X x N ·∇ ∈ ∈ second fundamental form of the embedding Σ ֒ by some authors (cf. [Sim83b, Section → M 7], where the tensor is denoted B, or [Lee97, Chapter 8], where the author uses the notation II) and we will use the same terminology, although in the literature in differential geometry (above all when working with embedded hypersurfaces, that is in case the codimension of the submanifold is k = 1) it is sometimes more customary to call A “shape operator” and to use “second fundamental form” for scalar products h(X,Y) = A(X,Y),η with a fixed normal h i vector field η (cf. [dC92, Chapter 6, Section 2]). Observethat,sincewehaveassumedΣtobeminimalin ,themeancurvatureH := tr(A) M is everywhere vanishing on Σ. The curvature endomorphism of the ambient manifold is denoted by R: we recall that M this is a tensor field on of type (3,1), whose action on vector fields is defined by M R(X,Y)Z := Z Z Z, X Y Y X [X,Y] ∇ ∇ −∇ ∇ −∇ where [X,Y] is the Lie bracket of the vector fields X and Y. Recall also that the Riemann tensor can be defined by setting Rm(X,Y,Z,W) := R(X,Y)Z,W h i for any choice of the vector fields X,Y,Z,W, and that the Ricci tensor is the trace of the curvature endomorphism with respect to its first and last indices, that is Ric(X,Y) is the trace of the linear map Z R(Z,X)Y. 7→ Observe that Σ has a natural structure of metric measure space: for any pair of points x,y Σ, d(x,y) will be their Riemannian geodesic distance, while measures and integrals ∈ will be computed with respect to the n-dimensional Hausdorff measure n defined in the H ambient space RM. Boldface characters will always be used to denote quantities which are related to theRiemannian geodesic distance: for instance, if x Σand r is a positive number, ∈ B (x) is the geodesic ball with center x and radius r, namely the set of points y Σ such r ∈ that d(y,x) < r. In the same fashion, if U and V are two subsets of Σ we will set dist(U,V):= inf d(x,y) : x U,y V . { ∈ ∈ } Finally, constants will be usually denoted by C. The precise value of C may change from linetolinethroughoutacomputation. Moreover, wewillwriteC(a,b,...)or C to specify a,b,... that C depends on previously introduced quantities a,b,.... 1.2. Multiple valued functions. In this subsection, we briefly recall the relevant defini- tions and properties concerning Q-valued functions. First introduced by Almgren in his groundbreaking Big Regularity Paper [Alm00], multiple valued functions have proved them- selves to be a fundamental tool to tackle the problem of the interior regularity of area min- imizing integral currents in codimension higher than one. The interested reader can see [DLS11] for a simple, complete and self-contained reference for Almgren’s theory of multi- ple valued functions, [DS15] for a nice presentation of their link with integral currents, and [DL16, De 15] for a nice survey of the strategy adopted in [DLS14, DLS16a, DLS16b] to re- visit Almgren’s program and obtain a much shorter proof of his celebrated partial regularity resultfor areaminimizingcurrentsin highercodimension. Otherremarkablereferenceswhere the theory of Dir-minimizing multiple valued functions plays a major role include the papers 8 SALVATORESTUVARD [DSS15a, DSS15b, DSS15c], where the authors investigate the regularity of suitable classes of almost-minimizing two-dimensional integral currents. 1.2.1. The metric space of Q-points. From now on, let Q 1 be a fixed positive integer. ≥ Definition 1.1 (Q-points). The space of Q-points in the Euclidean space RM is denoted (RM) and defined as follows: Q A Q (RM) := T = T : T RM for every l = 1,...,Q , (1.1) Q l l A  ∈   Xl=1  where T is the Dirac mass δ centeJredKat the point T RM. Hence, every Q-point T is in l Tl l ∈  fact a purely atomic measure of mass Q in RM. For the sake of notational simplicity, we will sometimes write instead of (RM) if J K Q Q A A there is no chance of ambiguity. The space (RM) has a natural structure of complete separable metric space. Q A Definition 1.2. If T = T and S = S , then the distance between T and S is denoted l l (T,S) and given by G P P Q J K J K (T,S)2 := min T S 2, (1.2) l σ(l) G σ∈PQl=1| − | X where is the group of permutations of 1,...,Q . Q P { } 1.2.2. Q-valued maps. Given an open subset Ω Σ, continuous, Lipschitz, Hölder and mea- ⊂ surable functions u: Ω (RM) can be straightforwardly defined taking advantage of the Q → A metric space structure of both the domain and the target. As for the spaces Lp(Ω; ), Q A 1 p , they consist of those measurable maps u: Ω Q(RM) such that u Lp := ≤ ≤ ∞ → A k k (u,Q 0 ) is finite. We will systematically use the notation u := (u,Q 0 ), so that Lp(Ω) kG k | | G u p = upd n J K k kLp ˆ | | H J K Ω for 1 p < and ≤ ∞ u L∞ = esssup u. k k | | Ω In spite of this notation, we remark here that, when Q > 1, (RM) is not a linear space: Q A thus, in particular, the map T T is not a norm. 7→ | | Any measurable Q-valued function can be thought as coming together with a measurable selection, as specified in the following proposition. Proposition 1.3 (Measurable selection, cf. [DLS11, Proposition 0.4]). Let B Σ be a n- ⊂ H measurable set and u: B (RM) be a measurable function. Then, there exist measurable Q → A functions u ,...,u : B RM such that 1 Q → Q u(x) = u (x) for a.e. x B. (1.3) l ∈ l=1 X It is possible to introduce a notion ofJdifferKentiability for multiple valued maps. Definition 1.4 (Differentiable Q-valued functions). A map u: Ω (RM) is said to be Q → A differentiable at x Ω if there exist Q linear maps λ : T Σ RM satisfying: l x ∈ → MULTIPLE VALUED JACOBI FIELDS 9 (i) (u(exp (ξ)),T u(ξ)) = o(ξ ) as ξ 0 for any ξ T Σ, where exp is the exponen- x x x G | | | | → ∈ tial map on Σ and Q T u(ξ) := u (x)+λ ξ ; (1.4) x l l · l=1 X (ii) λl = λl′ if ul(x) = ul′(x). J K Wewillusethenotation Du (x)forλ ,andformallysetDu(x) = Du(x) : observethat l l l l onecanregardDu(x)asanelementof (RM×n)assoonasabasisofT Σhasbeenfixed. For Q x A P any ξ T Σ, we define the directional derivative of u along ξ to be D u(x) := Du(x) ξ , x ξJ K l l ∈ · and establish the notation D u= D u . ξ l ξ l P Differentiable functions enjoy a chain rule formula. J K P Proposition 1.5 (Chain rules, cf. J[DLSK11, Proposition 1.12]). Let u: Ω (RM) be Q → A differentiable at x . 0 (i) Consider Φ: Ω˜ Ω such that Φ(y ) = x , and assume that Φ is differentiable at y . 0 0 0 → Then, u Φ is differentiable at y and 0 ◦ Q D(u Φ)(y )= Du (x ) DΦ(y ) . (1.5) 0 l 0 0 ◦ · l=1 X (ii) Consider Ψ: Ωx RMp Rq such thatJΨ is differentiableKat the point (x0,ul(x0)) for × → every l. Then, the map Ψ(x,u): x Ω Q Ψ(x,u (x)) (Rq) fulfills (i) of ∈ 7→ l=1 l ∈ AQ Definition 1.4. Moreover, if also (ii) holds, then P Q J K DΨ(x,u)(x ) = D Ψ(x ,u (x ))+D Ψ(x ,u (x )) Du (x ) . (1.6) 0 x 0 l 0 p 0 l 0 l 0 · l=1 X (iii) Consider a map F: (RMJ)Q Rq with the property that for any choiKce of Q points → (y ,...,y ) (RM)Q then 1 Q ∈ F(y ,...,y ) = F(y ,...,y ) 1 Q σ(1) σ(Q) for any permutation σ . Then, if F is differentiable at (u (x ),...,u (x )) the Q 1 0 Q 0 ∈ P composition F u (which is a well defined function Ω Rq because, thanks to the ◦ → above property, F is a well defined map on the quotient (RM) = (RM)Q/ ) is Q Q A P differentiable at x and 0 Q D(F u)(x )= D F(u (x ),...,u (x )) Du (x ). (1.7) ◦ 0 yl 1 0 Q 0 · l 0 l=1 X Rademacher’s theorem extends to the Q-valued setting, as shown in [DLS11, Theorem 1.13]: Lipschitz Q-valued functions are differentiable n-almost everywhere in the sense of H Definition 1.4. Moreover, for a Lipschitz Q-valued function the decomposition result stated in Proposition 1.3 can be improved as follows. Proposition 1.6 (Lipschitz selection, cf. [DS15, Lemma 1.1]). Let B Σ be measurable, ⊂ and assume u: B (RM) is Lipschitz. Then, there are a countable partition of B in Q → A measurable subsets B (i N) and Lipschitz functions ul: B RM (l 1,...,Q ) such i ∈ i i → ∈ { } that (a) u = Q ul for every i N, and Lip(ul) Lip(u) for every i,l; |Bi l=1 i ∈ i ≤ P J K 10 SALVATORESTUVARD (b) for every i N and l,l′ 1,...,Q , either ul ul′ or ul(x) = ul′(x) x B ; ∈ ∈ { } i ≡ i i 6 i ∀ ∈ i (c) for every i one has Du(x) = Q Dul(x) for a.e. x B . l=1 i ∈ i 1.2.3. Push-forwardthroughmultiplePvaluedfunctionsofC1 submanifolds. Ausefulfact,which J K willindeedbethestartingpointof ouranalysis of multivalued normalvariations of Σin , is M that it is possible to push-forward C1 submanifolds of the Euclidean space through Q-valued Lipschitz functions. Before giving the rigorous definition of a Q-valued push-forward, it will be useful to introduce some further notation. We will assume the reader to be familiar with the basic concepts and notions related to the theory of currents: standard references on this topicincludethetextbooks[Sim83b]and[KP08],andthemorecompletetreatise[Fed69]. The spaceofsmoothandcompactly supporteddifferentialn-formsinRM willbedenoted n(RM), D andT(ω)willbetheaction ofthen-currentT onω n(RM). IfT isacurrent, then∂T and ∈ D M(T) are its boundary and its mass respectively. If B RM is n-rectifiable with orientation ⊂ ξ~and multiplicity Θ L1(B;Z), then the integer rectifiable current T associated to the triple ∈ B,ξ~,Θ will be denoted T = B,ξ~,Θ . In particular, if Σ RM is an n-dimensional C1 ⊂ o(cid:16)riented(cid:17)submanifold with finite n-measure and orientation ξ~=ξ ξ (thus ξ~(x) is a 1 n H ∧···∧ continuous unit n-vector field onJΣ withK(ξ )n an orthonormal frame of the tangent bundle i i=1 Σ), and B Σ is a measurable subset, then we will simply write B instead of the more T ⊂ cumbersome B,ξ~,1 to denote the current associated to B. We remark that the action of B on a form ω n(RM) is given by J K ∈ D J K B (ω):= ω(x),ξ~(x) d n(x). J K ˆ h i H B Inparticular, then-current ΣJ isKobtained byintegration ofn-formsover Σintheusualsense of differential geometry: Σ (ω)= ω. 1 Since we will always deal with compact manifolds, Σ we continue to assume that Σ is c´ompact, in order to avoid some technicalities which are J K instead necessary when dealing with the non-compact case (see [DS15, Definition 1.2]). J K Definition 1.7 (Q-valued push-forward, cf. [DS15, Definition 1.3]). Let Σ be as above, B Σ a measurable subset and u: B (RM) a Lipschitz map. Then, the push-forward Q ⊂ → A of B through u is the current T := (ul) B , where B and ul are as in Proposition 1.6: u i,l i ♯ i i i that is, P Q J K T (ω):= ω ul(x) ,Dul(x) ξ~(x) d n(x) ω n(RM), (1.8) u ˆ i i ♯ H ∀ ∈ D Xi∈NXl=1 BiD (cid:16) (cid:17) E where Dul(x) ξ~(x) := D ul(x) D ul(x) for a.e. x B . i ♯ ξ1 i ∧···∧ ξn i ∈ i Itisstraightforward, usingthepropertiesof theLipschitz decomposition outlined inPropo- sition 1.6 and recalling the standard theory of rectifiable currents (cf. [Sim83b, Section 27]) and the area formula (cf. [Sim83b, Section 8]), to conclude the following proposition. Proposition 1.8 (Representation of the push-forward, cf. [DS15, Proposition 1.4]). The definition of the action of T in (1.8) does not depend on the chosen partition B , nor on the u i 1Observe that this convention is coherent with the use of JPK, P ∈ RM, to denote the Dirac delta δP, conceived as 0-dimensional current in RM.