How To Prove That A Multplcty Map Is A Natural Map

ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS CAILLO DE LELLIS AND EANUELE SPADARO Abstract. We prove several results on Almgren s multple valued functons and ther lnks to ntegral currents. In partcular, we gve a smple proof of the fact that a Lpschtz multple valued map naturally defnes an nteger rectfable current; we derve explct formulae for the boundary, the mass and the frst varatons along certan specfc vectorfelds; and explot ths connecton to derve a delcate reparametrzaton property for multple valued functons. These results play a crucal role n our new proof of the partal regularty of area mnmzng currents [5, 6, 7]. 0. Introducton It s known snce the poneerng work of Federer and Flemng [10] that one can naturally assocate an nteger rectfable current to the graph of a Lpschtz functon n the Eucldean space, ntegratng forms over the correspondng submanfold, endowed wth ts natural orentaton. It s then possble to derve formulae for the boundary of the current, ts mass and ts frst varatons along smooth vector-felds. oreover, all these formulae have mportant Taylor expansons when the current s suffcently flat. In ths paper we provde elementary proofs for the correspondng facts n the case of Almgren s multple valued functons (see [4] for the relevant defntons). The connecton between multple valued functons and ntegral currents s crucal n the analyss of the regularty of area mnmzng currents for two reasons. On the one hand, t provdes the necessary tools for the approxmaton of currents wth graphs of multple valued functon. Ths s a fundamental dea for the study of the regularty of mnmzng currents n the classcal sngle-vaued case, and t also plays a fundamental role n the proof of Almgren s partal regularty result (cf. [1, 5]). In ths perspectve, explct expressons for the mass and the frst varatons are necessary to derve the rght estmates on the man geometrc quanttes nvolved n the regularty theory (cf. [5, 6, 7]). On the other hand, the connecton can be exploted to nfer nterestng conclusons about the multple valued functons themselves. Ths pont of vew has been taken frutfully n many problems for the case of classcal functons (see, for nstance, [11, 12] and the references theren), and has been recently exploted n the multple valued settng n [3, 14]. The prototypcal example of nterest here s the followng: let f : R m Ω R n be a Lpschtz map and Gr(f) ts graph. If the Lpschtz constant of f s small and we change coordnates n R m+n wth an orthogonal transformaton close to the dentty, then the set Gr(f) s the graph of a Lpschtz functon f over some doman Ω also n the new system of coordnates. In fact t s easy to see that there exst sutable maps Ψ and Φ such that f(x) = Ψ ( x, f(φ(x)) ). In the multple valued 1

2 CAILLO DE LELLIS AND EANUELE SPADARO case, t remans stll true that Gr(f) s the graph of a new Lpschtz map f n the new system of coordnates, but we are not aware of any elementary proof of such statement, whch has to be much more subtle because smple relatons as the one above cannot hold. It turns out that the structure of Gr(f) as ntegral current gves a smple approach to ths and smlar ssues. Several natural estmates can then be proved for f, although more nvolved and much harder. The last secton of the paper s dedcated to these questons; more careful estmates obtaned n the same ven wll also be gven n [6], where they play a crucal role. ost of the conclusons of ths paper are already establshed, or have a counterpart, n Almgren s monograph [1], but we are not always able to pont out precse references to statements theren. However, also when ths s possble, our proofs have an ndependent nterest and are n our opnon much smpler. ore precsely, the materal of Sectons 1 and 2 s covered by [1, Sectons 1.5-1.7], where Almgren deals wth general flat chans. Ths s more than what s needed n [5, 6, 7], and for ths reason we have chosen to treat only the case of nteger rectfable currents. Our approach s anyway smpler and, nstead of relyng, as Almgren does, on the ntersecton theory of flat chans, we use rather elementary tools. For the theorems of Secton 3 we cannot pont out precse references, but Taylor expansons for the area functonal are ubqutous n [1, Chapters 3 and 4]. The theorems of Secton 4 do not appear n [1], as Almgren seems to consder only some partcular classes of deformatons (the squeeze and squash, see [1, Chapter 5]), whle we derve farly general formulas. Fnally, t s very lkely that the conclusons of Secton 5 appear n some form n the constructon of the center manfold of [1, Chapter 4], but we cannot follow the ntrcate arguments and notaton of that chapter. In any case, our approach to reparametrzons of multple valued maps seems more flexble and powerful, capable of further applcatons, because, as t was frst realzed n [4], we can use tools from metrc analyss and metrc geometry developed n the last 20 years. Acknowledgments The research of Camllo De Lells has been supported by the ERC grant agreement RA (Regularty for Area nmzng currents), ERC 306247. The authors are warmly thankful to Bll Allard for several enlghtenng conversatons and hs constant enthusastc encouragement; and very grateful to Luca Spolaor and atteo Focard for carefully readng a prelmnary verson of the paper and for ther very useful comments. Camllo De Lells s also very thankful to the Unversty of Prnceton, where he has spent most of hs sabbatcal completng ths and the papers [5, 6, 7]. 1. Q-valued push-forwards We use the notaton, for: the eucldean scalar product, the naturally nduced nner products on p-vectors and p-covectors and the dualty parng of p-vectors and p-covectors; we nstead restrct the use of the symbol to matrx products. Gven a C 1 m-dmensonal submanfold Σ R N, a functon f : Σ R k and a vector feld X tangent to Σ, we denote by D X f the dervatve of f along X, that s D X f(p) = (f γ) (0) whenever γ s a smooth curve on Σ wth γ(0) = p and γ (0) = X(p). When k = 1, we denote by f the vector feld tangent to Σ such that f, X = D X f for every tangent vector feld X. For general

ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 3 k, Df x : T x Σ R k wll be the lnear operator such that Df x X(x) = D X f(x) for any tangent vector feld X. We wrte Df for the map x Df x and sometmes we wll also use the notaton Df(x) n place of Df x. Havng fxed an orthonormal base e 1,... e m on T x Σ and lettng (f 1,..., f k ) be the components of f, we can wrte f = m =1 a e and Df for the usual Hlbert-Schmdt norm: m k Df 2 = D e f 2 = f 2 = a 2. =1 =, All the notaton above s extended to the dfferental of Lpschtz multple valued functons at ponts where they are dfferentable n the sense of [4, Defnton 1.19]: although the defnton n there s for eucldean domans, ts extenson to C 1 submanfolds Σ R N s done, as usual, usng coordnate charts. We wll keep the same notaton also when f = Y s a vector feld,.e. takes values n R N, the same Eucldean space where Σ s embedded. In that case we defne addtonally dv Σ Y := D e Y, e. oreover, when Y s tangent to Σ, we ntroduce the covarant dervatve D Σ Y x,.e. a lnear map from T x Σ nto tself whch gves the tangental component of D X Y. Thus, f we denote by p x : R N T x Σ the orthogonal proecton onto T x Σ, we have D Σ Y x = p x DY (x). It follows that D Σ Y X = X Y, where we use for the connecton (or covarant dfferentaton) on Σ compatble wth ts structure as Remannan submanfold of R N. Such covarant dfferentaton s then extended n the usual way to general tensors on Σ. When dealng wth C 2 submanfolds Σ of R N we wll denote by A the followng tensor: A x as a blnear map on T x Σ T x Σ takng values on T x Σ (the orthogonal complement of T x Σ) and f X and Y are vector felds tangent to Σ, then A(X, Y ) s the normal component of D X Y, whch we wll denote by DX Y. A s called second fundamental form by some authors (cf. [13, Secton 7], where the tensor s denoted by B) and we wll use the same termnology, although n dfferental geometry t s more customary to call A shape operator and to use second fundamental form for scalar products A(X, Y ), η wth a fxed normal vector feld (cf. [8, Chapter 6, Secton 2] and [15, Vol. 3, Chapter 1]). In addton, H wll denote the trace of A (.e. H = A(e, e ) where e 1,..., e m s an orthonormal frame tangent to Σ) and wll be called mean curvature. 1.1. Push-forward through multple valued functons of C 1 submanfolds. In what follows we consder an m-dmensonal C 1 submanfold Σ of R N and use the word measurable for those subsets of whch are H m -measurable. Any tme we wrte an ntegral over (a measurable subset of) Σ we understand that ths ntegral s taken wth respect to the H m measure. We recall the followng lemma whch, even f not stated explctely n [4], s contaned n several arguments theren. Lemma 1.1 (Decomposton). Let Σ be measurable and F : A Q (R n ) Lpschtz. Then there are a countable partton of n bounded measurable subsets ( N) and Lpschtz functons f : R n ( {1,..., Q}) such that (a) F = Q =1 f for every N and Lp(f ) Lp(F ), ;

4 CAILLO DE LELLIS AND EANUELE SPADARO (b) N and, {1,..., Q}, ether f f or f (x) f (x) x ; (c) we have DF (x) = Q =1 Df (x) for a.e. x. Proof. The proof s by nducton on Q. For Q = 1 t s obvous. Assume the statement for any Q < Q and fx F and. Note that, wthout loss of generalty, we can assume that s bounded. We set 0 := {x : y = y(x) R n wth F (x) = Q y }. Clearly, 0 s measurable because t s the countermage of a closed subset of A Q (R n ). oreover, y : 0 R n s Lpschtz. We then set f 0 = y for every {1,..., Q}. Next, consder x 0. By [4, Proposton 1.6] there exst a ball B x, two postve numbers Q 1 and Q 2, and two Lpschtz Q l -valued functons g l : B x A Ql (R n ) for l = 1, 2 such that F Bx = g 1 + g 2. We can apply the nductve hypothess to g 1 and g 2, and conclude that F Bx can be reduced to the form as n (a) and (b) when restrcted to a (sutably chosen) countable partton of B x nto measurable sets. Snce Σ s paracompact, we can fnd a countable cover {B x } of \ 0, from whch (a) and (b) follow. The last statement can be easly verfed at every Lebesgue pont x where F and all the f s are dfferentable. When F : Σ R n s a proper Lpschtz functon and Σ R N s orented, the current S = F n R n s gven by S(ω) = ω(f (x)), DF (x) e(x) dh m (x) ω D m (R n ), where e(x) = e 1 (x)... e m (x) s the orentng m-vector of Σ and DF (x) e = (DF x e 1 )... (DF x e m ), (cf. [13, Remark 26.21(3)]; as usual D m (Ω) denotes the space of smooth m-forms compactly supported n Ω). Usng the Decomposton Lemma 1.1 t s possble to extend ths defnton to multple valued functons. To ths purpose, we gve the defnton of proper multple valued functons. Defnton 1.2 (Proper Q-valued maps). A measurable F : A Q (R n ) s called proper f there s a measurable selecton F 1,..., F Q as n [4, Defnton 1.1] (.e. F = F ) such that (F ) 1 (K) s compact for every compact K R n. It s then obvous that f there exsts such a selecton, then every measurable selecton shares the same property. We warn the reader that the termnology mght be slghtly msleadng, as the condton above s effectvely stronger than the usual properness of maps takng values n the metrc space (A Q (R n ), G), even when F s contnuous: the standard noton of properness would not ensure the well-defnton of the multple-valued push-forward. Defnton 1.3 (Q-valued push-forward). Let Σ R N be a C 1 orented manfold, Σ a measurable subset and F : A Q (R n ) a proper Lpschtz map. Then, we defne the push-forward of through F as the current T F =, (f ), where and f are

ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 5 as n Lemma 1.1: that s, T F (ω) := Q ω(f (x)), Df (x) e(x) dh m (x) N =1 } {{ } T (ω) ω D m (R n ). (1.1) We frst want to show that T s well-defned. Snce F s proper, we easly deduce that T (ω) Lp(F ) ω H m ((f ) 1 )(spt(ω)) <. On the other hand, upon settng F (x) := f (x) for x, we have (f ) 1 (spt(ω)) = (F ) 1 (spt(ω)) and (f ) 1 (spt(ω)) (f ) 1 (spt(ω)) = for, thus leadng to Q T (ω) Lp(F ) ω H m ((F ) 1 (spt(ω))) < +., =1 Therefore, we can pass the sum nsde the ntegral n (1.1) and, by Lemma 1.1, get Q T F (ω) = ω(f l (x)), DF l (x) e(x) dh m (x) ω D m (R n ). (1.2) l=1 In partcular, recallng the standard theory of rectfable currents (cf. [13, Secton 27]) and the area formula (cf. [13, Secton 8]), we have acheved the followng proposton. Proposton 1.4 (Representaton of the push-forward). The defnton of the acton of T F n (1.1) does not depend on the chosen partton nor on the chosen decomposton {f }, (1.2) holds and, hence, T F s a (well-defned) nteger rectfable current gven by T F = (Im(F ), Θ, τ) where: (R1) Im(F ) = x spt(f (x)) = Q N =1 f ( ) s an m-dmensonal rectfable set; (R2) τ s a Borel untary m-vector orentng Im(F ); moreover, for H m -a.e. p Im(F ), we have Df (x) e(x) 0 for every,, x wth f (x) = p and τ(p) = ± Df (x) e(x) Df (x) e(x) ; (1.3) (R3) for H m -a.e. p Im(F ), the (Borel) multplcty functon Θ equals Df Θ(p) := τ, (x) e(x) Df (x). e(x),,x:f (x)=p 1.2. Push-forward of Lpschtz submanfolds. As for the classcal push-forward, Defnton 1.3 can be extended to domans Σ whch are Lpschtz submanfolds usng the fact that such Σ can be chopped nto C 1 peces. Recall ndeed the followng fact. Theorem 1.5 ([13, Theorem 5.3]). If Σ s a Lpschtz m-dmensonal orented submanfold, then there are countably many C 1 m-dmensonal orented submanfolds Σ whch cover H m -a.s. Σ and such that the orentatons of Σ and Σ concde on ther ntersecton.

6 CAILLO DE LELLIS AND EANUELE SPADARO Defnton 1.6 (Q-valued push-forward of Lpschtz submanfolds). Let Σ R N be a Lpschtz orented submanfold, Σ a measurable subset and F : A Q (R n ) a proper Lpschtz map. Consder the {Σ } of Theorem 1.5 and set F := F Σ. Then, we defne the push-forward of through F as the nteger rectfable current T F := T F. The aboved defnton can be extended to Q-valued pushforwards of general rectfable currents n a straghtforward way: however ths wll never be used n the papers [5, 6, 7] and thus goes beyond the scope of our work. The followng concluson s a smple consequence of Theorem 1.5 and classcal arguments n geometrc measure theory (cf. [13, Secton 27]). Lemma 1.7. Let, Σ and F be as n Defnton 1.6 and consder a Borel untary m- vector e orentng Σ. Then T F s a well-defned nteger rectfable current for whch all the conclusons of Proposton 1.4 hold. As for the classcal push-forward, T F s nvarant under blpschtz change of varables. Lemma 1.8 (Blpschtz nvarance). Let F : Σ A Q (R n ) be a Lpschtz and proper map, Φ : Σ Σ a blpschtz homeomorphsm and G := F Φ. Then, T F = T G. Proof. The lemma follows trvally from the correspondng result for classcal push-forwards (see [9, 4.1.7 & 4.1.14]), the Decomposton Lemma 1.1 and the defnton of Q-valued pushforward. We wll next use the area formula to compute explctely the mass of T F. Followng standard notaton, we wll denote by JF (x) the Jacoban determnant of DF,.e. the number DF (x) e = det((df (x)) T DF (x)) Lemma 1.9 (Q-valued area formula). Let Σ, and F = F be as n Defnton 1.6. Then, for any bounded Borel functon h : R n [0, [, we have h(p) d T F (p) h(f (x)) JF (x) dh m (x). (1.4) Equalty holds n (1.4) f there s a set of full measure for whch DF (x) e(x), DF (y) e(y) 0 x, y and, wth F (x) = F (y). (1.5) If (1.5) holds the formula s vald also for bounded real-valued Borel h wth compact support. Proof. Let h : R n [0, [ be a Borel functon. Consder a decomposton as n the Decomposton Lemma 1.1 and the nteger rectfable currents T of (1.1). By the classcal area formula, see [13, Remark 27.2], we have h(p) d T (p) h(f (x))jf (x) dhm (x). (1.6) Summng ths nequalty over and and usng Lemma 1.1(c), we easly conclude (1.4). When (1.5) holds, we can choose τ of Proposton 1.4 such that the dentty (1.3) has always the + sgn. Defne Θ (p) := H 0 ({x : f (x) = p}. We then conclude from Proposton

ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 7 1.4(R3) that Θ(p) =, Θ (p) for H m -a.e. p Im(F ). On the other hand, agan by [13, Remark 27.2], equalty holds n (1.6) and, moreover, we have the denttes T = Θ H m Im(f ), T F = ΘH m Im(F ). Ths easly mples the second part of the lemma and hence completes the proof. A partcular class of push-forwards are gven by graphs. Defnton 1.10 (Q-graphs). Let Σ, and f = f be as n Defnton 1.6. Defne the map F : A Q (R N+n ) as F (x) := Q =1 (x, f (x)). T F s the current assocated to the graph Gr(f) and wll be denoted by G f. Observe that, f Σ, f and F are as n Defnton 1.10, then the condton (1.5) s always trvally satsfed. oreover, when Σ = R m the well-known Cauchy-Bnet formula gves m (JF ) 2 = 1 + (det A) 2, k=1 A k (DF ) where k (B) denotes the set of all k k mnors of the matrx B. Lemma 1.9 gves then the followng corollary n the case of Q-graphs Corollary 1.11 (Area formula for Q-graphs). Let Σ = R m, R m and f be as n Defnton 1.10. Then, for any bounded compactly supported Borel h : R m+n R, we have ( m ) 1 h(p) d G f (p) = h(x, f (x)) 1 + (det A) 2 2 dx. (1.7) 2. Boundares k=1 A k (DF ) In the classcal theory of currents, when Σ s a Lpschtz manfold wth Lpschtz boundary and F : Σ R N s Lpschtz and proper, then (F Σ ) = F Σ (see [9, 4.1.14]). Ths result can be extended to multple-valued functons. Theorem 2.1 (Boundary of the push-forward). Let Σ be a Lpschtz submanfold of R N wth Lpschtz boundary, F : Σ A Q (R n ) a proper Lpschtz functon and f = F Σ. Then, T F = T f. The man buldng block s the followng small varant of [4, Homotopy Lemma 1.8]. Lemma 2.2. There s c(q, m) > 0 such that, for every closed cube C R m centered at x 0 and every F Lp(C, A Q (R n )), we can fnd G Lp(C, A Q (R n )) satsfyng: () G C = F C =: f, Lp(G) c Lp(F ) and G(F, G) L c Lp(F ) dam(c); () there are Lpschtz mult-valued maps G and f (wth {1,..., J}) such that G = J =1 G, f = J =1 f and G G = (x 0, a ) G f for some a R n. Proof. The proof of () s contaned n [4, Lemma 1.8]. Concernng (), the proof s contaned n the nductve argument of [4, Lemma 1.8], t suffces to complement the arguments there wth the followng fact: f C = [ 1, 1] m, u Lp( C, A Q (R n )) and

8 CAILLO DE LELLIS AND EANUELE SPADARO G(x) = ( ) x x u s the cone-lke extenson of u to C (where x = sup x x ), then G G = 0 G u. The proof of ths clam s a smple consequence of the Decomposton Lemma 1.1 and the very defnton of G u. Consder, ndeed, a countable measurable partton = C and Lpschtz functons u wth u = u. Accordng to our defntons, G u =, (u ) =:, T. Consder now for each the set R := {λx : x, λ ]0, 1]} and defne G (λx) := λu (x) for every x and λ ]0, 1]. The sets R are a measurable decomposton of C \ {0} and we have G = G R. Therefore, settng S := (G ) R, we have G G =, S. On the other hand, by the classcal theory of currents S = 0 T (see [9, Secton 4.1.11]). Snce ((S ) + (T )) <, the desred clam follows. Proof of Theorem 2.1. The proof s by nducton on the dmenson m. Snce every Lpschtz manfold can be trangulated and the statement s nvarant under blpschtz homemorphsms, t suffces to prove the theorem when Σ = [0, 1] m. Next, gven a classcal Lpschtz map Φ : R N R k, let Φ F be the multple-valued map Φ(F ) (cf. [4, Secton 1.3.1]). If F s a classcal Lpschtz map, then T Φ F = Φ F Σ = Φ T F (cf. [9, 4.1.14]). The same dentty holds for Q-valued map, as the Decomposton Lemma 1.1 easly reduces t to the sngle-valued case. Then, f p : R m R m+n R m+n s the orthogonal proecton on the second components, we have p G F = T F. Gven the classcal commutaton of boundary and (sngle-valued) push-forward (see [9, Secton 4.1.14]) we are then reduced to provng he dentty G F = G f. We turn therefore to the case G F. The startng step m = 1 s an obvous corollary of the Lpschtz selecton prncple [4, Proposton 1.2]. Indeed, for F Lp([0, 1], A Q (R n )), there exst functons F Lp([0, 1], R n ) such that F = F. Therefore, T F = T F and T F = T F = ( F (1) F (0) ) = T f. For the nductve argument, consder the dyadc decomposton at scale 2 l of [0, 1] m : [0, 1] m = Q k,l, wth Q k,l = 2 l (k + [0, 1] m ). k {0,...,2 l 1} m In each Q k,l, let u k,l be the cone-lke extenson gven by Lemma 2.2 of f k,l := F Qk,l. Denote by u l the Q- functon on [0, 1] m whch concdes wth u k,l on each Q k,l. Obvously the u l s are equ-lpschtz and converge unformly to F by Lemma 2.2 (). Set T l := G ul = k G u k,l. By the nductve hypothess G fk,l = 0. Snce ( p T ) = T p T (see [13, Secton 26]), Lemma 2.2 mples G uk,l = G fk,l. Consderng that the boundary faces common to adacent cubes come wth opposte orentatons, we conclude T l = G f. By Corollary 1.11, lm sup l ((T l ) + ( T l )) < and so the compactness theorem for ntegral currents (see [13, Theorem 27.3]) guarantees the exstence of an ntegral current T whch s the weak lmt of a subsequence of {T l } (not relabeled). It suffces therefore to show that:

ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 9 (C) f Ω R m s an open set and u l s a sequence of Lpschtz Q-valued maps on Ω such that u l converge unformly to some F and T l := G ul converge to an ntegral current T, then T = G F. We wll prove (C) by nducton over Q: the case Q = 1 s classcal (see for nstance [11, Theorem 2, Secton 3.1 n Chapter 3] and [11, Proposton 2, Secton 2.1 n Chapter 3]). We assume (C) holds for every Q < Q and want to prove t for Q. Fx a sequence as n (C). Clearly T s supported n the rectfable set Gr(F ). Fx an orthonormal bass e 1,..., e m of R m and extend t to an orthonormal bass of R m+n wth postve orentaton. Set e = e 1... e m. Thanks to the Lpschtz regularty of F, Gr(F ) can be orented by m-planes τ wth the property that τ, e c > 0, where the constant c depends on Lp(F ). We have T = (Gr(F ), τ, Θ) and G F = (Gr(F ), τ, Θ): we ust need to show that Θ = Θ H m -a.e. on Gr(F ). As observed n Lemma 1.1 there s a closed set 0 and a Lpschtz functon f 0 such that: F (x) = Q f 0 (x) for every x 0 ; F splts locally on Ω = Ω\ 0 nto (Lpschz) functons takng less than Q values. Usng the nducton hypothess, t s trval to verfy that T Ω R n = G F Ω R n. Thus we ust need to show that Θ(x, f 0 (x)) = Θ(x, f 0 (x)) for H m -a.e. x 0. Consder the orthogonal proecton p : R m+n R m. By the well-known formula for the pusforward of currents (see [9, Lemma 4.1.25]), we have p T = Θ Ω and p G F = Θ Ω, where Θ (x) = Θ(x, y) and Θ (x) = Θ(x, y). (x,y) Gr(F ) (x,y) Gr(F ) Therefore Θ (x) = Θ(x, f 0 (x)) and Θ (x) = Θ(x, f 0 (x)) for H m -a.e. x 0. On the other hand, by the defnton of G F and the very same formula for the push-forward (.e. [9, Lemma 4.1.25]) t s easy to see that p G F = Q Ω = p T l. Snce p T l converges to p T, we conclude that Θ Q Θ H m -a.e. on Ω, whch n turn mples Θ(x, f 0 (x)) = Θ(x, f 0 (x)) for a.e. x 0. Ths completes the proof of the nductve step. 3. Taylor expanson of the area functonal In ths secton we compute the Taylor expanson of the area functonal n several forms. To ths am, we fx the followng notaton and hypotheses. Assumpton 3.1. We consder the followng: () an open submanfold R m+n of dmenson m wth H m () <, whch s the graph of a functon ϕ : R m Ω R n wth ϕ C 3 c; A and H wll denote, respectvely, the second fundamental form and the mean curvature of ; (U) a regular tubular neghborhood U of,.e. the set of ponts {x + y : x, y T x, y < c 0 }, where the thckness c 0 s suffcently small so that the nearest pont proecton p : U s well defned and C 2 ; the thckness s supposed to be larger than a fxed geometrc constant;

10 CAILLO DE LELLIS AND EANUELE SPADARO (N) a Q-valued map F : A Q (R m+n ) of the form Q Q F (x) = x + N (x), =1 =1 where N : A Q (R m+n ) satsfes x + N (x) U, N (x) T x for every x and Lp(N) c. We recall the notaton η F := 1 Q F, for every multple valued functon F = F. Theorem 3.2 (Expanson of (T F )). If, F and N are as n Assumpton 3.1 and c s smaller than a geometrc constant, then (T F ) = Q H m () Q η N + H, 1 DN 2 2 ( ) + P 2 (x, N ) + P 3 (x, N, DN ) + R 4 (x, DN ), (3.1) where P 2, P 3 and R 4 are C 1 functons wth the followng propertes: () n P 2 (x, n) s a quadratc form on the normal bundle of satsfyng P 2 (x, n) C A(x) 2 n 2 x, n T x ; (3.2) () P 3 (x, n, D) = L (x, n)q (x, D), where n L (x, n) are lnear forms on the normal bundle of and D Q (x, D) are quadratc forms on the space of (m + n) (m + n)-matrces, satsfyng L (x, n) Q (x, D) C A(x) n D 2 x, n T x, D ; () R 4 (x, D) = D 3 L(x, D), for some functon L wth Lp(L) C, whch satsfes L(x, 0) = 0 for every x and s ndependent of x when A 0. oreover, for any Borel functon h : R m+n R, ( ) h d T F h F C A h F N + h ( DN 2 + A N 2 ), (3.3) and, f h(p) = g(p(p)) for some g, we have h d T F Q (1 H, η N + 1 2 DN 2 ) g C ( A 2 N 2 + DN 4) g. (3.4) In partcular, as a smple corollary of the theorem above, we have the followng. Corollary 3.3 (Expanson of (G f )). Assume Ω R m s an open set wth bounded measure and f : Ω A Q (R n ) a Lpschtz map wth Lp(f) c. Then, (G f ) = Q Ω + 1 Df 2 + R 4 (Df ), (3.5) 2 where R 4 C 1 satsfes R 4 (D) = D 3 L(D) for L wth Lp( L) C and L(0) = 0. Ω Ω

ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 11 Proof. The corollary s reduced to Theorem 3.2 by smply settng = Ω {0}, N = N (x) := (0, f (x)) and F (x) = F (x) = (x, f (x)). Snce n ths case A vanshes, (3.1) gves precsely (3.5). Proof of Theorem 3.2. We wll n fact prove the statement for (T F V ), where V s any Borel subset of. Under ths generalty, by the decomposton Lemma 1.1, t s enough to consder the case F V = G, where each G = F V = x + N V s a (one-valued!) Lpschtz map. Next observe that (1.5) obvously holds f c s suffcently small. Therefore, (T F V ) = ((F ) V ), and, snce η N = 1 Q N, the formula (3.1) follows from summng the correspondng denttes ((F ) V ) = H m (V ) + H, N + 1 DN 2 V 2 V ( ) + P 2 (x, N ) + P 3 (x, N, DN ) + R 4 (x, DN ). (3.6) V To smplfy the notaton we drop the subscrpt n the proof of (3.6). Usng the area formula, we have that (F V ) = DF ξ dh m, where ξ = ξ 1... ξ m s the smple m-vector assocated to an orthonormal frame on T. By smple multlnear algebra DF ξ = det, where s the m m matrx gven by k = DF ξ, DF ξ k = ξ + DN ξ, ξ k + DN ξ k V = δ k + DN ξ, ξ k + DN ξ k, ξ + DN ξ, DN ξ k. (3.7) } {{ } } {{ } a k b k Set a = (a k ), b = (b k ) and denote by 2 (a + b) and 3 (a + b), respectvely, the sum of all 2 2 and that of all 3 3 mnors of the matrx (a + b); smlarly denote by R(a + b) the sum of all k k mnors wth k 4. Then, det = 1 + tr (a + b) + 2 (a + b) + 3 (a + b) + R(a + b). (3.8) Observe that the entres of a are lnear n DN and those of b are quadratc. Thus, 2 (a + b) = 2 (a) + 2 (b) + C 2 (a, b), (3.9) 3 (a + b) = 3 (a) + C 4 (a, b), (3.10) where C 2 (a, b) s a lnear combnaton of terms of the form a k b lm and C 4 (a, b) s a polynomal n the entres of DN satsfyng the nequalty C 4 (a, b) C DN 4. Recall the

12 CAILLO DE LELLIS AND EANUELE SPADARO Taylor expanson 1 + τ = 1 + τ τ 2 + τ 3 + g(τ), where g s an analytc functon wth 2 8 16 g(τ) τ 4. Wth the ad of (3.8), (3.9) and (3.10) we reach the followng concluson: DF ξ tr (a + b) + 2 (a) + C 2 (a, b) + 3 (a) = 1 + + 2 (tr a)2 + 2 tr a tr b + 2 tr a 2 (a) (tr a)3 + + R 4, (3.11) 8 16 where R 4 s an analytc functon of the entres of DN whch satsfes R 4 (DN) C DN 4. Observe next that tr b = k DN ξ k, DN ξ k = DN 2. oreover, DN ξ, ξ k = ξ ( N, ξ k ) N, ξ ξ k = N, A(ξ, ξ k ). Thus, by the symmetry of the second fundamental form, we have a k = 2 A(ξ, ξ k ), N and tr a = 2 H, N. We then can rewrte DF ξ DN 2 = 1 H, N + + 2(a) (tr a)2 2 } 2 {{ 8 } P 2 + + C 2(a, b) + 3 (a) tr a (tr b + 2(a)) (tr a)3 + } 2 {{ 4 16 } +R 4. (3.12) P 3 Integratng (3.12) we reach (3.6). It remans to show that P 2, P 3 and R 4 satsfy (), () and (). If A = 0, then s flat and the frame ξ 1,..., ξ m can be chosen constant, so that R 4 wll not depend on x. Next, each b k s a quadratc polynomal n the entres of DN, wth coeffcents whch are C 2 functons of x. Instead each a k can be seen as a lnear functon n DN wth coeffcents whch are C 2 functons of x, but also as a lnear functon L k of N, wth a C 1 dependence on x. In the latter case we have the bound L k (x, n) A(x) n. Therefore the clams n () and () follow easly. Fnally, snce R 4 s an analytc functon of the entres of DN satsfyng R 4 (DN) C DN 4, the representaton n () follows from the elementary consderaton that R 4(D) s a Lpschtz functon vanshng at the orgn. D 3 Fnally, observe that the argument above mples (3.4) when g s the ndcator functon of any measurable set and the general case follows from standard measure theory. The dentty (3.3) follows easly from the same formulas for DF ξ, usng ndeed cruder estmates. 3.1. Taylor expanson for the excess n a cylnder. The last results of ths secton concerns estmates of the excess n dfferent systems of coordnates, n partcular wth respect to tlted planes and curvlnear coordnates. Proposton 3.4 (Expanson of a curvlnear excess). There exst a dmensonal constant C > 0 such that, f, F and N are as n Assumpton 3.1 wth c small enough, then T F (x) (p(x)) 2 d T F (x) DN 2 C ( A 2 N 2 + DN 4 ), (3.13)

ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 13 where T F and are the unt m-vectors orentng T F and T, respectvely. Proof. Let p and defne (p) = ξ 1... ξ m for some orthonormal frame ξ 1,..., ξ m for T and T F (F (p)) = ζ ζ wth ζ = (ξ 1 + DN p ξ 1 )... (ξ m + DN p ξ m ). Our assumptons mply p(f (p)) = p. Usng the Q-valued area formula and obvous computatons we get T F p 2 d T F (x) = ζ ζ 2 ( ζ = 2 ζ ) ζ,. As already computed n the proof of Theorem 3.2, ζ = Q Q H, η N + DN 2 2 + O( A 2 N 2 + DN 4 ). If we next defne Bk := ξ, ξ k + DN ξ k = δ k N, A(ξ, ξ k ), we then get ζ, = det B = Q Q H, η N + O( A 2 N 2 ). Hence the clamed formula follows easly. Next we compute the excess of a Lpschtz graph wth respect to a tlted plane. Theorem 3.5 (Expanson of a cylndrcal excess). There exst dmensonal constants C, c > 0 wth the followng property. Let f : R m A Q (R n ) be a Lpschtz map wth Lp (f) c. For any 0 < s, set A := B s D(η f) and denote by τ the orented untary m-dmensonal smple vector to the graph of the lnear map y A y. Then, we have G f τ 2 d G f G(Df, Q A ) 2 C Df B s B 4. (3.14) s C s Proof of Theorem 3.5. Argung as n the prevous proofs, thanks to Lemma 1.1, we can wrte f = f and process local computatons (when needed) as f each f were Lpschtz. oreover, we have that τ = ξ ξ wth ξ = (e 1 + A e 1 )... (e m + A e m ). Here and for the rest of ths proof, we dentfy R m and R n wth the subspaces R m {0} and {0} R n of R m+n, respectvely: ths ustfes the notaton e + A e for e R m and A e R n. Next, we recall that ξ = ξ, ξ = det(δ + A e, A e ) = 1 + 1 2 A 2 + O( A 4 ).

14 CAILLO DE LELLIS AND EANUELE SPADARO By Corollary 1.11 we also have E := G f τ 2 d G f = 2 (G f ) 2 G f, τ d G f C s = 2 Q B s + ( Df 2 + O( Df 4 )) 2 (e 1 + Df e 1 )... (e m + Df e m ), τ. B s On the other hand A e, e k = 0 = Df e, e k. Therefore, (e 1 + Df e 1 )... (e m + Df e m ), τ = ξ 1 det(δ k + Df e, A e k ) ) 1 = (1 + A 2 2 + ( O( A 4 ) 1 + Df : A + O( Df 2 A 2 ) ). Recallng that A Cs m Df C ( s m Df 4) 1 4, we then conclude ( ) E = Df 2 + Q B s A 2 2 Df : A + O Df 4 B s B s B s ( ) ( ) = Df A 2 + O Df 4 = G(Df, Q A ) 2 + O Df 4. B s B s B s B s 4. Frst varatons In ths secton we compute the frst varatons of the currents nduced by multple valued maps. These formulae are ultmately the lnk between the statonarty of area mnmzng currents and the partal dfferental equatons satsfed by sutable approxmatons. We use here the followng standard notaton: gven a current T n R N and a vector feld X C 1 (R N, R N ), we denote the frst varaton of T along X by δt (X) := d dt t=0 (Φ t T ), where Φ :] η, η[ U R N s any C 1 sotopy of a neghborhood U of spt(t ) wth Φ(0, x) = x for any x U and d dε ε=0 Φ ε = X (n what follows we wll often use Φ ε for the map x Φ(ε, x)). It would be more approprate to use the notaton δt (Φ) (see, for nstance, [9, Secton 5.1.7]), but snce the currents consdered n ths paper are rectfable, t s well known that the frst varaton depends only on X and s gven by the formula δt (X) = dv T X d T, (4.1) where dv T X = D e X, e for any orthonormal frame e 1,..., e m wth e 1... e m = T (see [9, 5.1.8] and cf. [13, Secton 2.9]). We begn wth the expanson for the frst varaton of graphs. In what follows, A : B wll denote the usual Hlbert Schmdt scalar product of two k matrces.

ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 15 Theorem 4.1 (Expanson of δg f (X)). Let Ω R m be a bounded open set and f : Ω A Q (R n ) a map wth Lp(f) c. Consder a functon ζ C 1 (Ω R m, R n ) and the correspondng vector feld χ C 1 (Ω R n, R m+n ) gven by χ(x, y) = (0, ζ(x, y)). Then, δg ( ) f(χ) Dx ζ(x, f ) + D y ζ(x, f ) Df : Df C Dζ Df 3. (4.2) Ω The next two theorems deal wth general T F as n Assumpton 3.1. However we restrct our attenton to outer and nner varatons, where we borrow our termnology from the elastcty theory and the lterature on harmonc maps. Outer varatons result from deformatons of the normal bundle of whch are the dentty on and map each fber nto tself, whereas nner varatons result from composng the map F wth sotopes of. Theorem 4.2 (Expanson of outer varatons). Let, U, p and F be as n Assumpton 3.1 wth c suffcently small. If ϕ C 1 c () and X(p) := ϕ(p(p))(p p(p)), then δt F (X) = where Err 3 C ( ϕ DN 2 + ) (N Dϕ) : DN Q ϕ H, η N + } {{ } Err 1 Err 2 C Ω 3 Err (4.3) =2 ϕ A 2 N 2 (4.4) ( ϕ ( DN 2 N A + DN 4) + Dϕ ( DN 3 N + DN N 2 A )). (4.5) Let Y be a C 1 vector feld on T wth compact support and defne X on U settng X(p) = Y (p(p)). Let {Ψ ε } ε ] η,η[ be any sotopy wth Ψ 0 = d and d dε ε=0 Ψ ε = Y and defne the followng sotopy of U: Φ ε (p) = Ψ ε (p(p)) + (p p(p)). Clearly X = d dε ε=0 Φ ε. Theorem 4.3 (Expanson of nner varatons). Let, U and F be as n Assumpton 3.1 wth c suffcently small. If X s as above, then ( DN 2 δt F (X) = dv Y ) 3 DN : (DN D Y ) + Err, (4.6) 2 =1 where Err 3 C ( Err 1 = Q H, η N dv Y + D Y H, η N ), (4.7) Err 2 C A ( 2 DY N 2 + Y N DN ), (4.8) ( Y A DN 2( N + DN ) + DY ( A N 2 DN + DN 4)). (4.9)

16 CAILLO DE LELLIS AND EANUELE SPADARO 4.1. Proof of Theorem 4.1. Set Φ ε (x, y) := (x, y + ε ζ(x, y)). For ε suffcently small Φ ε s a dffeomorphsm of Ω R n d nto ntself. oreover, Φ dε ε ε=0 = χ. Let f ε = f + ε ζ(x, f ). Snce (Φ ε ) G f = G fε, we can apply Corollary 3.3 to compute δg f (χ) = d (G fε ) (3.5) = d 1 ( D(f + ε ζ) 2 + dε ε=0 dε ε=0 2 R 4 (D(f + ε ζ)) ) ( ) d = Q Dx ζ(x, f ) + D y ζ(x, f ) Df : Df + R4 (Df + εdζ). dε ε=0 Snce R 4 () = 3 L() for some Lpschtz L wth L(0) = 0, we can estmate as follows: d R4 ( + εζ) dε ε=0 CL() 2 Dζ + C 3 Lp(L) Dζ C 3 Dζ, thus concludng the proof. 4.2. Proof of Theorem 4.2. Consder the map Φ ε (p) = p + εx(p). If ε s suffcently small, Φ ε maps U dffeomorphcally n a neghborhood of and we obvously have δt F (X) = d ((Φ dε ε) T F ) ε=0. Next set F ε (x) = x + N (x)(1 + ε ϕ(x)) and observe that (Φ ε ) T F = T Fε. Thus we can apply Theorem 3.2 to get: ( δt F (X) = ϕ DN 2 + ) (N Dϕ) : DN Q ϕ H, η N } {{ } =:Err 1 d + P 2 (x, N (1 + εϕ)) dε ε=0 } {{ } =:Err 2 d ( + P3 (x, N (1 + εϕ), D(N (1 + εϕ))) + R 4 (x, D(N (1 + εϕ))) ). dε ε=0 } {{ } =:Err 3 Snce n P 2 (x, n) s a quadratc form, we have P 2 (x, N (1 + εϕ)) = (1 + εϕ) 2 P 2 (x, N ) and thus (4.4) follows from (3.2). Next, by Theorem 3.2(), we have the bound d P 3 (x, N (1 + εϕ), D(N (1 + εϕ))) dε ε=0 C A(x) ( Dϕ N 2 DN + ϕ N DN 2). Fnally, takng nto account Theorem 3.2(): d R 4 (x, D(N (1 + εϕ))) dε ε=0 C ( DN 3 + DN 3 Lp(L) ) ( N Dϕ + DN ϕ). Puttng together the last two nequaltes we get (4.5).

ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 17 4.3. Proof of Theorem 4.3. Set F ε (x) = x + N (Ψ 1 ε (x)). Clearly, Φ ε T F = T Fε. Fx an orthonormal frame e 1,..., e m on T and let e = e 1... e m. By Lemma 1.9, δt F (X) = d (T Fε ) = d (DF ε, ) e. dε ε=0 dε ε=0 Fx {1,..., Q}. Usng the chan rule [4, Proposton 1.12], we have: (DF ε, ) e = w 1 (ε, x)... w m (ε, x) =: w(ε, x), where w (ε, x) = e (x) + DN Ψ 1 ε (x) DΨ 1 ε x e (x). Set v (ε, x) = w (ε, Ψ ε (x)). Snce Ψ 0 s the dentty, we obvously have v(0, ) = DF e. If we denote by JΨ ε (x) the Jacoban determnant of the transformaton Ψ ε, we can change varable n the ntegral to conclude: d (DF ε, ) e = d v(ε, x) JΨ ε (x) dε ε=0 dε ε=0 = (DF ) e d JΨ ε + v(0, x) 1 ε v(0, x), v(0, x) dε ε=0 = (DF ) e dv Y + ε v(0, x), (DF ) e } {{ } } {{ } I,1 I,2 + ε v(0, x), (DF ) e ( DF e 1 1 ). } {{ } I,3 Thus, δt F (X) = I,1 + I,2 + I,3 =: I 1 + I 2 + I 3 and we wll next estmate these three terms separately. Step 1. Estmate on I 1. By the Q-valued area formula of Lemma 1.9 and (3.4) n Theorem 3.2, I 1 = Q dv Y + 1 DN 2 dv Y Q H, η N dv Y + Err 2 where Err C ( A 2 N 2 + DN 4 ) dv Y. Snce dv Y = 0 (recall that Y Cc 1 ()), we easly conclude that I 1 = 1 3 DN 2 dv Y Q H, η N dv Y + Err, (4.10) 2 where the Err s satsfy the estmates (4.8) and (4.9). Step 2. Estmate on I 2. Set ζ (x) := ε v(0, x), (DF ) e = ε v(0, x), v(0, x) = 1 2 =2 d dε v(ε, x) 2. ε=0

18 CAILLO DE LELLIS AND EANUELE SPADARO Snce v(ε, x) 2 s ndependent of the orthonormal frame chosen, havng fxed a pont x, we can mpose D e = 0 at x. By multlnearty ε v(0, x) = v 1 (0, x)... ε v (0, x)... v m (0, x). (4.11) We next compute ε v (0, x) = ( ( ) ) e (Ψ ε (x)) + DN ε x DΨ 1 ε ε=0 ε e ε=0 Ψε(x) (Ψ ε (x)) = D Y e (x) + DN x [Y, e ](x), (4.12) where [Y, e ] s the Le bracket. On the other hand, snce D e (x) = 0, we have D Y e (x) = A(e, Y ) and [Y, e ](x) = e Y (x). Recall that v (0, ) = e + DN e. By the usual computatons n multlnear algebra, t turns out that ζ = det, where the entres of the m m matrx are gven by: αβ = e α + DN e α, e β + DN e β = δ αβ + O( A N ) + O( DN 2 ) for β, α = e α + DN e α, A(e, Y ) DN e Y. (The entres for α are computed as n the proof of Theorem 3.2). Denote by n α the (m 1) (m 1) mnor whch s obtaned by deletng the α row and the column. We then easly get the followng estmates: n C( DN 2 + A N ) for α, (4.13) oreover, observe that α n = 1 + O( DN 2 + A N ). (4.14) α = DN e α, DN e Y e α, DN e Y + A(e α, Y ), DN e = DN e α, DN e Y + A(e α, e Y ), N + A(e α, Y ), DN e. (4.15) We therefore conclude from (4.13), (4.14) and (4.15) that ζ (x) = det = ( 1) +α α n α α = ( DN e, DN e Y + A(e, e Y ), N + A(e, Y ), DN e ) ( + O DY ( DN 4 + A 2 N 2) ) + Y ( A DN 3 + A 2 N DN ). (4.16) Summng over and ntegratng, we then acheve I 2 = DN : (DN D Y ) + J 2 + Err 2 + Err 3, (4.17)

ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 19 where Err 2, Err 3 are estmated as n (4.8), (4.9), and ( J 2 = Q A(e, e Y ), η N + A(e, Y ), D e η N ). In order to treat ths last term, we consder the vector feld Z = A(e, Y ), η N e. Z s ndependent of the choce of the orthonormal frame e : therefore, to compute ts dvergence at a specfc pont x we can assume D e = 0. We then get dv Z = ( ) A(e, Y ), D e η N + De A(e, Y ), η N + A(e, e Y ), η N, where the tensor DX A(U, Y ) s defned as (D X (A(U, Y ))) A( X U, Y ) A(U, X Y ), (recall that (D X W ) denotes the normal component of D X W ). The Codazz-anard equatons (cf. [16, Chapter 7.C, Corollary 15] mply the symmetry of D A. Thus, De A(e, Y ), η N = DY A(e, e ), η N = DY H, η N. (4.18) Summarzng (and recallng that η N s normal to ), dv Z = ( A(e, Y ), D e η N + A(e, e Y ), η N ) + D Y H, η N. (4.19) Snce Z s compactly supported n, ntegratng (4.19) and usng the dvergence theorem we conclude 0 = Q 1 J 2 + D Y H, η N. We thus get I 2 = DN : (DN D Y ) Q D Y H, η N + Err 2 + Err 3. Step 3. Estmate on I 3. From the proof of Theorem 3.2, (cf. (3.11) and (3.12)) we conclude 1 (DF ) e C ( DN 2 + A N ). To show that I 3 can be estmated wth Err 2 and Err 3 observe that, by (4.16) we have ε v(0, x), (DF ) e = ζ (x) C DN 2 DY + C A DY N + C A DN Y. 5. Reparametrzng multple valued graphs In ths secton we explot the lnk between currents and multple valued functons n the opposte drecton, n order to gve condtons under whch Q-valued graphs can be sutably reparametrzed and to establsh relevant estmates on the parametrzaton. We fx the short-hand notaton e = e 1... e m+n, e m = e 1... e m and e n = e m+1... e m+n, where e 1,..., e m, e m+1,..., e m+n s the standard bass of R m R n. We wll often use the notaton π 0 and π 0 for R m {0} and {0} R n.

20 CAILLO DE LELLIS AND EANUELE SPADARO Theorem 5.1 (Q-valued parametrzatons). Let Q, m, n N and s < r < 1. Then, there are constants c 0, C > 0 (dependng on Q, m, n and r ) wth the followng property. Let ϕ, s and U be as n Assumpton 3.1 wth Ω = B s and let f : B r A Q (R n ) be such that ϕ C 2 + Lp(f) c 0 and ϕ C 0 + f C 0 c 0 r. (5.1) Set Φ(x) := (x, ϕ(x)). Then, there are maps F and N as n Assumpton 3.1(N) such that T F = G f U and Lp(N) C ( D 2 ϕ C 0 N C 0 + Dϕ C 0 + Lp(f) ), (5.2) 1 2 Q N(Φ(p)) G(f(p), Q ϕ(p) ) 2 Q N(Φ(p)) p B s, (5.3) η N(Φ(p)) C η f(p) ϕ(p) + CLp(f) Dϕ(p) N(Φ(p)) p B s. (5.4) Fnally, assume p B s and (p, η f(p)) = ξ + q for some ξ and q T ξ. Then, G(N(ξ), Q q ) 2 Q G(f(p), Q η f(p) ). (5.5) For further reference, we state the followng mmedate corollary of Theorem 5.1, correspondng to the case of a lnear ϕ. Proposton 5.2 (Q-valued graphcal reparametrzaton). Let Q, m, n N and s < r < 1. There exst postve constants c, C (dependng only on Q, m, n and r ) wth the followng s property. Let π 0 and π be m-planes wth π π 0 c and f : B r (π 0 ) A Q (π0 ) wth Lp(f) c and f cr. Then, there s a Lpschtz map g : B s (π) A Q (π ) wth G g = G f C s (π) and such that the followng estmates hold on B s (π): g C 0 Cr π π 0 + C f C 0, (5.6) Lp(g) C π π 0 + CLp(f). (5.7) In fact the proof of Theorem 5.1 wll gve a more precse nformaton about the map F, namely ts pontwse values can be determned wth a natural geometrc algorthm. Defnton 5.3 (ultplcty n Q-valued maps). Gven a Q-valued map F, we say that a pont p has multplcty k n F (x) f we can wrte F (x) = k p + Q k =1 p where p p for every,.e. f p has multplcty k when treatng F (x) as a 0-dmensonal ntegral current. Lemma 5.4 (Geometrc reparametrzaton). The values of F n Theorem 5.1 can be determned at any pont p as follows. Let κ be the orthogonal complement of T p. Then, Gr(f) (p + κ) s nonempty, conssts of at most Q ponts and every q Gr(f) (p + κ) has n F (p) the same multplcty of p π 0 (q) n f(p π0 (q)). 5.1. Exstence of the parametrzaton. The next lemma s a natural outcome of the Ambroso-Krchhem approach to the theory of currents [2]. Followng [9, Secton 4.3], f T s a flat m-dmensonal current n U and h : U R k a Lpschtz map wth k m, we denote by T, h, y the slce of T wth respect to h at the pont y (well-defned for a.e. y R k ). Snce we deal wth normal currents, the equvalence of the classcal Federer- Flemng theory and the modern Ambroso-Krchhem theory (cf. [2, Theorem 11.1]) allows us to use all the results of the paper [2].

ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 21 Lemma 5.5. Consder a C 2 nectve open curve γ :]a, b[ R N, l = γ(]a, b[), a regular tubular neghborhood U(l) and the map q := γ 1 p, where p s the assocated C 1 normal proecton p : U(l) l. Let T be an ntegral 1-dmensonal current n U(l) wth T = 0 such that, for a.e. p ]a, b[, the slce F (p) := T, q, p s a sum of Q (not necessarly dstnct) Drac masses P. If the measure µ(a) := T (q 1 (A)) s absolutely contnuous, then F W 1,1 (]a, b[, A Q (R N )) n the sense of [4, Defnton 0.5] and G(F (p), F (p )) Cµ([p, p ]) for a.e. p, p. Proof. Consder the metrc space I 0 of 0-dmensonal ntegral currents endowed wth the flat norm F as defned n [2, Secton 7]. By [2, Proof of Theorem 8.1] the map p F (p) s a I 0 -valued functon of bounded varaton n the sense of [2, Defnton 7.1], that s: there s a countable dense set F I 0 such that, for every S F, the map Φ S (p) := F (S, F (p)) s a real-valued functon of bounded varaton; DΦ S (A) CLp(q) T (q 1 (A)) + C q C 0 T (q 1 (A)) for every Borel set A and a dmensonal constant C. On the other hand, T = 0 and the measure A µ(a) := T (q 1 (A)) s absolutely contnuous wth respect to the Lebesgue measure. By a smple densty argument, t holds Φ S (p) Φ S (q) C µ([p, q]) S I 0 and a.e. p, q ]a, b[. (5.8) Observe that by assumpton F (p) takes values n A Q (R N ) for a.e. p and, for S = S, R = R A Q (R N ), t s well known that F (S, R) = mn π P Q S R σ() G(S, R) CF (S, R). Then, t follows from (5.8) that G(S, F (p)) G(S, F (q)) CF (F (p), F (q)) C µ([p, q]) for every S A Q (R N ). By [4, Defnton 0.5], ths concludes the proof. The lemma can be used to nfer, n a rather straghtforward way, the exstence of the parametrzaton F n Theorem 5.1 Proof of Theorem 5.1: Part I. After rescalng we can assume, wthout loss of generalty, r = 1. Ths also easly shows that the constants depend only on the rato r. We start s wth a procedure to dentfy the Q-valued functon F. By (5.1), G f (B 1 R n ) must be supported n a neghborhood of sze 4 c 0 of Φ(B 1 ). Therefore, f the constant c 0 s chosen accordngly, the boundary of T := G f p 1 () s actually supported n p 1 ( ) and the constancy theorem gves p T = k for some k Z. Frst we show that k = Q. Consder the functons ϕ t := tϕ for t [0, 1], the manfolds t := Gr(ϕ t ) and the correspondng proectons p t. It s smple to verfy that the map ( t S t := (p t ) Gf (p 1 t ( t )) ) s contnuous n the space of currents. The constancy theorem gves S t = Q(t) t for some nteger Q(t) and snce S 0 = Q R m {0}, t follows that S 1 = p T = Q. Defne for smplcty q T q := G f, p, q. The nteger rectfable current G f s represented by the trple (Im(G), τ, Θ) as n Proposton 1.4. The slcng theory gves then the followng propertes for H m -a.e. p (see [9, 4.3.8]):

22 CAILLO DE LELLIS AND EANUELE SPADARO () T p conssts of a fnte sum of Drac masses N p =1 k δ q ; () q Gr(f) and k = Θ(q ) for every ; () f v s the contnuous untary m-vector orentng p 1 (p) compatbly wth the orentaton of, the sgn of k s sgn( T (q ) v(q ), e ). By the bounds on ϕ and f, T (x) s close to e m, whle v s close to e n. Therefore, each k turns out to be postve. On the other hand, snce p T = Q, then k = Q. Ths shows that p F (p) := k q defnes a Q-valued functon. Next we show the Lpschtz contnuty of F. Fx a coordnate drecton n R m, wthout loss of generalty e 1, and consder the map U z Λ(z) := P p(z), where P : R m+n R m 1 s the orthogonal proecton P (x 1,..., x m+n ) = (x 2,..., x m ). Consder the correspondng slce Tȳ := T, Λ, ȳ for ȳ R m 1. For H m 1 -a.e. ȳ P (), Tȳ s a rectfable 1-dmensonal current wth ( Tȳ) U = 0 (see [9, Secton 4.3.1]). If we slce further Tȳ wth respect to the map pȳ := x 1 p, we conclude that for a.e. ȳ and a.e. p lȳ we must have Tȳ, pȳ, p = F (p) (cf. [2, Lemma 5.1]). Applyng the coarea formula to the rectfable set G f shows also that, f c 0 s suffcently small, then T (p 1 ȳ (A)) C A, where C s a geometrc constant (and denotes the Lebesgue 1-dmensonal measure); cf. [9, Theorem 4.3.8]. Defne ]a, b[= {t : (t, ȳ) B s }, l := {ϕ(t, ȳ) : t ]a, b[} and γ(t) := ϕ(t, ȳ) It s easy to see that on spt( Tȳ the map pȳ concdes wth the map q of Lemma 5.5. Therefore the map ]a, b[ F (t, ȳ) s Lpschtz (up to a null-set). Argung n the same way for each coordnate, we conclude that one can redefne F on a set of measure zero n such a way that F s Lpschtz: we wll keep the notaton F for such Lpschtz map. Defne next N(x) = F (x) x. We then see that, by constructon, N satsfes Assumpton 3.1(N). Fx next coordnates on (for nstance usng Φ as chart). By Proposton 1.4 and the bounds on f and ϕ, we deduce that dp, G f c > 0 and dp, T F c > 0, for a sutable geometrc constant c (where we use the notaton dp = dp 1... dp m and p 1,..., p m are the components of p n the partcular chart chosen on ). Hence, f T F G f p 1 (), then necessarly T F dp G f dp, whch s a contradcton to T, p, y = T, p, y for a.e. y (cf. [2, (5.7) and Theorem 5.6]). Part II. To prove (5.2) consder frst pars of ponts p, q wth the followng property: (AE) let σ = Φ([p π0 (p), p π0 (q)]), F σ = F wth each F Lpschtz (cf. [4, Proposton 1.2]), and consder the correspondng curves γ = F (σ): then, for H 1 - a.e. y γ, γ (y) belongs to the tangent plane T y G f. We clam that (AE) mples: N(p) N(q) C( D 2 ϕ C 0 N C 0 + Lp(f) + Dϕ C 0) p π0 (p) p π0 (q). (5.9) By standard measure theoretc arguments, (AE) holds for a set of pars (p, q) of full measure n. Wth a smple densty argument we then conclude the valdty of (5.9) for every par p, q. Denote by d the geodesc dstance on. Snce p π0 (p) p π0 (q) d(p, q), we then conclude the Lpschtz estmate (5.2).

ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 23 Let us turn to (5.9). We parameterze σ by arc-length s : [0, l] σ and for every defne n(t) := F (s(t)) s(t). Clearly, n s Lpschtz and we clam that: n (t) C ( ) D 2 ϕ C 0 n C 0 + Lp(f) + Dϕ C 0 for a.e. t. Observe that s (t)+n (t) = γ s (t)+n (t) (F (s(t))) whch, for a.e. t, belongs to T F (s(t))gr(f). The angle θ between γ (F (s(t)) and the plane p 1 (s(t)) can then be estmated by π 2 θ ( C Lp(f) + Dϕ C 0). (5.10) Let p T and p be the proectons to the tangent and normal planes to n F (s(t)). Then, f c 0 s chosen small enough to have n (t) 1, we get p (n (t)) = p (n (t) + s (t)) = n (t) + s (t) p ( γ (F (s(t))) 2 cos θ (5.10) C ( Lp(f) + Dϕ C 0). (5.11) In order to compute the tangental component, let ν 1,..., ν n be an orthonormal frame on the normal bundle. It can be chosen so that Dν C 0 C D 2 ϕ C 0 for every (see Lemma A.1). From n(t) := λ (t)ν (s(t)), wth λ (t) := n(t) ν (s(t)) Lpschtz functons, we get ( p T (n (t)) = p T λ (t)ν (s(t)) + λ (t) d ) dt ν (s(t)) = ( ) d λ (t)p T dt ν (s(t)), whch mples p T (n (t)) λ C 0 Dν C 0 C n C 0 D 2 ϕ C 0. (5.12) Puttng together (5.12) and (5.11), we get (5.9). 5.2. Valdty of the geometrc algorthm. Before completng the proof of Theorem 5.1 we show Lemma 5.4, whch ndeed wll be used the derve the remanng estmates n Theorem 5.1. Proof of Lemma 5.4.. By the representaton formula n Proposton 1.4, snce the support of the push-forward va a Lpschtz map s the mage of the map and we already proved T F = G f U, we then conclude that Im(F ) = Gr(f) U as sets. Thus, to complete the proof of Lemma 5.4 we ust have to show the rule for determnng the multplcty of a pont q (p + κ) Gr(f) n F (p). Ths rule follows easly from the area formula when Lp(f), Lp(N) and Lp(ϕ) are smaller than a geometrc constant, snce under such assumpton the Taylor expansons for the mass gven by Theorem 3.2 and Corollary 3.3 mply the followng facts: f y has multplcty k n f(x), then k 1 2 lm nf r 0 G f (B ρ ((x, y))) ω m ρ m lm sup r 0 G f (B ρ ((x, y))) ω m ρ m k + 1 2 ;

24 CAILLO DE LELLIS AND EANUELE SPADARO f p has multplcty k n F (x), then k 1 2 lm nf r 0 T F (B ρ (p)) ω m ρ m lm sup r 0 We can now conclude the proof of Theorem 5.1. T F (B ρ (p)) ω m ρ m k + 1 2. Proof of Theorem 5.1: Part III. We frst deal wth (5.3) and (5.4). Observe frst that, thanks to Lemma 5.4, the value of N at the pont (p, ϕ(p)) does not change f we replace ϕ wth ts frst order Taylor expanson. oreover, upon translaton we can further assume p = 0 and ϕ(0) = 0. We moreover fx the notaton π := {(x, Dϕ(x) x) : x π 0 } = T 0 Gr(ϕ) and denote by κ the orthogonal complement of π. Wth a slght abuse of notaton, the same pont p R m+n s then represented by a par (x, y) π 0 π0 and a par (x, y ) π κ. Concernng (5.3), snce the role of the two systems can be reversed, t suffces to show only one nequalty, namely f (0) 2 Q N(0). (5.13) Let f(0) = P, q := p π (P ) and N(q ) = Q,. There s then a () such that (q, Q,() ) π κ s the same pont as (0, P ) π 0 π 0. Observng that q C Dϕ 0 P, we then get P q + Q,() q + N(0) + G(N(0), N(q )) N(0) + (1 + Lp(N)) q N(0) + C(1 + Lp(N)) Dϕ 0 P. (5.14) We use now (5.2) wth ϕ lnear: Lp(N) C( Dϕ 0 + Lp(f)) Cc 0. We thus conclude P N(0) + C(1 + c 0 C)c 0 P. However, the constant C n the last nequalty s only geometrc and does not depend on c 0. Thus, f c 0 s chosen suffcently small, we conclude P 2 N(0). Summng upon, we then reach f(0) 2Q 1 2 N(0). We now pass to (5.4), keepng the assumpton f(0) = P and wrtng N(0) = F (0) = p. Set p π0 (p ) = (x, 0) and p π 0 (p ) = (0, y ). The angle θ between p and p π 0 (p ) s estmated by C Dϕ(0), because the p s are elements of κ. Thus, x p sn θ C Dϕ(0) N(0) =: ρ. (5.15) Consder also that p π 0 : κ π0 s a lnear nvertble map and n fact we can assume that the operator norm of ts nverse, whch we denote by L, s bounded by 2. Thus η N(0) 2 y and t suffces to estmate y P + C Lp(f)ρ. (5.16) To ths am, we notce that, f we set h = Lp(f) ρ, we can decompose f(0) as f(0) = T (where T A Q and Q 1 +... + Q J = Q) so that () d(t ) 4 Q h, where d(s) := max, s s s the dameter of S = s cf. [4]; () z w > 4 h for all z T and w T wth.

ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 25 To prove ths clam we order the P s and partton them n subcollectons T 1,..., T k wth the followng algorthm. T 1 contans P 1 and any other pont P l for whch there exsts a chan P (1),..., P (l) spt(t ) of ponts wth (1) = 1, (l) = l and P (l) P (l 1) 4 h. Clearly d(t 1 ) 4 Q h and f spt(t ) = spt(t 1 ) we are fnshed. Otherwse we use the procedure above to defne T 2 from spt(t ) \ spt(t 1 ), observng that q p > 4 h for any par of elements q spt(t 1 ) and p spt(t ) \ spt(t 1 ). By the choce of the constants, t then follows that the functon f separates nto J Lpschtz functons f : B ρ A Q (R n ) wth f(x) = J =1 f (x) and Lp(f ) Lp(f). Consder the correspondng graphs Gr(f ). Observe that, by the geometrc algorthm, N(0) contans ponts from each of these sets and moreover such ponts have, n N(0), the same multplcty that they have n f. Ths means that the ponts p such that N(0) = p Q l=1 p l. spt(n(0)) there exsts a pont P k(,l) sptf (0) such that y l P k(,l) G(f (p π0 (p l )), f (0)) Lp(f) p π0 (p l ) h. Thus Q J Q y = y l =1 l=1 J Q J P l =1 l=1 + y l P l =1 l=1 Q J ( ) P + y l P k(,l) + P k(,l) P l P + C h. can actually be also grouped n J famles {p 1,..., p Q } so that N(0) = J =1 Note that, by the defnton of the dstance G, for each p l =1 l=1 Fnally, for what concerns (5.5), observe that, wthout loss of generalty, we can assume q = 0 by smply shftng to q+: Lemma 5.4 mples that the map N gven by Theorem 5.1 appled to q + satsfes N (ξ +q) = N (ξ) q and so thus G(N (ξ +q), Q 0 ) = G(N(ξ), q ) Assumng q = 0 we have ξ = (p, η f(p)) = (p, ϕ(p)) and thus the estmate matches the left hand sde of (5.3). Appendx A. Trvalzng normal bundles In ths and the forthcomng papers the followng procedure wll be often used. Consder, ϕ and Φ as n Assumpton 3.1. We then construct a standard orthonormal frame on the normal bundle of as follows: (Tr1) we let e m+1,..., e m+n be the standard orthonormal base of {0} R n ; (Tr2) for any p we let κ p be the orthogonal complement of T p and denote by p κp the orthogonal proecton onto t; (Tr3) for any {1,..., n} and any p we generate the frame ν 1 (p),..., ν n (p) applyng the Gram-Schmdt orthogonalzaton procedure to p κp (e m+1 ),..., p κp (e m+n ). We record then the followng lemma. Lemma A.1 (Trvalzaton of the normal bundle of ). If Dϕ C 0 s smaller than a geometrc constant, then ν 1,..., ν n s an orthonormal frame spannng κ p at every p. Consder ν as functon of x Ω usng the nverse of Φ as chart. For every α + k 0

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