Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane

RECTIFIAB I LITY OF SETS OF FINITE PERI METER IN CARNOT GROUPS: EXISTENCE OF A T ANGENT HYPER PLANE LUIGI AMBR OSIO, BRUCE KLEINER, AND ENRICO LE DONNE Abstract. W e consider sets o f lo cally finite p erimeter in Ca rnot groups. W e show that if E is a set of lo cally finite perimeter in a Carnot group G then, for a lmo st every x ∈ G with resp ect to the per imeter measure o f E , some tangent of E at x is a vertical halfspa ce. This is a partial extension of a theor em of F r anchi-Serapioni-Serra Cassano in step 2 Carnot groups: they show in [18, 19] that, for a lmost every x , E has a unique tangent at x , a nd this tangent is a v ertical halfspac e . 1. Introduction The differen tiabilit y prop erties of functions and the rectifiabilit y prop erties of sets a re classical themes of Real Analysis and G eometric Measure Theory , with many m utual con- nections. In the c on text of stratified Carnot g roups, the first pr oblem has been s olv ed, within the category of L ipsc hitz maps, in a deep work of P ansu [37 ]; here w e are in terested in the second problem, in the clas s of se ts E of lo cally finite p erimeter: if we denote b y X 1 , . . . , X m an orthonormal basis of the horizontal la yer of the Lie algebra g of left- in v aria n t v ector fields of the Carnot group G , this class of sets is defined b y the pro p ert y that the distributional deriv ativ es X 1 1 E , . . . , X m 1 E are represe n ta ble b y Radon measures in G . This notion, whic h extends the classical one deve lop ed and deeply studied b y D e Giorg i in [12] and [13] ( see also [3]), is compatible with the Carnot-Cara th ´ eo dory (subriemannian) dis- tance d induced by X 1 , . . . , X m ; in this contex t the total v ariation | D 1 E | of the R m -v alued measure ( X 1 1 E , . . . , X m 1 E ) pla ys the role of surface measure asso ciated to d . Our interes t in this topic w as also motiv ated by the recen t pap ers [7], [8], whe re sets of finite p erimeter in Carnot groups (and in particular in the Heisen b erg groups) are used to study a new notion of differen tiabilit y for maps with v alues in L 1 , with the aim of finding examples of spaces whic h cannot b e bi- Lipsc hitz em b edded in to L 1 . The first basic prop erties of the class of sets of finite p erime ter (and of B V functions as w ell), suc h as compactness, g lobal and lo cal isop erimetric inequalities, ha ve b een pro ve d in [21]; then, in a series of pap ers [18, 19 ], F ra nc hi, Serapioni and Serra Cassano made a more precise analysis of this class of sets, first in the Heisen b erg groups H n and then in all step 2 Carnot groups (using also some measure-theoretic prop erties pro ved , in a mor e general con t ext, in [1], see also Theorem 4.16). As in the work of De Giorgi, the crucial problem is Date : Octo ber 2 4 , 20 18. 1991 Mathematics Subje ct Classific ation. 28A75; 4 9Q15; 58 C35. Key wor ds and phr ases. Rectifiabilit y , Carno t gro ups, Cacciop oli set, sets of finite p erimeter. The sec o nd author was pa rtially supp orted b y NSF gra nt DMS-07 01515 . 1 2 LUIGI AMBROSIO, BRUCE KLEINER, AND ENRICO LE DONN E the analysis of ta ngen t sets to E at a p oin t ¯ x , i.e. all limits lim i →∞ δ 1 /r i ( ¯ x − 1 E ) where ( r i ) ↓ 0 and con v ergence o ccurs lo cally in me asure ( here δ r : G → G denote t he in t rinsic dilations of the group). In [19] it is prov ed that for | D 1 E | -a.e. ¯ x there exis ts a unit v ector ν E ( ¯ x ) ∈ S m − 1 , that w e shall call horizon tal no rmal, suc h that (1.1) m X i =1 ν E , i ( ¯ x ) X i 1 E ≥ 0 and m X i =1 ξ i ( ¯ x ) X i 1 E = 0 ∀ ξ ⊥ ν E ( ¯ x ) . W e shall call these sets with constan t horizon tal nor mal (iden tified, in the co ordinates relativ e to t he basis X 1 , . . . , X m , b y the v ector ν E ( ¯ x )): the question is whethe r (1.1) implies additional info rmation on the deriv ative of E a long v ector fields Y that do not b elong to the horizon tal lay er: eve n t hough m < n = dim( g ), this can b e exp ected, ha ving in mind that the Lie algebra generated by X 1 , . . . , X m is the whole o f g . The main result o f [19] is the pro of that, in all step 2 groups, (1.1) implies [ X i , X j ] 1 E = 0 for all i, j = 1 , . . . , m . As a conseq uence, up to a left tra nslation E is really , when seen in exp onen tial co ordinates, an halfspace: ( x ∈ R n : m X i =1 ν E , i ( ¯ x ) x i ≥ 0 ) . W e shall call it v ertical halfspace, k eeping in mind that there is no dep endence on the co ordinates x m +1 , . . . , x n . This fact leads to a complete classification of the ta ngen t sets and has relev an t consequence s, as in the classical theory , on the represen tatio n of | D 1 E | in terms of t he spherical Hausdorff measure and on the rectifiabilit y , in a suitable intrinsic sense, of the measure-theoretic b oundary of E , see [19] for more precise informatio ns. On the other ha nd, still in [19], it is pr o v ed that f or general Carnot gro ups the conditions (1.1) do not c haracterize v ertical half spaces: an explicit example is pro vided in a step 3 group of Engel t yp e (see also Section 7). Basically , b ecause of this obstruction, the results of [19] are limited to step 2 groups. The classification and eve n the regularit y prop erties of sets E with a constant horizon tal normal is a c hallenging and, so far, completely op en question. Ho w ev er, recen tly w e fo und a w a y to b ypass this difficult y and, in this pap er, we sho w the follo wing result: Theorem 1.2. Supp ose E ⊆ G has lo c al ly fi n ite p erim eter. Then, for | D 1 E | -a.e. ¯ x ∈ G a vertic al halfsp ac e H b elongs to the tangents to E at ¯ x . Of course Theorem 1.2 do es no t provide ye t a complete solution of the rectifiability prob- lem: indeed, ev en though the direction ν E ( ¯ x ) of the halfspace H dep ends o n ¯ x only , w e kno w that ¯ x − 1 E is close on an infinitesimal sequence o f scales to H , but w e are not able to sho w that this happens on all sufficien tly small scales. What is still miss ing is some monotonicit y/stabilit y argumen t that singles out halfspaces as the o nly p ossible tangents, wherev er they are tangen t (se e a lso the discussion in Remark 5.5 ). In a similar con text, namely the rectifiabilit y of measures ha ving a spherical densit y , this is prec isely the phe- nomenon disco v ered by P reiss in [39]: we to ok some ideas f rom this paper, adapting them to the setting of Carnot groups, to obta in our result. F or t hese reasons, the complete solution T ANGENT HYPERPLA NE I N CARN OT GROUPS 3 of the rectifiabilit y problem seems to b e related to the follo wing question (w e denote by v ol G the Haa r measure of the group and b y e the iden tit y of the group): let E ⊂ G b e a set with a constant horizon tal normal ν ∈ S m − 1 and let H b e a vertical halfspace with the same horizontal normal; if lim inf R → + ∞ v ol G  ( E ∆ H ) ∩ B R ( e )  v ol G  B R ( e )  = 0 , is it true that E is a v ertical halfspace? How ev er, as p ointe d out to us b y Vittone, the answ er to this question is negativ e, see (7.5), so that new ideas seem to b e needed to pro v e the uniqueness, at | D 1 E | -a.e. p oin t, o f the tangen t set. In order to illustrate the main ide as b ehind the pro o f of our result, let us call regular directions of E the vec tor fields Z in the Lie algebra g suc h that Z 1 E is represen table by a Radon measure, and in v arian t directions those for whic h t he measure is 0. Our strategy of pro of rests mainly on the follo wing observ ations: the first one (Prop osition 4.7) is that the adjoin t op erator Ad exp( Y ) : g → g maps regular directions in to regular directions whenev er Y is an in v arian t direction. If X := m X i =1 ν E , i ( ¯ x ) X i ∈ g , w e lo ok at the v ector space spanned b y Ad exp( Y ) ( X ) , as Y v aries among the in v aria n t directions, and use this f act to sho w tha t an y set with constan t horizontal normal m ust ha v e a regular direction Z not b elonging t o the ve ctor space spanned b y the inv arian t directions and X (whic h con tains at least the horizontal lay er). This is prov ed in Prop osition 2.17 in purely geometric terms in g eneral Lie gropus, and Prop o sition 2 .18 provides a more explicit expression o f the new regular directions generated, in Carnot groups, with this pro cedure. Then, the se cond main observ ation is that if a regular direction Z f or a set F has no comp onen t in the horizontal la y er, then the ta ngen ts to F a t ¯ x are in v arian t a long a new direction dep ending on Z for most p oints ¯ x ; this follow s ( Lemma 5.8) by a simple scaling argumen t, taking into accoun t that the L ie algebra dilations δ r shrink more, as r ↓ 0, in the non-horizontal directions. Therefore, at man y p oints, a tangen t to a set with constant horizon tal normal has a new in v ariant direction. Ha ving gained this new direction, this pro cedure can b e restarted: the a djoin t can b e used to generate a new regular direction, then a tangen t will ha v e a new in v ariant direction, and so on. In this w ay we sho w in Theorem 5.2 that, if we iterate the tangen t op erator sufficien tly man y time s (the n um b er dep ending on the Lie algebra stratification only) w e do get a v ertical halfspace. This means t hat w e consider a tangent set E 1 to E at ¯ x , then a ta ngen t E 2 to E 1 at a suitable p o in t ¯ x 1 in the supp ort of | D 1 E 1 | , and so on. A t this stage w e b o rro w some ideas from [39] to conclude that, at | D 1 E | -a.e. p oint ¯ x , iterated tangen ts are tangent to the initial set: this is accomplishe d in Section 6 and leads to the pro of of Theorem 1.2. Ac kno wledgemen ts. W e thank V. Magnani and A. Martini f or some useful commen ts on a preliminary v ersion of this pap er. 4 LUIGI AMBROSIO, BRUCE KLEINER, AND ENRICO LE DONN E 2. Main notions 2.1. V ector fields, div ergence, X -deriv ativ e. Throughout this section, w e will denote b y M a smooth differentiable manifold with top ological dimension n , endo w ed with a n - differen tial v olume form v ol M (ev entually M will b e a L ie gro up G , a nd vol M the rig h t Haar measure). F or x ∈ M , the fib er T x M o f the t angen t bundle T M is a deriv ation of germs of C ∞ functions at x (i.e., an R - linear application from C ∞ ( x ) → R that satisfies the Leibnitz rule). If F : M → N is smo oth and x ∈ M , we shall denote by dF x : T x M → T F ( x ) N its differen tial, defined as follows: the pull back op erator u 7→ F ∗ x ( u ) := u ◦ F maps C ∞ ( F ( x )) in t o C ∞ ( x ); thus, for v ∈ T x M w e ha v e that dF x ( v )( u ) := v ( u ◦ F )( x ) , u ∈ C ∞ ( F ( x )) defines an elemen t of T F ( x ) N . W e denote by Γ( T M ) the linear space of smo oth vec tor fields, i.e. smoot h sections of the tangen t bundle T M ; w e will t ypically use the notation X , Y , Z to denote them. W e use the not ation [ X , Y ] f := X ( Y f ) − Y ( X f ) for the Lie brac k et, that induces on Γ( T M ) an infinite-dimensional Lie alg ebra structure. If F : M → N is smo ot h and in v ertible a nd X ∈ Γ( T M ), the push forw ard ve ctor field F ∗ X ∈ Γ( T N ) is defined by the iden tit y ( F ∗ X ) F ( x ) = dF x ( X x ). Equiv alen tly , (2.1) ( F ∗ X ) u := [ X ( u ◦ F )] ◦ F − 1 ∀ u ∈ C ∞ ( M ) . The push-fo rw ard comm utes with the Lie brac ket, namely (2.2) [ F ∗ X , F ∗ Y ] = F ∗ [ X , Y ] ∀ X, Y ∈ Γ( T M ) . If F : M → N is smo oth and σ is a smo oth curv e on M , then (2.3) dF σ ( t ) ( σ ′ ( t )) = ( F ◦ σ ) ′ ( t ) , where σ ′ ( t ) ∈ T σ ( t ) M and ( F ◦ σ ) ′ ( t ) ∈ T F ( σ ( t )) N a re the tangen t v ector fields along the tw o curv es, in M a nd N . If u ∈ C ∞ ( M ), iden tifying T u ( p ) R with R itself, giv en X ∈ Γ( T M ), w e ha v e du p ( X ) = X p ( u ) . No w w e use the volume form to define the div ergence as follo ws: (2.4) Z M X u d v ol M = − Z M u div X d v ol M ∀ u ∈ C ∞ c ( M ) . When ( M , g ) is a Riemannian manifold and v ol M is the v olume form induced b y g , then an explicit expression of this differen tia l op erator can b e obtained in terms o f the comp onents of X , and (2.4) corresponds to the divergen ce theorem on manifo lds. W e w on’t need either a Riemannian structure o r an explicit expression o f div X in the sequel, and fo r this reason w e ha ve chos en a definition base d o n (2.4): this emphasizes the dep endence of div X on v ol M only . By applying this iden t it y to a dive rgence-free v ector field X , w e obtain (2.5) Z M uX v d vol M = − Z M v X u d v ol M ∀ u, v ∈ C ∞ c ( M ) . T ANGENT HYPERPLA NE I N CARN OT GROUPS 5 This motiv ates the fo llo wing classical definition. Definition 2.6 (X-distributional deriv ativ e) . L et u ∈ L 1 lo c ( M ) and let X ∈ Γ( T M ) b e diver genc e-fr e e. We denote by X u the distribution h X u, v i := − Z M uX v d v ol M , v ∈ C ∞ c ( M ) . If f ∈ L 1 lo c ( M ) , we write X u = f if h X u , v i = R M v f d v ol M for al l v ∈ C ∞ c ( M ) . A nal- o gously, if µ is a R adon me a sur e in M , we write X u = µ if h X u , v i = R M v dµ for al l v ∈ C ∞ c ( M ) . According to (2.5) (still v alid when u ∈ C 1 ( M )), the distributional definition of X u is equiv alen t to the classical one whenev er u ∈ C 1 ( M ). In Euclidean spaces, the X -deriv ativ e of characteris tic functions of nice domains can b e easily compute d ( and o f course the result could b e exte nded to manifolds, but w e w on’t need t his extension). 2.2. X -deriv ativ e of nice functions and domains. If u is a C 1 function in R n , t hen X u can b e calculated as the scalar pro duct betw een X and the gradien t of u : (2.7) X u = h X, ∇ u i . Assume that E ⊂ R n is lo cally the sub-lev el set o f the C 1 function f and that X ∈ Γ( T R n ) is dive rgence-free. Then, for any v ∈ C ∞ c ( R n ) w e can apply the Gauss–Green fo rm ula to the v ector field v X , whose div ergence is X v , to obtain Z E X v dx = Z ∂ E h v X , ν eu E i d H n − 1 , where ν eu E is the unit (Euclidean) outer normal to E . This pro v es that X 1 E = −h X , ν eu E i H n − 1 x ∂ E . Ho w ev er, w e ha ve an explicit form ula for the unit (Euclide an) outer normal to E , it is ν eu E ( x ) = ∇ f ( x ) / |∇ f ( x ) | , so, b y (2.7), h X , ν eu E i = h X , ∇ f |∇ f | i = h X , ∇ f i |∇ f | = X f |∇ f | . Th us (2.8) X 1 E = − X f |∇ f | H n − 1 x ∂ E . 6 LUIGI AMBROSIO, BRUCE KLEINER, AND ENRICO LE DONN E 2.3. Flow of a v ec t or field. G iv en X ∈ Γ( T M ) w e can consider the asso ciated flow , i.e., the solution Φ X : M × R → M of the following ODE (2.9)      d dt Φ X ( p, t ) = X Φ X ( p,t ) Φ X ( p, 0) = p. Notice that the smoo thness of X ensures uniqueness, and therefore the semigroup prop erty (2.10) Φ X ( x, t + s ) = Φ X (Φ X ( x, t ) , s ) ∀ t, s ∈ R , ∀ x ∈ M but not global e xistence; it will b e guaran teed, ho w ev er, in all cases considered in this pap er. W e ob viously ha v e (2.11) d dt ( u ◦ Φ X )( p, t ) = ( X u )(Φ X ( p, t )) ∀ u ∈ C 1 ( M ) . An ob vious consequence o f this iden t it y is t hat, for a C 1 function u , X u = 0 implies that u is constant along the flow, i.e. u ◦ Φ X ( · , t ) = u for all t ∈ R . A similar statemen t holds ev en for distributional deriv ative s along v ector fields: for simplicit y let us state and pro v e this result for divergenc e-free v ector fields only . Theorem 2.12. L et u ∈ L 1 lo c ( M ) b e satisfying X u = 0 in the sense of distributions. Then , for al l t ∈ R , u = u ◦ Φ X ( · , t ) v ol M -a.e. i n M . Pr o of. Let g ∈ C 1 c ( M ); we need to sho w that the map t 7→ R M g u ◦ Φ X ( · , t ) d v ol M is inde- p enden t of t . Indeed, the semigroup prop ert y (2.10), a nd the fact that X is divergenc e-free yield Z M g u ◦ Φ X ( · , t + s ) d vol M − Z M g u ◦ Φ X ( · , t ) d v ol M = Z M ug ◦ Φ X ( · , − t − s ) d vol M − Z M ug ◦ Φ X ( · , − t ) d v ol M = Z M ug ◦ Φ X (Φ X ( · , − s ) , − t ) d v o l M − Z M ug ◦ Φ X ( · , − t ) d v ol M = − s Z M uX ( g ◦ Φ X ( · , − t )) d v ol M + o ( s ) = o ( s ) .  R emark 2.1 3 . W e notice also that the flo w is vol M -measure preserving (i.e. v ol M (Φ X ( · , t ) − 1 ( A )) = v ol M ( A ) for all Borel sets A ⊆ M and t ∈ R ) if and only if div X is equal to 0. Indeed, if f ∈ C 1 c ( M ), the measure preserving prop erty giv es that R M f (Φ X ( x, t )) d v o l M ( x ) is inde- p enden t of t . A time differen tiatio n and (2.11) then giv e 0 = Z M d dt f (Φ X ( x, t )) d v o l M ( x ) = Z M X f (Φ X ( x, t )) d v o l M ( x ) = Z M X f ( y ) d v ol M ( y ) . Therefore R M f div X d v ol M = 0 for all f ∈ C 1 c ( M ), and X is divergenc e-free. The pro o f of the conv erse implication is similar, and analo gous to the one o f Theorem 2.12. T ANGENT HYPERPLA NE I N CARN OT GROUPS 7 2.4. Lie groups. Let G b e a Lie group, i.e. a differentiable n -dimensional manifold with a smo oth gr oup op eration. W e shall denote b y e the iden tity of the group, by R g ( h ) := hg the right translation, and b y L g ( h ) := g h the left translation. W e shall also denote b y v o l G the volume form a nd, at the same time, the righ t-in v arian t Haar measure. F orced to mak e a choice , w e follow the ma jor it y of the literature fo cusing o n the left in v ar ian t v ector fields . i.e. the v ector fields X ∈ Γ( T G ) suc h that ( L g ) ∗ X = X , so that ( dL g ) x X = X L g ( x ) for all x ∈ G . In differen tial terms, w e hav e X ( f ◦ L g )( x ) = X f ( L g ( x )) ∀ x, g ∈ G . Thanks to (2.2) with F = L g , the class of left in v arian t v ector fields is easily se en to be closed under t he Lie brack et, and w e shall denote b y g ⊆ Γ( T G ) the Lie algebr a of left in v ar ian t v ector fields. W e will t ypically use the notations U, V , W to denote subspaces of g . Note that, after fixing a ve ctor v ∈ T e G , w e can construct a left in v arian t v ector field X defining X g := ( L g ) ∗ v for an y g ∈ G . This construction is an isomorphism b et w een the set g of all left in v arian t v ector fields a nd T e G , and pro v es that g is a n -dimensional subspace of Γ( T G ). Let X ∈ g and let us denote, as usual in the theory , by exp( tX ) the flo w of X at t ime t starting from e ( that is, exp( tX ) := Φ X ( e, t ) = Φ tX ( e, 1)); then, the curv e g exp( tX ) is the flo w starting at g : indeed, since X is left inv arian t, setting for simplicit y γ ( t ) := exp( tX ) and γ g ( t ) := g γ ( t ), w e ha v e d dt γ g ( t ) = d dt ( L g ( γ ( t ))) = ( dL g ) γ ( t ) d dt γ ( t ) = ( dL g ) γ ( t ) X = X γ g ( t ) . This implies that Φ X ( · , t ) = R exp( tX ) and so the flow preserv es the rig h t Haar measure, and the left translation preserv es the flow lines. By Remark 2 .13 it fo llo ws that all X ∈ g are divergenc e-free, and Theorem 2.12 g iv es (2.14) f ◦ R exp( tX ) = f ∀ t ∈ R ⇐ ⇒ X f = 0 whenev er f ∈ L 1 lo c ( G ). Before stating the next prop osition, w e recall the definition of the adjoin t. F or k ∈ G , t he conjugation map C k : G → G g 7→ C k ( g ) := k g k − 1 (2.15) is the comp osition of L k with R k − 1 . The adjoint op erator k 7→ Ad k maps G in GL ( g ) as follo ws: Ad k ( X ) := ( C k ) ∗ X , so that Ad k ( X ) f ( x ) = X ( f ◦ C k )( C − 1 k ( x )). (2.16) The definition is w ell p osed b ecause Ad k ( X ) is left in v arian t whenev er X is left in v arian t: for all g ∈ G w e ha ve indeed Ad k ( X ) ( f ◦ L g )( x ) = X ( f ◦ L g ◦ C k )( k − 1 xk ) = X ( f ◦ R k − 1 ◦ L g k )( k − 1 xk ) = X ( f ◦ R k − 1 )( g xk ) . On the other hand Ad k ( X ) f ( L g ( x )) = X ( f ◦ C k )( k − 1 g xk ) = X ( f ◦ R k − 1 )( g xk ) . 8 LUIGI AMBROSIO, BRUCE KLEINER, AND ENRICO LE DONN E Prop osition 2.17. Assume that G is a c onne cte d, s imply c onne cte d nilp otent Lie gr oup. L et g ′ b e a Lie sub algebr a of g satisfying dim( g ′ ) + 2 ≤ dim ( g ) , and ass ume that W := g ′ ⊕ { R X } gener ates the whol e Lie algebr a g for some X / ∈ g ′ . Then, ther e exists k ∈ exp( g ′ ) s uch that Ad k ( X ) / ∈ W . Pr o of. Not e that g ′ is a finite-dimensional sub-alg ebra and that exp is, unde r the simple connectednes s assumption, a homeomorphism, hence K := exp( g ′ ) is a closed (prop er) Lie subgroup o f G . Therefore, w e can consider the quotien t manifold G / K , in fact the homo- geneous space of right cose ts: it consists of the equiv alence classes of G induced b y the relation x ∼ y ⇐ ⇒ y − 1 x ∈ K . W e shall denote b y π : G → G / K the canonical pro jection. The natural top ology of G / K is determined b y the requiremen t that π should b e con tinuous and op en. Let m denote some v ector space of g suc h that g = g ′ ⊕ m . The sub-manifo ld exp( m ) is referred as a lo cal cross section for K at the origin, and it can be used to giv e a differen tiable structure to G / K . In fact, let Z 1 , . . . , Z r b e a basis of m , then the mapping ( x 1 , . . . , x r ) 7→ π ( g exp ( x 1 Z 1 + . . . + x r Z r )) is a homeomorphism of an op en set of R r on to a neigh b or ho o d of g K in G / K . Then it is easy (se e [24 ] for details) to see that with these c harts, G / K is an analytic manifold. In particular, π restrict to exp( m ) is a lo cal diffeomor phism into G / K and dπ ( X ) 6 = 0 since the pro jection of X on m is non zero. Notice that, b y our assumption on the dimension o f g ′ , the top ological dimension of G / K is at least 2. No w, if the statement w ere false, taking into accoun t that Ad k ( g ′ ) ⊆ g ′ , w e w ould hav e Ad k ( W ) ⊆ W for all k ∈ K . By the definition of adjoin t op erator as comp osition of the differen tials of righ t and left translations, the ab o v e w ould b e equiv alen t to sa y that ( R k ) ∗ (( L k − 1 ) ∗ ( Y )) ∈ W ∀ Y ∈ W , k ∈ K . Since the v ector fields in W are left in v arian t (i.e. ( L g ) ∗ Y = Y for a ll Y ∈ W ), this condition w ould say that W is K -r igh t inv a rian t, and we can write this condition in the form d ( R k ) x ( W x ) ⊂ W xk for all x ∈ G and k ∈ K . No w, let us consider the subs paces dπ x ( W x ) of T π ( x ) G / K : they are all 1-dimensional, thanks to the fact t hat dim( W ) = 1 + dim( g ′ ), and they dep end only on π ( x ): indeed, K -righ t inv ariance and the iden tit y π ◦ R k = π giv e dπ x ( Y x ) = dπ xk ( d ( R k ) x ( Y x )) ∈ dπ xk ( W xk ) for all Y ∈ W a nd k ∈ K . Therefore w e can define a (smo oth) 1- dimensional distribution W / K in G / K by ( W / K ) y := dπ x ( W x ), where x is an y elemen t of π − 1 ( y ). In particular W / K w ould b e tangent to a 1 -dimensional foliation F of G / K that has at least co dimension 1, since G / K has at least dimension 2 . Letting F ′ b e the foliation o f G whose lea ves are the in verse images via π o f lea ves of F , we find that still F ′ has co dimension a t least 1, and W is ta ngen t to the lea ve s of F ′ . But this con t radicts the f act t hat W generates g : in f act, the only sub-manifold to which W could b e tangen t is all the manifold G .  T ANGENT HYPERPLA NE I N CARN OT GROUPS 9 In the follow ing prop osition we provide a c har acterization of the v ector space spanned b y Ad exp( Y ) ( X ) , where Y v aries in a Lie subalgebra of g . This impro ve d v ersion of Prop osi- tion 2.18 w as p ointed out to us by V. Magna ni. Prop osition 2.18. L et g b e a nilp otent Lie alg e br a, let g ′ ⊂ g b e a Lie algebr a and let X ∈ g . T hen span  { Ad exp( Y ) ( X ) : Y ∈ g ′ }  = [ g ′ , X ] + [ g ′ , [ g ′ , X ]] + [ g ′ , [ g ′ , [ g ′ , X ]]] + · · · . Pr o of. Let us denote by S the space span  { Ad exp( Y ) ( X ) : Y ∈ g ′ }  . Ob viously S con ta ins X and all v ector fields Ad exp( rY ) ( X ) for r ≥ 0 and Y ∈ g ′ . Now, denoting b y L ( g ) the linear maps from g to g , let us r ecall the formula (see [29], page 54) Ad exp( Y ) = e ad Y , where ad · : g → End( g ) is the op erator ad Y ( X ) = [ Y , X ] and the exp onen tial e A is defined for a n y A ∈ L ( g ), by e A := ∞ P i =0 A i /i ! ∈ L ( g ). Therefore (2.19) Ad exp( Y ) X = X + [ Y , X ] + 1 2 [ Y , [ Y , X ]] + · · · Let ν be the dimension of g ′ and let ( Y 1 , . . . , Y ν ) b e a basis of g ′ . T aking in to account the iden tity (2.19 ), for all Y = P ν 1 r j Y j ∈ g ′ , we define Φ( r 1 , . . . , r ν ) := Ad exp( P ν 1 r j Y j ) X − X = s − 1 X k =1 1 k !  ν X j = 1 r j ad Y j  k X = s − 1 X k =1 1 k ! ν X j 1 ,...,j k =1 r j 1 · · · r j k  ad Y j 1 · · · ad Y j k  X ∈ S . Since this p o lynomial tak es its v alues in S , it turns o ut that all its co effi cien ts b elong to S . In particular, w e hav e ad Y i ( X ) = ∂ r i Φ(0) ∈ S and  ad Y i ad Y j + ad Y j ad Y i  X = 2 ∂ r i ∂ r j Φ(0) ∈ S . The Ja cobi iden t it y can b e read as ad U ad W − ad W ad U = ad [ U,W ] , so that  ad Y i ad Y j + ad Y j ad Y i  X = 2 ad Y i ad Y j X + ad[ Y j , Y i ] X . It follo ws that  ad Y i ad Y j  X ∈ S , and this pro v es that [ g ′ , X ] + [ g ′ , [ g ′ , X ]] ⊂ S . By induction, let us suppose that u k − 1 := [ g ′ , X ] + [ g ′ , [ g ′ , X ]] + · · · + [ g ′ , [ g ′ , · · · , [ g ′ | {z } ( k − 1) times , X ] · · · ] ⊂ S for some k ≥ 3. In general we ha ve (2.20) ∂ r i 1 · · · ∂ r i k Φ(0) = 1 k ! X σ  ad Y j σ (1) · · · ad Y j σ ( k )  X ∈ S, where the sum runs on all p erm utations σ of k elemen ts. By the Jacobi iden tit y  ad Y j σ (1) · · · ad Y j σ ( k )  X −  ad Y j η (1) · · · ad Y j η ( k )  X ∈ u k − 1 10 LUIGI AMBROSIO, BRUCE KLEINER, AND ENRICO LE DONN E if σ ◦ η − 1 is a transposition. Then, b y the inductiv e assumption, w e can iterate transp ositions in  ad Y j σ (1) · · · ad Y j σ ( k )  X t o w rite it as  ad Y j 1 · · · ad Y j k  X + W σ with W σ ∈ S . Then, from (2.20) w e g et  ad Y j 1 · · · ad Y j k  X ∈ S . This show s that  ad Y j 1 · · · ad Y j k  X ∈ S , so that u k ⊂ S , and this pro v es the inclusion [ g ′ , X ] + [ g ′ , [ g ′ , X ]] + [ g ′ , [ g ′ , [ g ′ , X ]]] + . . . ⊂ span { Ad exp Y X | Y ∈ g ′ } Observing t hat the opp osite inclusion trivially holds, w e are led to our claim.  2.5. Carnot groups. A Carnot g roup G of step s ≥ 1 is a connected, simply connected Lie gro up whose Lie algebra g admits a step s stratification: t his means t hat we can write g = V 1 ⊕ · · · ⊕ V s with [ V j , V 1 ] = V j + 1 , i ≤ j ≤ s , V s 6 = { 0 } and V s +1 = { 0 } . W e k eep the notation n = P i dim V i for the top ological dimension of G , and denote b y Q := s X i =1 i dim V i the so-called homo gene ous d i m ension of G . W e denote by δ λ : g → g the fa mily of inhomo- geneous dila tions defined b y δ λ ( s X i =1 v i ) := s X i =1 λ i v i λ ≥ 0 where X = s P i =1 v i with v i ∈ V i , 1 ≤ i ≤ s . The dilations δ λ b elong to G L ( g ) a nd are uniquely determined by the homogeneit y conditions δ λ X = λ k X ∀ X ∈ V k , 1 ≤ k ≤ s. W e den ote b y m the dime nsion of V 1 and w e fix an inner pro duct in V 1 and an or thonormal basis X 1 , . . . , X m of V 1 . This basis of V 1 induces the so- called Carnot-Caratheo dory left in v ar ian t distance d in G , defined as follows : d 2 ( x, y ) := inf ( Z 1 0 m X i =1 | a i ( t ) | 2 dt : γ (0) = x, γ (1) = y ) , where the infim um is made among all Lipsc hitz curv es γ : [0 , 1] → G suc h tha t γ ′ ( t ) = P m 1 a i ( t )( X i ) γ ( t ) for a.e. t ∈ [0 , 1 ] (the so-called horizontal curv es). F or Carnot groups, it is well kno wn that the map exp : g → G is a diffeomorphism, so an y elemen t g ∈ G can represen ted as exp( X ) for some unique X ∈ g , and therefore uniquely written in the for m (2.21) exp( s X i =1 v i ) , v i ∈ V i , 1 ≤ i ≤ s. T ANGENT HYPERPLA NE I N CARN OT GROUPS 11 This represen tation allo ws to define a family indexed b y λ ≥ 0 of intrinsic dilations δ λ : G → G , b y δ λ  exp( s X i =1 v i )  := exp  s X i =1 λ i v i  (i.e. exp ◦ δ λ = δ λ ◦ exp.) W e ha v e k ept t he same notation δ λ for b oth dilations (in g and in G ) b ec ause no ambiguit y will arise. Obvious ly , δ λ ◦ δ η = δ λη , a nd the Ba k er-Campb ell-Ha usdorff form ula giv es δ λ ( xy ) = δ λ ( x ) δ λ ( y ) ∀ x, y ∈ G . Moreo v er, the Carnot-Cara theo dory distance is w ell-b ehav ed under these dilatio ns, namely d ( δ λ x, δ λ y ) = λd ( x, y ) ∀ x, y ∈ G . Besides δ λ ◦ exp = exp ◦ δ λ , another useful relation b et w een dilations in G a nd dilations in g is δ λ X = ( δ λ ) ∗ X , namely (2.22) X ( u ◦ δ λ )( g ) = ( δ λ X ) u ( δ λ g ) ∀ g ∈ G , λ ≥ 0 . W e ha ve indeed X ( u ◦ δ λ )( g ) = d dt u ◦ δ λ ( g exp( tX ) )     t =0 = d dt u ( δ λ g δ λ exp( tX ) )     t =0 = d dt u ( δ λ g exp( tδ λ X ))     t =0 = ( δ λ X ) u ( δ λ g ) . 3. M easure-the oretic tools In this s ection w e specify the notions of con vergenc e used in this pap er (at the lev el of sets and of measures), and p oin t out some useful fa cts concerning Radon measures. The results quoted without an explicit reference are all quite standard, and can b e found for instance in [3], and those concerning Hausdorff measures in metric spaces in [14] or [4]. Haar, Leb esgue and Hausdorff measures. Carnot groups a re nilp oten t and so uni- mo dular, therefore the right and left Haar measures coincide, up t o constan t m ultiples. W e fix one of them and denote it by v ol G . W e shall denote b y H k (resp. S k ) the Hausdorff (resp. spherical Hausdorff ) k -dimensional measure; these measure s dep end on the distance, and, unless otherwis e stated, to build the m w e will use the Carnot- Caratheo dory distance in G and the Euclidean distance in Euclide an spaces. Using the left translation and scaling inv ariance of the Carnot-Caratheo dory distance one can easily c heck that the Haar measures of G a re a constant m ultiple of the spherical Hausdorff measure S Q and of H Q . In exp onen tia l co ordinates, all these measures ar e a constan t m ultiple of the Leb esgue measure L n in R n , namely v ol G  { exp( n X i =1 x i X i ) : ( x 1 , . . . , x n ) ∈ A }  = c L n ( A ) for all Borel sets A ⊆ R n for some constan t c . Using this fact, one can easily prov e that (3.1) v ol G ( δ λ ( A )) = λ Q v ol G ( A ) 12 LUIGI AMBROSIO, BRUCE KLEINER, AND ENRICO LE DONN E for all Borel sets A ⊆ G . The follo wing implication will b e useful: for µ nonnegativ e Radon measure, t > 0 and B ⊆ G Borel, w e ha v e lim sup r ↓ 0 µ ( B r ( x )) ω k r k ≥ t ∀ x ∈ B = ⇒ µ ( B ) ≥ t S k ( B ) , where ω k is the Leb esgue measure of the unit ball in R k (it app ears as a no rmalization constan t in the definitions of H k and S k , in order to ensure the identit y H k = S k = L k in R k ). In particular w e obtain that (3.2) { x ∈ G : lim sup r ↓ 0 µ ( B r ( x )) r k > 0 } is σ -finite with resp ect to S k . Characteristic functions, conv ergence in measure. F or an y set E w e shall denote b y 1 E the c haracteristic function of E (1 on E , 0 on G \ E ); within the class of Borel sets of G , the con v ergence w e consider is the so-called lo c al c onver g enc e in me asur e (equiv a len t to the L 1 lo c con vergenc e of the c haracteristic functions), na mely: E h → E ⇐ ⇒ v ol G  K ∩ [( E h \ E ) ∪ ( E \ E h )]  = 0 for all K ⊆ G compact. Radon measures and their con v ergence. The class M ( G ) of Radon measures in G coincides with the class o f 0 order distributions in G , namely those distributions T such that, for an y b ounded op en set Ω ⊆ G t here exists C (Ω) ∈ [0 , + ∞ ) satisfying |h T , g i| ≤ C (Ω) sup | g | ∀ g ∈ C 1 c (Ω) . These distributions can b e uniquely extended to C c ( G ), and their action can b e represen ted, thanks to Riesz theorem, through a n in t egral with resp ect to a σ -a dditiv e se t function µ defined o n b ounded Borel sets. Thanks to t his fact, the action of these distributions can b e extended ev en up t o b ounded Borel functions with compact suppo rt. W e will ty pically use b oth viewp oints in this pap er (for instance the first o ne pla ys a role in the definition of distributional deriv ative, while the second one is ess en tial to obtain differen tiation results). If µ is a nonnegativ e Radon measure w e shall denote supp µ := { x ∈ G : µ ( B r ( x )) > 0 ∀ r > 0 } . The only con v ergence we use in M ( G ) is the w eak ∗ one induce d b y the dualit y with C c ( G ), namely µ h → µ if lim h →∞ Z G u dµ h = Z G u dµ ∀ u ∈ C c ( G ) . Push-forw ard. If f : G → G is a prop er Borel map, then f − 1 ( B ) is a b ounded Borel set whenev er B is a b o unded Borel set. The push-forward measure f ♯ µ is then defined b y f ♯ µ ( B ) := µ ( f − 1 ( B )) . In integral terms, t his definition corresp onds to Z G u d f ♯ µ := Z G u ◦ f dµ whenev er the in tegra ls mak e sense (for instance u Borel, bo unded and compactly supp orted). T ANGENT HYPERPLA NE I N CARN OT GROUPS 13 V ector-v alued Radon measures. W e will also consider R m -v alued Radon measures, represen table as ( µ 1 , . . . , µ m ) with µ i ∈ M ( G ). The total variation of | µ | of an R m -v alued measure µ is the smallest nonnegative measure ν defined on Borel sets of G suc h that ν ( B ) ≥ | µ ( B ) | for all b ounded Borel set B ; it can b e explic itly defined by | µ | ( B ) := sup ( ∞ X i =1 | µ ( B i ) | : ( B i ) Bo rel partitio n of B , B i b ounded ) . Push forw ard and conv ergence in M m ( G ) can b e defined comp onent wise. Useful relations b et w een conv ergence and tota l v ariation are: (3.3) lim inf n →∞ | µ n | ( A ) ≥ | µ | ( A ) fo r all A ⊆ G op en, (3.4) sup n →∞ | µ n | ( K ) < + ∞ for all K ⊆ G compact, whenev er µ n → µ in M m ( G ). Asymptotically doubling measures. A nonnegative Radon measure µ in G is said to b e a s ymptotic al ly doubling if lim sup r ↓ 0 µ ( B 2 r ( x )) µ ( B r ( x )) < + ∞ for µ -a.e. x ∈ G . F or asymptotically doubling measures all the standard r esults of Leb esgue differentiation theory hold: for instance, for any Borel set A , µ -a.e. p oin t x ∈ A is a density p oin t of A , namely lim r ↓ 0 µ ( A ∩ B r ( x )) µ ( B r ( x )) = 1 . The same result holds for any set A , provided w e replace µ by the outer measure µ ∗ , defined for any A ⊆ G by µ ∗ ( A ) := inf { µ ( B ) : B Borel, B ⊇ A } . It follow s directly from the definition that µ ∗ is subadditiv e. Moreo v er, let ( B n ) b e a minimizing sequence and let B the in tersection of all sets B n : then B is a Borel set, B ⊇ A and µ ∗ ( A ) = µ ( B ). F urthermore, fo r all Borel sets C w e ha v e µ ∗ ( A ∩ C ) = µ ( B ∩ C ) (if not, adding the strict inequality µ ∗ ( A ∩ C ) < µ ( B ∩ C ) to µ ∗ ( A \ C ) ≤ µ ( B \ C ) w ould giv e a contradiction). Cho osing C = B r ( x ), with x densit y po in t of B , w e obtain lim r ↓ 0 µ ∗ ( A ∩ B r ( x )) µ ( B r ( x )) = lim r ↓ 0 µ ( B ∩ B r ( x )) µ ( B r ( x )) = 1 . This pro ves tha t t he se t of p oin ts o f A that are not den sit y points is con tained in a µ – negligible Borel set. W e will also b e using in the pro of of Theorem 6.4 the fact that µ ∗ is c ountably subadditiv e, namely µ ∗ ( A ) ≤ P i µ ∗ ( A i ) f or all seq uences ( A i ) with A ⊆ ∪ i A i . W e recall the follo wing result, prov ed in Theorem 2.8.17 of [14]: Theorem 3.5 (Differen tiation) . Assume that µ is asymptotic al ly doubling and ν ∈ M ( G ) is absolutely c ontinuous wi th r esp e ct to µ . Then the limit f ( x ) := lim r ↓ 0 ν ( B r ( x )) µ ( B r ( x )) 14 LUIGI AMBROSIO, BRUCE KLEINER, AND ENRICO LE DONN E exists and is finite for µ -a.e. x ∈ supp µ . In addition, f ∈ L 1 lo c ( µ ) and ν = f µ , i. e . ν ( B ) = R B f dµ for al l b ounde d Bor el sets B ⊆ G . The pro of giv en in [14] co v ers m uch more general situations; the re ader already acquainte d with the theory of differen tiation with resp ect to doubling measures can easily realize that the results ex tend to asymptotically doubling ones b y consider the lo c alize d (in G × (0 , + ∞ )) maximal op erators: M B ,r ν ( x ) := sup s ∈ (0 ,r ) ν ( B s ( x )) µ ( B s ( x )) , x ∈ B , where ν is an y nonnegativ e Radon measure in G . Thank s to the asymptotic doubling prop ert y , one can find a family of Borel sets B h ⊆ supp µ whose union co v ers G , constan t s C h ≥ 1 and radii r h > 0 suc h that µ ( B 3 r ( x )) ≤ C h µ ( B r ( x )) for x ∈ B h and r ∈ (0 , r h ). F or the op erators M B h ,r h , the uniform doubling prop erty o n B h and a co v ering lemma yield the w eak L 1 estimate µ ( E ∩ { M B h ,r h ν > t } ) ≤ t − 1 C h ν ( E ) (f or E ⊆ B h Borel, t > 0). This leads to the differen tiation result on a ll B h , a nd then µ -a.e. on G . 4. Sets of locall y finite pe rimeter In this section w e recall a few useful f acts ab o ut sets of finite p erimeter, considering also sets whose deriv ativ e along non-horizon tal directions is a measure. Definition 4.1 (R egular and in v aria n t directions) . L et f ∈ L 1 lo c ( G ) . We shal l d e note by Reg( f ) the ve ctor subsp ac e of g made by ve ctors X such that X f is r epr esentable by a R adon me asur e. We shal l denote by Inv( f ) the subsp ac e of Reg( f ) c orr esp o n ding to the ve ctor fie l d s X such that X f = 0 , and by In v 0 ( f ) the subset made by homo gene ous dir e ctions, i.e. In v 0 ( f ) := In v ( f ) ∩ s [ i =1 V i . Notice that, according to (2.14), f ◦ R exp( tX ) = f ∀ t ∈ R , X ∈ Inv( f ) . W e will mostly consider regula r and in v arian t directions o f c har acteristic functions, there- fore w e set Reg( E ) := Reg ( 1 E ) , In v( E ) := In v ( 1 E ) , Inv 0 ( E ) := In v 0 ( 1 E ) . W e can now naturally define halfspaces b y requiring inv a riance along a codimension 1 space of directions, and monotonicit y alo ng the remaining direction; if this direction is horizon tal, we call these sets ve rtic al halfsp ac es . Definition 4.2 (V ertical halfspaces) . We say that a Bor el set H ⊆ G is a v ertical halfspace if In v 0 ( H ) ⊇ ∪ s 2 V i , V 1 ∩ In v 0 ( H ) is a c o d imension one subsp ac e of V 1 and X 1 H ≥ 0 for some X ∈ V 1 , with X 1 H 6 = 0 . T ANGENT HYPERPLA NE I N CARN OT GROUPS 15 Since (4.3) span  In v 0 ( H )  = s M i =1 V i ∩ In v 0 ( H ) , w e can equiv alen tly sa y that H is a n halfspace if span( In v 0 ( H )) is a co dimension 1 subs pace of g , V 1 ∩ span(Inv 0 ( H )) is a co dimension 1 subspace of V 1 and X 1 H ≥ 0 for some X ∈ V 1 : indeed, (4 .3) forces, whenev er the co dimension is 1, all subspaces V i ∩ In v 0 ( H ) to coincide with V i , with just o ne exception. Let u s recall that m denotes the dimension of V 1 , and that X 1 , . . . , X m is a giv en orthonor- mal ba sis of V 1 . With this notation, v ertical halfspaces can b e characterize d as follows: Prop osition 4.4 (Characterization of v ertical halfspaces) . H ⊆ G is a vertic al halfsp ac e if and only if ther e exist c ∈ R and a unit ve ctor ν ∈ S m − 1 such that H = H c,ν , wher e (4.5) H c,ν := exp  { m X i =1 a i X i + s X i =2 v i : v i ∈ V i , a ∈ R m , m X i =1 a i ν i ≤ c }  . Pr o of. Let us denote b y ν ∈ S m − 1 the unique v ector suc h that the v ector Y = P i ν i X i is orthogonal to all in v aria n t directions in V 1 . Let us w ork in exp onen tial co ordinates, with the function ( x 1 , . . . , x n ) 7→ exp( n X i =1 x i v i ) , and le t ˜ H ⊂ R n b e the set H in these coo rdinates. Here ( v 1 , . . . , v n ) is a basis of g compatible with the stratificatio n: this means that, if m i are the dime nsions of V i , with 1 ≤ i ≤ s , l 0 = 0 and l i = P i 1 m j , the n v l i − 1 +1 , . . . , v l i is a basis of V i . By the Bak er- Campb ell-Hausdorff form ula, in these co o rdinates the v ector fields v i corresp ond to ∂ x i for l s − 1 + 1 ≤ i ≤ l s = n , and Theorem 2.12 g iv es that 1 ˜ H do es not dep end on x l s − 1 +1 , . . . , x n . F or l s − 2 + 1 ≤ i ≤ l s − 1 the vec tor fields v i − ∂ x i , still in these co o rdinates, are giv en b y the sum of p olynomials m ultiplied b y ∂ x j , with l s − 1 + 1 ≤ j ≤ l s . As a consequenc e ∂ x i 1 ˜ H = 0 and we can apply Theorem 2.12 again to obtain that 1 ˜ H do es not dep end on x l s − 2 +1 , . . . , x l s − 1 either. Con tinuing in this w a y we obtain that 1 ˜ H dep ends on ( x 1 , . . . , x m 1 ) only . F ur thermore, P i ξ i ∂ x i 1 ˜ H is equal to 0 if ξ ⊥ ν , and it is nonnegative if ξ = ν . Then, a classical Euclidean argumen t (it appears in De Gio rgi’s rectifiabilit y proof [13], se e also the pro of of this result in Theorem 3 .59 of [3]) sho ws that 1 ˜ H dep ends o n P m 1 ν i x i only , a nd it is a monoto ne function of this quan tity . This immediately giv es (4.5).  R emark 4.6 . An analogous computation in exponential co ordinates sho ws that In v ( f ) = g if and only if f is equiv alen t to a constan t. In the next prop o sition we p oin t out useful stability prop erties of R eg( f ) and Inv( f ). Prop osition 4.7. L et f ∈ L 1 lo c ( G ) . Then Reg( f ) , Inv( f ) , In v 0 ( f ) a r e invari a nt under left tr anslations, and In v 0 ( f ) is invariant under intrinsic dilations. Mor e ove r: (i) In v ( f ) is a Lie sub alge b r a of g and [In v 0 ( f ) , Inv 0 ( f )] ⊂ Inv 0 ( f ) ; 16 LUIGI AMBROSIO, BRUCE KLEINER, AND ENRICO LE DONN E (ii) If X ∈ In v ( f ) and k = exp ( X ) , then Ad k maps Reg ( f ) into R eg( f ) and Inv( f ) into In v ( f ) . Mor e pr e cisely (4.8) Ad k ( Y ) f = ( R k − 1 ) ♯ Y f ∀ Y ∈ Reg ( f ) . Pr o of. The pro of o f the in v ariance is simple , so we omit it. (i) W e simply no tice that for all X, Y ∈ In v ( f ) w e ha ve Z G f [ X , Y ] g d v ol G = −h X f , Y g i + h Y f , X g i = 0 ∀ g ∈ C ∞ c ( G ) . The second stated prop ert y follow s by the f act that [ V i , V j ] ⊂ V i + j . (ii) L et Y ∈ Reg( f ) a nd Z = Ad k ( Y ). F or g ∈ C ∞ c ( G ) and k ∈ G w e hav e (taking into accoun t the left in v ariance of Y ) Z g ( x ) = Y ( g ◦ C k )( C − 1 k ( x )) = Y ( g ◦ R k − 1 )( L k ◦ C k − 1 ( x )) = Y ( g ◦ R k − 1 )( R k ( x )) . Therefore ( Z g ) ◦ R k − 1 = Y ( g ◦ R k − 1 ) and a c hange of v ariables giv es Z G f Z g d v ol G = Z G f ◦ R k − 1 Y ( g ◦ R k − 1 ) d v ol G . No w, if k = exp( X ) with X ∈ In v( f ), w e hav e f ◦ R k − 1 = f , and this giv es (4.8 ).  R emark 4.9 . Let X ∈ Reg( f ) and a ssume that X f ≥ 0; then, com bining (2.19) with (4.8 ), w e obtain X f + s − 1 X i =1 t i i ! ad i Y ( X ) f ≥ 0 ∀ t ∈ R , ∀ Y ∈ Inv( f ) . Since t can b e c hosen arbitrarily large, this implies that ad s − 1 Y ( X ) f ≥ 0 ∀ Y ∈ Inv( f ) . In particular, if s is eve n, b y applying the same inequalit y with − Y in place of Y we get (4.10) ad s − 1 Y ( X ) ∈ In v( f ) . Definition 4.11 (Sets of lo cally finite p erime ter) . T he main obje ct of investigation of this p ap er is the class of sets of lo cally finite p erimeter , i.e. those Bor e l sets E such that X 1 E is a R adon me a sur e for any X ∈ V 1 . Still using the ort honormal basis o f V 1 , fo r f ∈ L 1 lo c ( G ) with X i f ∈ M ( G ) we can define the R m -v alued R adon measure (4.12) D f := ( X 1 f , . . . , X m f ) . Tw o v ery basic prop erties that will pla y a role in the sequel are: (4.13) D f = 0 = ⇒ f is (equiv a len t to) a constant (4.14) sup n Z Ω | f n | d v ol G + | D f n | (Ω) < + ∞ ∀ Ω ⋐ G = ⇒ ( f n ) relativ ely compact in L 1 lo c . The pro of of the first one can b e obtained combining Prop osition 4.7 ( that gives that In v ( f ) = g with Remark 4.6). The second one has b een pro ved in [21]. T ANGENT HYPERPLA NE I N CARN OT GROUPS 17 Definition 4.15 (De Giorgi’s reduced boundary) . L et E ⊆ G b e a set of lo c al ly finite p erimeter. We de note by F E the se t of p oints x ∈ supp | D 1 E | wher e: (i) the limit ν E ( x ) = ( ν E , 1 ( x ) , . . . , ν E , m ( x )) := lim r ↓ 0 D 1 E ( B r ( x )) | D 1 E | ( B r ( x )) exists; (ii) | ν E ( x ) | = 1 . The following result has b een obtained in [1]. Theorem 4.16. L et E ⊆ G b e a set of lo c al ly finite p erimeter. Then | D 1 E | is asymptotic al ly doubling, and mor e pr e cisely the fol lowing pr op erty holds: for | D 1 E | -a.e. x ∈ G ther e exists ¯ r ( x ) > 0 satisfying (4.17) l G r Q − 1 ≤ | D 1 E | ( B r ( x )) ≤ L G r Q − 1 ∀ r ∈ (0 , ¯ r ( x )) , with l G and L G dep ending on G only. As a c onse quenc e | D 1 E | is c onc entr ate d on F E , i.e., | D 1 E | ( G \ F E ) = 0 . Actually the result in [1 ] is v alid in a ll Ahlfors Q -regular metric spaces for whic h a P oincar ´ e inequalit y ho lds (in this con text, ob viously including all Lie groups, still the measure | D 1 E | mak es sense, see [32]); ( 4.17) also implies that the measure | D 1 E | can also b e b ounded fro m ab ov e and b elo w by the spherical Hausdorff measure S Q − 1 , namely (4.18) l G ω Q − 1 S Q − 1 ( A ∩ F E ) ≤ | D 1 E | ( A ) ≤ L G ω Q − 1 S Q − 1 ( A ∩ F E ) for all Borel sets A ⊆ G (since H k ≤ S k ≤ 2 k H k , similar inequalities hold with H Q − 1 ). In general doubling metric spaces, wh ere no natural dimension Q exists, the asymptotic doubling prop erty of | D 1 E | and a suitable repre sen tatio n of it in terms of Hausdorff me asures ha v e b een o btained in [2]. 5. Itera ted t angents are halfsp aces In this section w e sho w that if w e iterate sufficien tly many times the tangen t op erator we do get a v ertical halfspace. Let us b egin with a precise definition of tangen t set. Definition 5.1 (T angen t set) . L et E ⊆ G b e a set of lo c al ly finite p erime ter and x ∈ F E . We denote by T an( E , x ) al l limit p oints, in the top olo gy of lo c al c onver genc e in me asur e, of the tr anslate d and r esc ale d family of sets { δ 1 /r ( x − 1 E ) } r > 0 as r ↓ 0 . If F ∈ T an( E , x ) we sa y that F is tangent to E at x . We also set T an( E ) := [ x ∈ F E T an( E , x ) It is also useful to consider iter ate d ta ngen ts; to this aim, still for x ∈ F E , w e define T an 1 ( E , x ) := T an( E , x ) and T an k +1 ( E , x ) := [  T an( F ) : F ∈ T an k ( E , x )  . The result w e shall prov e in this section is an in termediate step tow ards Theorem 1.2: 18 LUIGI AMBROSIO, BRUCE KLEINER, AND ENRICO LE DONN E Theorem 5.2. L et E ⊆ G b e a set with lo c al ly finite p erimeter. Then, for | D 1 E | -a.e. x ∈ G we have (with the notation (4.5) ) H 0 ,ν E ( x ) ∈ T an k ( E , x ) with k := 1 + 2 ( n − m ) . Notice tha t, b y Theorem 4.16, w e need only to consider p oints x ∈ F E . Our starting p oin t is the following prop osition, o btained in [19], sho wing that the tangen t set at p oin ts in the reduce d b oundary is alw a ys inv aria n t along co dimension 1 subspace of V 1 , and monotone along the remaining hor izon ta l direction. Prop osition 5.3. L et E ⊆ G b e a se t of lo c a l ly finite p eri m eter. Then, f o r al l ¯ x ∈ F E the fol lowing p r op erties hold: (i) 0 < lim inf r ↓ 0 | D 1 E | ( B r ( ¯ x )) /r Q − 1 ≤ lim sup r ↓ 0 | D 1 E | ( B r ( ¯ x )) /r Q − 1 < + ∞ ; (ii) T an( E , ¯ x ) 6 = ∅ and, for al l F ∈ T an( E , ¯ x ) , we hav e that e ∈ supp | D 1 F | and ν F = ν E ( ¯ x ) | D 1 F | -a.e. in G . In p articular V 1 ∩ In v 0 ( F ) c oinci d es w ith the c o dim ension 1 subsp ac e of V 1 ( m X i =1 a i X i : m X i =1 a i ν E , i ( ¯ x ) = 0 ) and, setting, X x := P m i =1 ν E , i ( ¯ x )( X i ) x ∈ g , X 1 F is a nonne g a tive R ad on me asur e. In groups o f step 2, in [19] it is prov ed that constancy of ν E c har acterizes ve rtical sub- spaces. W e pro vide here a differen t pro of of this fact, based on the prop erties of the adjo in t op erator, and in par ticular on Remark 4.9. Prop osition 5.4. L et E ⊂ G b e a set with lo c al ly fin i te p erimeter, and assume that ν E is (e quivalent to) a c onstant. Then, if G is a step 2 gr oup, E is a vertic al halfsp ac e. Pr o of. Let us denote b y ξ t he constan t v alue of ν E , and set X := P i ξ i X i . Then X 1 E ≥ 0 and Inv( E ) con ta ins all v ectors Y = P i η i X i with η ∈ R m p erp endicular to ξ . F r om (4.1 0) w e get [ Y , X ] 1 E = 0 for an y Y ∈ In v ( E ) ∩ V 1 , and since these comm utat ors, together with the comm utat ors { [ Y 1 , Y 2 ] : Y i ∈ Inv( E ) ∩ V 1 } , span the whole of V 2 , the pro of is ac hiev ed.  R emark 5.5 . The follow ing simple example, that w e learned f rom F. Serra Cassano, sho ws that the sign condition is esse n tia l for the v alidity of the classification res ult, ev en in the first Heisen b erg group H 1 . Cho osing exp onen tial co ordinates ( x, y , t ), and t he v ector fields X 1 := ∂ x + 2 y ∂ t and X 2 := ∂ y − 2 x∂ t , the function f ( x, y , t ) := g ( t + 2 xy ) (with g smo oth) satisfie s X 1 f = 4 y g ′ ( t + 2 xy ) and X 2 f = 0. Therefore the sets E t := { f < t } are X 2 -in v arian t and are not halfspaces. The same example can b e used to sho w that there is no lo cal v ersion of Prop osition 5.4, b ecause the sets E t lo cally ma y satisfy X 1 1 E t ≥ 0 or X 1 1 E t ≤ 0 (dep ending on the sign of g ′ and y ), but a re not lo cally half spaces. The non-lo calit y app ears also in our argumen t: indee d, the pro of of (4.10) depends on the sign condition o f Ad exp( tX 2 ) ( X 1 ) 1 E with t arbit rarily large, and this is the right translate, b y exp( tX 2 ), of X 1 1 E . The pro of g iv en in [1 9] dep ends, instead, on the p ossibilit y of jo ining T ANGENT HYPERPLA NE I N CARN OT GROUPS 19 t w o differen t p oints in H 1 b y following in tegral lines of X 2 in b oth directions, and in tegral lines of X 1 in just one direction: a n insp ection of the pro of rev eals that these paths can not b e confined in a b ounded region, ev en if the initial and final p oint are confined within a small region. In this sense, Prop o sition 5.4 could b e considered as a kind of Liouville theorem. Let f ∈ L 1 lo c ( G ) and X ∈ g ; then, for all r > 0 w e ha ve the iden tity (5.6) δ 1 /r X ( f ◦ δ r ) = r − Q ( δ 1 /r ) ♯ ( X f ) in the sense of distributions. Indeed, writing in brief X r := δ 1 /r X , if g ∈ C ∞ c ( G ), from (2.22) w e get X r ( g ◦ δ r ) = ( X g ) ◦ δ r ; a s a conseq uence (3.1) give s h X r ( f ◦ δ r ) , g i = − Z G ( f ◦ δ r ) X r g d v o l G = − r − Q Z G f ( X r g ) ◦ δ 1 /r d v ol G (5.7) = − r − Q Z G f X ( g ◦ δ 1 /r ) d v ol Q = h r − Q ( δ 1 /r ) ♯ ( X f ) , g i . The first crucial lemma sho ws tha t if X ∈ Reg ( E ) b elongs to ⊕ s 2 V i , then the ta ngen ts to E at | D 1 E | -a.e. x a re inv arian t under Y , where Y is the “ higher degree part” of X induced b y the stratification of g . The underlying reason for this fa ct is tha t the in t rinsic dilations b eha v e quite differen tly in the X direction and in the horizontal direction. Lemma 5.8. L et F b e a set with lo c al ly finite p erimeter, X ∈ Reg( F ) , µ = X 1 F and assume that X = P l i =2 v i with v i ∈ V i and l ≤ s . The n, for | D 1 F | -a.e. x , v l ∈ In v 0 ( L ) for al l L ∈ T an( F, x ) . Pr o of. ¿F rom (3.2) w e kno w that the set N of p oints x suc h that lim sup r ↓ 0 r 2 − Q | µ | ( B r ( x )) is p ositiv e is σ -finite with resp ect to S Q − 2 , and therefore S Q − 1 -negligible and | D 1 F | -negligible (recall (4.18)). W e will pro v e that the statemen t holds at any x ∈ ( F F ) \ N and w e shall assume, up to a left tra nslation, that x = e . Giv en an y g ∈ C 1 c ( G ), let R b e suc h that supp( g ) ( B R ( e ); ( 5.6) with f = 1 F giv es Z G 1 δ 1 /r F X r g d v o l G = r l − Q Z G g ◦ δ 1 /r dµ with X r := r l δ 1 /r X , so that X r → v l as r ↓ 0. No w, notice that l ≥ 2, and that the righ t hand side can b e b ounded with sup | g | r l − Q | µ | ( B Rr ( e )) = O ( r l − Q ) o ( r Q − 2 ) = o (1) . So, pa ssing to the limit as r ↓ 0 along a suitable seque nce, w e obtain tha t v l 1 L = 0 for all L ∈ T an( F, e ).  The in v ariance of Inv 0 under left tr anslations and scaling sho ws that In v 0 ( F ) contains In v 0 ( E ) for all F ∈ T a n( E ). Let us define co dimension of In v 0 ( E ) in g a s the co dimension of its linear span; w e know that this co dimension is at least 1 (b ecause the co dimension within V 1 is 1) for al l tangen t sets, and it is equal to 1 precisely for vertical halfspaces, thanks to Prop osition 4.4. 20 LUIGI AMBROSIO, BRUCE KLEINER, AND ENRICO LE DONN E The second crucial lemma shows that, when the co dime nsion of In v 0 ( E ) in g is at least 2, a double tangen t strictly incre ases, at | D 1 E | -a.e. p oint, the set Inv 0 ( E ). The strategy is to find first a tangen t set F with Reg( F ) ) span(Inv 0 ( E )) (this is based on the geometric Prop osition 2.17 and Prop osition 4.7) a nd then on the application of the previous lemma, whic h turns a r egular direction of F in t o an inv aria n t homogeneous direction of a tangen t to F . Lemma 5.9 (Improv emen t of Inv 0 ( E )) . L et E ⊆ G b e a set of lo c al ly finite p erimeter and assume that dim  span(In v 0 ( E ))  ≤ n − 2 . Then, for al l ¯ x ∈ F E , Inv 0 ( L ) ) In v 0 ( E ) for some L ∈ T an 2 ( E , ¯ x ) . Pr o of. ( Step 1) W e sho w first the existence of Z ∈ g \ [span( In v 0 ( E )) + V 1 ] suc h that Z ∈ Reg( F ) for all F ∈ T an( E , ¯ x ). T o this aim, w e apply Prop osition 2.17 with g ′ := span(In v 0 ( E )) (recall that, b y Proposition 4.7(i), g ′ is a Lie algebra) and X := P m 1 ν E , i ( ¯ x ) X i to obtain Y ∈ g ′ suc h that Z := Ad exp( Y ) ( X ) / ∈ span(Inv 0 ( E )) ⊕ { R X } = span(Inv 0 ( E )) + V 1 . Then, sin ce In v 0 ( F ) con tains In v 0 ( E ) for all F ∈ T an( F , ¯ x ), w e hav e that Y ∈ In v( F ), therefore Prop osition 4.7(ii) sho ws that Z ∈ Reg ( F ) for all F ∈ T an( E , ¯ x ). (Step 2) Now , let F ∈ T an ( E , ¯ x ), Z / ∈ span (In v 0 ( E )) + V 1 giv en b y the previous step, and set µ = Z 1 F . P ossibly remo ving from Z its horizon tal comp onent we can write Z = v i 1 + · · · + v i l with i j ≥ 2 a nd v i j ∈ V i j . Then, v i k / ∈ Inv 0 ( E ) for at least one k ∈ { 1 , . . . , l } , and let us choose the larg est o ne with this prop ert y . Then, setting Z ′ = v i 1 + · · · + v i k , since v i j ∈ Inv 0 ( E ) ⊆ In v 0 ( F ) f or all k < j ≤ l , w e still hav e Z ′ 1 F = µ . By Lemma 5.8 we can find L ∈ T an( F ) with v i k 1 L = 0, i.e. v i k ∈ Inv 0 ( L ). Since v i k / ∈ Inv 0 ( E ), w e hav e prov ed that Inv 0 ( L ) strictly con tains In v 0 ( E ).  Pro of of Theorem 5.2. Recall that m = dim( V 1 ). Sets in T an( E , ¯ x ) are in v arian t, thanks to Prop osition 5.3, in at least m − 1 dir ections. Let us define i k := max  dim  span(In v 0 ( F ))  : F ∈ T an k ( E , ¯ x )  . Then i 1 ≥ m − 1 and w e pro v ed in Lemma 5.9 that i k +2 > i k as long a s there exists F ∈ T an k ( E , ¯ x ) with dim  span(In v 0 ( F ))  ≤ n − 2. By iterating k times, with k ≤ 2( n − m ), the tang en t op erator w e find F ∈ T a n k ( E , ¯ x ) with dim  span(In v 0 ( F ))  ≥ n − 1. W e kno w from Prop o sition 5.3 that e ∈ supp | D 1 F | , that the co dime nsion of In v 0 ( F ) is exactly 1, and precisely that V 1 ∩ In v 0 ( F ) = ( m X i =1 a i X i : m X i =1 a i ν E , i ( ¯ x ) = 0 ) and tha t P i ν E , i ( ¯ x ) X i 1 F ≥ 0 . Therefore Prop o sition 4.4 giv es F = H 0 ,ν E ( ¯ x ) . T ANGENT HYPERPLA NE I N CARN OT GROUPS 21 6. Itera ted t angents are t angent In this section we complete the pro o f of Theorem 1.2. T aking in to accoun t the statemen t of Theorem 5.2, we need only to pro ve the follo wing result. Theorem 6.1. L et E ⊆ G b e a set with lo c al ly finite p erimeter. Then, for | D 1 E | -a.e. x ∈ G we have ∞ [ k =2 T an k ( E , x ) ⊆ T an( E , x ) . In turn, this result follo ws b y an analogous one in volving tangents t o measures, pro v ed in [39] in the Euclidean case; w e j ust adapt the argumen t to Carnot groups and to v ector- v alued measures. In the seque l w e shall denote b y I x,r ( y ) := δ 1 /r ( x − 1 y ) the comp osition δ 1 /r ◦ L x − 1 . W e sa y that a measure µ ∈ M m ( G ) is asymptotic al ly q -r e gular if (6.2) 0 < lim inf r ↓ 0 | µ | ( B r ( x )) r q ≤ lim sup r ↓ 0 | µ | ( B r ( x )) r q < + ∞ for | µ | -a.e. x ∈ G . Notice tha t asymptotically q -regular measures are asymptotically doubling, and that t he p erimeter measure | D 1 E | is asymptotically ( Q − 1)-regular, thanks to Theorem 4.16. Definition 6.3 (T angen ts to a measure) . L et µ ∈ M m ( G ) b e asymptotic al ly q -r e g ular. We shal l deno te by T an( µ , x ) the family of al l me a s ur es ν ∈ M m ( G ) that ar e we ak ∗ limit p oints as r ↓ 0 of the family of m e asur es r − q ( I x,r ) ♯ µ . Theorem 6.4. L et µ ∈ M m ( G ) b e asymptotic al ly q -r e gular. Then, for | µ | - a .e. x , the fol lowing p r op erty hold s : T an( ν , y ) ⊆ T an ( µ, x ) ∀ ν ∈ T an( µ, x ) , y ∈ supp | ν | . The connection b et wee n Theorem 6.1 and Theorem 6.4 rests on the follo wing observ ation: (6.5) L ∈ T an( F , x ) ⇐ ⇒ D 1 L ∈ T an( D 1 F , x ) \ { 0 } for all x ∈ F F . The implication ⇒ in (6.5 ) is easy , b ecause a simple scaling arg umen t giv es (6.6) L = lim i →∞ δ 1 /r i ( x − 1 F ) = ⇒ D 1 L = lim i →∞ r 1 − Q i ( I x,r i ) ♯ D 1 F . Therefore L ∈ T an( F, x ) implies D 1 L ∈ T an( D 1 F , x ); clearly D 1 L 6 = 0 because x ∈ F F . No w we prov e the harder implication ⇐ in (6 .5): assume, up to a left translation, that x = e , and that D 1 L 6 = 0 is the w eak ∗ limit of r 1 − Q i ( I e,r i ) ♯ D 1 F , with r i ↓ 0; no w, set F i := δ 1 /r i F , so that D 1 F i = r 1 − Q i ( I e,r i ) ♯ D 1 F , and b y the compactness prop erties of sets of finite p erimeter (see (4.14)) a ssume with no lo ss of generalit y that F i → L ′ lo cally in measure, so t hat L ′ ∈ T an( F, e ). Then r 1 − Q i ( I e,r i ) ♯ D 1 F = D 1 F i w eakly ∗ con verge to D 1 L ′ : indeed, the con v ergence in the sense of distributions is ob vious, a nd since the tota l v ariations are lo cally uniformly b ounded, we ha ve w eak ∗ con vergenc e as w ell. It follow s that D 1 L = D 1 L ′ . Since 1 L − 1 L ′ has zero horizon tal distributional deriv ativ e, b y (4.13) it must b e (equiv alen t to) a constan t; this can happ en only whe n eithe r L = L ′ or L = G \ L ′ ; but the second 22 LUIGI AMBROSIO, BRUCE KLEINER, AND ENRICO LE DONN E p ossibilit y is ruled out b ecause it w ould imply t hat D 1 L = − D 1 L ′ and that D 1 L = 0. This pro v es that L = L ′ ∈ T an( F, e ). Pr o of. ( of Theorem 6.1) At any p oin t x ∈ F E where t he prop erty stated in Theorem 6.4 holds with µ = D 1 E w e ma y consider any F ∈ T an( E , x ) and L ∈ T an( F , y ) for some y ∈ F F ; then, b y ( 6.5) w e kno w that D 1 F ∈ T an( D 1 E , x ) and D 1 L ∈ T an( D 1 F , y ) \ { 0 } ; as a consequenc e, Theorem 6.4 give s D 1 L ∈ T an( D 1 E , x ) \ { 0 } , hence ( 6.5) again giv es that L ∈ T an( E , x ). This pro v es that T a n 2 ( E , x ) ⊆ T an( E , x ), and therefore T an 3 ( E , x ) ⊆ T an 2 ( E , x ), and so on.  The rest o f this section is dev oted to the pro of of Theorem 6.4. W e will follo w with minor v arian ts ( b ecause w e ar e dealing with v ector-v alued measures) the pro of giv en in Mattila’s b o ok [3 1]. Before pro ce eding to the pro of of Theorem 6.4 w e state a simple lemma. Lemma 6.7. Assume that A ⊂ G and a ∈ A is a de nsity p oint for A r e lative to | µ | ∗ , i.e. (6.8) lim r ↓ 0 | µ | ∗ ( B r ( a ) ∩ A ) | µ | ( B r ( a )) = 1 . If, for some r i ↓ 0 and λ i ≥ 0 the me asur es λ i ( I a,r i ) ♯ µ we akly ∗ c onver ge to ν , then lim i →∞ d ( aδ r i y , A ) r i = 0 ∀ y ∈ supp | ν | . Pr o of. Let τ := d ( y , e ) a nd let us argue by con tradiction. If the statemen t w ere false, τ w ould b e p ositiv e and there w ould exist ǫ ∈ (0 , τ ) suc h that d ( aδ r i y , A ) > ǫr i for infinitely man y v alues of i . P ossibly extracting a subsequence , let us assume that this happ ens for all i : w e kno w that (6.9) B ǫr i ( aδ r i y ) ⊆ G \ A and since ǫ < τ we hav e (6.10) B ǫr i ( aδ r i y ) ⊆ B τ r i ( aδ r i y ) ⊆ B 2 τ r i ( a ) . No w use, in this order, the definition of densit y p o in t, (6.9), (6.10) and ( 3.3) to get 1 = lim i →∞ | µ | ∗ ( B 2 τ r i ( a )) ∩ A ) | µ | ( B 2 τ r i ( a )) ≤ lim sup i →∞ | µ | ( B 2 τ r i ( a ) \ B ǫr i ( aδ r i y ) ) | µ | ( B 2 τ r i ( a )) = lim sup i →∞ | µ | ( B 2 τ r i ( a ))) − | µ | ( B ǫr i ( aδ r i y ) ) | µ | ( B 2 τ r i ( a )) = 1 − lim inf i →∞ | µ | ( B ǫr i ( aδ r i y ) ) | µ | ( B 2 τ r i ( a )) = 1 − lim inf i →∞ ( I a,r i ) ♯ | µ | ( B ǫ ( y )) ( I a,r i ) ♯ | µ | ( B 2 τ ( e )) ≤ 1 − lim inf i →∞ | λ i ( I a,r i ) ♯ µ | ( B ǫ ( y )) lim sup i →∞ | λ i ( I a,r i ) ♯ µ | ( B 2 τ ( e )) ≤ 1 − | ν | ( B ǫ ( y )) lim sup i →∞ | λ i ( I a,r i ) ♯ µ | ( B 2 τ ( e )) . But, | ν | ( B ǫ ( y )) > 0 b ecause y ∈ supp | ν | , and the lim sup is finite b y (3.4 ). This con tradic- tion concludes the pro of of the lemma.  T ANGENT HYPERPLA NE I N CARN OT GROUPS 23 Pro of of Theorem 6.4 . F or ν, ν ′ ∈ M m ( G ), define d R ( ν, ν ′ ) := sup  Z G φ dν − Z G φ dν ′ : φ ∈ D R  , where D R := { φ ∈ C c ( B R ( e )) : sup | φ | ≤ 1 and | φ ( x ) − φ ( y ) | ≤ d ( x, y ) ∀ x, y ∈ G } . It is well kno wn, and easy to che c k, that d R induces the w eak ∗ con vergenc e in all b ounded sets of M m ( B R ( e )). W e define a distance ¯ d in M m ( G ) b y ¯ d ( µ, ν ) := ∞ X R =1 2 − R min  1 , d R ( µ, ν )  . Let x b e a p oint where the limsup in (6.2 ) is finite; no w w e c hec k that, for all infinitesimal sequence s ( r i ) ⊂ (0 , + ∞ ), we hav e (6.11) ν = w eak ∗ − lim i →∞ r − q i ( I x,r i ) ♯ µ ⇐ ⇒ lim i →∞ ¯ d ( ν, r − q i ( I x,r i ) ♯ µ ) = 0 . The implication ⇒ is o b vious, because ¯ d -con v ergence is equiv alen t to d R -con v ergence for all R , and all we akly ∗ -con v ergent sequences are lo cally uniformly b ounded (see (3.4)). The implication ⇐ is analogous, but it dep ends on our c ho ice o f x , whic h ensures the prop ert y sup i ∈ N r − q i | ( I x,r i ) ♯ µ | ( B R ( e )) = sup i ∈ N r − q i | µ | ( B Rr i ( x )) ≤ R q lim sup r ↓ 0 | µ | ( B r ( x )) r q < + ∞ . This property ensures tha t r − q i ( I x,r i ) ♯ µ is b ounded in all M m ( B R ( e )) for all R > 0, and enables to pass f rom d R -con v ergence to w eak ∗ con vergenc e in all balls B R ( e ). Thanks to the equiv alenc e stated in (6.1 1), b y a diagonal argumen t it suffices to prov e that, fo r | µ | -a.e. x , the followin g prop erty holds: for all ν ∈ T an( µ, x ), y ∈ supp | ν | and r > 0 w e hav e r − q ( I y , r ) ♯ ν ∈ T an( µ, x ). But since t he op eration σ 7→ r − q ( I e,r ) ♯ σ is easily seen to map T an( µ, x ) in to T an ( µ, x ), and I y , r = I e,r ◦ I y , 1 , we need j ust to sho w that: (*) for | µ | - a.e. x the fo llo wing prop ert y holds: for all ν ∈ T a n( µ, x ) and all y ∈ supp | ν | , we ha v e ( I y , 1 ) ♯ ν ∈ T an( µ, x ). Heuristically , this prop erty holds at “Leb esgue” p oin ts o f the multiv a lued map x 7→ T an( µ, x ), thanks to the iden tit y (6.12) I δ 1 /r ( x − 1 y ) , 1 ◦ I x,r = I y , r . Indeed, this identit y implies that tang en ts to µ at x on the scale r are close to t angen ts to µ at y on the scale r when d ( x, y ) ≪ r . Let us consider the set R of p oin t s where the prop erty (*) fa ils: for all x ∈ R there exist a measure ν ∈ T an( µ, x ) and a point y ∈ supp | ν | suc h tha t ( I y , 1 ) ♯ ν / ∈ T an( µ, x ). T his implies, thanks to t he implication ⇐ in (6.11), the existence of integers z , k ≥ 1 suc h that the measure ( I y , 1 ) ♯ ν is 1 /k fa r (relativ e to ¯ d ) from the set r − q ( I x,r ) ♯ µ : r ∈ (0 , 1 /z ) } . Set A z ,k := { x ∈ G : ∃ ν ∈ T an( µ, x ) , ∃ y ∈ supp | ν | suc h that d (( I y , 1 ) ♯ ν, r − q ( I x,r ) ♯ µ ) > 1 /k , ∀ r ∈ (0 , 1 /z )  . 24 LUIGI AMBROSIO, BRUCE KLEINER, AND ENRICO LE DONN E Since R is contained in the union of these sets, to conclude the pro of it suffices to sho w that | µ | ∗ ( A z ,k ) = 0 for any z , k ≥ 1. Supp ose b y con tradiction | µ | ∗ ( A z ,k ) 6 = 0 for some z , k ≥ 1 and let us fix these t w o parameters; it is not difficult to c hec k that we can co v er the space M m ( G ) with a family { B l } of sets satisfying (6.13) ¯ d ( ν, ν ′ ) < 1 2 k ∀ ν, ν ′ ∈ B l . Let us no w consider the sets A z ,k ,l := { x ∈ G : ∃ ν ∈ T an( µ, x ) , ∃ y ∈ supp | ν | suc h that ( I y , 1 ) ♯ ν ∈ B l , d (( I y , 1 ) ♯ ν, r − q ( I x,r ) ♯ µ ) > 1 /k , ∀ r ∈ (0 , 1 /z )  . Since ∪ l A z ,k ,l con t ains A z ,k and | µ | ∗ is countably subadditiv e, at least one of t hese sets satisfies | µ | ∗ ( A z ,k ,l ) > 0. Let us fix l with this prop ert y , and let us denote A z ,k ,l b y A . Since | µ | ∗ ( A ) > 0 and | µ | is asymptotically do ubling, we can find a ∈ A whic h is a densit y p oin t of A relative to | µ | ∗ . ¿F rom no w on also t he p oint a will be fixed, and so an asso ciated measure ν a ∈ T an( µ, a ), a p oint y a ∈ supp | ν a | satisfying ( I y a , 1 ) ♯ ν a ∈ B l and (6.14) d (( I y a , 1 ) ♯ ν a , r − q ( I a,r ) ♯ µ ) > 1 k , ∀ r ∈ (0 , 1 /m ) . W e can also write ν a = lim i →∞ r − q i ( I a,r i ) ♯ µ , for suitable r i ↓ 0, and clearly (6.14) implies that y a 6 = e . Let us consider the p oints a · δ r i y a and their distance from A and tak e a i ∈ A suc h that dist( aδ r i y a , a i ) ≤ dist( aδ r i y a , A ) + r i /i . Lemma 6.7 yields that dist ( aδ r i y a , a i ) = o ( r i ) as i → ∞ , a nd so δ 1 /r i ( a − 1 a i ) → y a . No w, ( 6.12) shows that I δ 1 /r i ( a − 1 a i ) , 1 ◦ I a,r i = I a i ,r i , so that lim i →∞ r − q i ( I a i ,r i ) ♯ µ = lim i →∞ r − q i ( I δ 1 /r i ( a − 1 a i ) , 1 ) ♯ ( I a,r i ) ♯ µ = lim i →∞ ( I δ 1 /r i ( a − 1 a i ) , 1 ) ♯  r − q i ( I a,r i ) ♯ µ  = ( I y a , 1 ) ♯ ν a . So, we can fix i sufficien tly large such that r i < 1 /z and (6.15) d ( r − q i ( I a i ,r i ) ♯ µ, ( I y a , 1 ) ♯ ν a ) < 1 2 k . Since a i ∈ A = A z ,k ,l , w e can find a measure ν ′ ∈ T an( µ, a i ) a nd a p oint y ′ ∈ supp | ν ′ | with ( I y ′ , 1 ) ♯ ν ′ ∈ B l suc h that 1 k < d ( r − q i ( I a i ,r i ) ♯ µ, ( I y ′ , 1 ) ♯ ν ′ ) . By applying the tria ngle inequalit y w e obtain 1 k < d ( r − q i ( I a i ,r i ) ♯ µ, ( I y a , 1 ) ♯ ν a ) + d (( I y a , 1 ) ♯ ν a , ( I y ′ , 1 ) ♯ ν ′ ) < 1 2 k + 1 2 k , where w e used (6.1 5) and our c ho ice (6.13) of B l . The con t radiction ends the pro of o f the theorem. T ANGENT HYPERPLA NE I N CARN OT GROUPS 25 7. The Engel cone example In this section we revisit the example in [19] of a set with a constan t normal whic h is not a v ertical halfspace, a nd w e sho w why the improv emen t pro cedure do es not w ork, at least at some p oints , in this case. 7.1. The E ngel group. Let us recall the definition of Engel Lie algebra and group. Let E b e the Carnot g roup whose Lie a lgebra is g = V 1 ⊕ V 2 ⊕ V 3 with V 1 = span { X 1 , X 2 } , V 2 = { R X 3 } and V 3 = { R X 4 } , t he only non zero comm utation relations b eing (7.1) [ X 1 , X 2 ] = − X 3 , [ X 1 , X 3 ] = − X 4 . An explicit represen tation of the v ector fields in R 4 is: X 1 = ∂ 1 , X 2 = ∂ 2 − x 1 ∂ 3 + x 2 1 2 ∂ 4 , X 3 = ∂ 3 − x 1 ∂ 4 , X 4 = ∂ 4 . Clearly E is a Carnot g roup with step s = 3, top ological dimension n = 4, ho mogeneous dimension Q = 2 · 1 + 1 · 2 + 1 · 3 = 7 , and dimension of the hor izon ta l la yer m = 2. F rom no w on, w e shall use the co ordinates ab ov e to denote t he elemen ts of the group. 7.2. A c one in the Engel group. F or a n y α > 0, let P = P α : R 4 → R b e the p olynomial P ( x ) = α x 3 2 + 2 x 4 , whose gr adien t is ∇ P ( x ) =  0 , 3 αx 2 2 , 0 , 2  . In particular all leve l sets { P = c } of P are ob viously graphs of smo oth functions dep ending on ( x 1 , x 2 , x 3 ). The deriv ativ e of P is particularly simple along the v ector fields of the horizon tal lay er: indeed, w e hav e X 1 P ( x ) = ∂ 1 ( αx 3 2 + 2 x 4 ) = 0 and X 2 P ( x ) = [ ∂ 2 − x 1 ∂ 3 + x 2 1 2 ∂ 4 ]( αx 3 2 + 2 x 4 ) = 3 αx 2 2 + x 2 1 ≥ 0 . Hence (7.2) X 1 P ( x ) = 0 , X 2 P ( x ) = x 2 1 + 3 α x 2 2 ∀ x ∈ R 4 . W e define C := { x ∈ R 4 : P ( x ) ≤ 0 } , whose b oundary ∂ C is the set { P = 0 } . No tice that, due to the (intrins ic) homogeneit y of degree 3 of the p olynomial, the set C is a cone, i.e. δ r C = C for a ll r > 0. 26 LUIGI AMBROSIO, BRUCE KLEINER, AND ENRICO LE DONN E W e shall denote b y ν eu C ( x ) = ∇ P ( x ) / |∇ P ( x ) | the unit ( Euclidean) outer normal to C . W e also hav e the expansion |∇ P | ( x ) = q 4 + 9 α 2 x 4 2 = 2 + 9 2 α 2 x 4 2 + O ( d 4 ( x, 0)) . Thanks to Subsection 2 .2 the set C has lo cally finite p erime ter, and more precisely we hav e the form ula (2.8) (throughout this section H k is the Hausdorff measure induced b y the Euclidean distance) (7.3) Z 1 C = − Z P |∇ P | H 3 x ∂ C ∀ Z ∈ g . In particular (7.2) and (7.3) g iv e D 1 C = ( X 1 1 C , X 2 1 C ) = (0 , 1) X 2 1 C = − x 2 1 + 3 α x 2 2 |∇ P ( x ) | (0 , 1) H 3 x ∂ C . It fo llo ws that (7.4) | D 1 C | = x 2 1 + 3 α x 2 2 |∇ P ( x ) | H 3 x ∂ C and that the horizontal normal, tha t is the v ector field ν C = (0 , 1), is constant, so that al l p oin ts of supp | D 1 C | b elong to F C . Since w e prov ed in Lemma 5.8 that non-horizontal regular directions Z for E giv e rise, after blow-up, to in v ariant directions, at least at p oin ts ¯ x where | Z 1 E | ( B r ( ¯ x )) /r Q − 2 is infin- itesimal as r ↓ 0, and since the cone is self-similar under blo w-up at ¯ x = 0, it m ust happ en that | Z 1 C | ( B r (0)) /r Q − 2 is not infinitesimal as r ↓ 0 for an y non-horizontal regular directions Z (actually , for the cone C , al l directions are regular). Let us sho w explicitly this f act for Z := Ad exp( X 1 ) ( X 2 ): taking in to accoun t the comm utato r relatio ns (7.1) and Z := Ad exp( X 1 ) X 2 = X 2 + [ X 1 , X 2 ] + 1 2 [ X 1 , [ X 1 , X 2 ]] = X 2 − X 3 + 1 2 X 4 = ∂ 2 − x 1 ∂ 3 + x 2 1 2 ∂ 4 − ∂ 3 + x 1 ∂ 4 + ∂ 4 = ∂ 2 − (1 + x 1 ) ∂ 3 + (1 + x 1 ) 2 2 ∂ 4 . W e can now compute the deriv ativ e along the v ector field Z : Z P ( x ) =  ∂ 2 − (1 + x 1 ) ∂ 3 + (1 + x 1 ) 2 2 ∂ 4  ( αx 3 2 + 2 x 4 ) = 3 αx 2 2 + (1 + x 1 ) 2 = 1 + O ( d ( x, 0)) . In tuitively , the quotien t | Z 1 C | ( B r (0)) / | D 1 C | ( B r (0)) tends to + ∞ as r ↓ 0 b ecause of the relations (7.3) and (7.4), and the fact that Z P (0) 6 = 0 (notice that the factor |∇ P | is close T ANGENT HYPERPLA NE I N CARN OT GROUPS 27 to 2 near to the orig in). Let us make a more precise analysis: according to the ball-b o x theorem, balls B r (0) are comparable t o the b o xes Q r := [ − r , r ] 2 × [ − r 2 , r 2 ] × [ − r 3 , r 3 ] , so w e will compute the density on these b ox es, ra ther than on balls. W e shall assume, for the sake of simplicit y , that α ∈ (0 , 2]. The homog eneit y of C and t he fact that 0 ∈ F C giv e | D 1 C | ( Q r ) = cr 6 for some p ositiv e constan t c . The function x 4 = − αx 3 2 := g ( x 1 , x 2 , x 3 ) , whose gr aph is ∂ C , has absolute v alue strictly less t han r 3 , thus Q r ∩ ∂ C is the graph of g on the “basis” [ − r , r ] 2 × [ − r 2 , r 2 ] of the b o x Q r . Moreo ve r, since g has zero gradien t at the origin, H 3 ( Q r ∩ ∂ C ) = Z [ − r,r ] 2 × [ − r 2 ,r 2 ] p 1 + |∇ g | 2 d L 3 ∼ Z [ − r,r ] 2 × [ − r 2 ,r 2 ] 1 d L 3 = L 3 ([ − r , r ] 2 × [ − r 2 , r 2 ]) = 8 r 4 . F rom ( 7.3) w e o btain | Z 1 C | ( Q r ) = 4 r 4 + o ( r 4 ) and w e conclude that | Z 1 C | ( B r (0)) / | D 1 C | ( B r (0)) ∼ r − 2 . 7.3. A coun terexample to asymptotic stabilit y of halfspaces. Let us cons ider the set (7.5) E := { ( x 1 , x 2 , x 3 , x 4 ) ∈ E : x 2 + ar ctan( x 4 ) > 0 } . Since X 1 1 E = 0 and X 2 1 E ≥ 0 this set has a constan t horizon tal norma l, and clearly it is not an halfspace. On the other hand, it is not difficult to c hec k that the inclusions n x 2 > π 2 o ⊆ E ⊆ n x 2 > − π 2 o imply that E is asymptotic at infinit y to the halfpspace { x 2 > 0 } . 7.4. Ot her constant normal sets in t he Engel gr oup. W e prese n t here another fa mily of sets that ha v e constan t horizon tal normal. This time w e hav e a dep endence on t w o parameters a, b ∈ R . Let P a,b : R 4 → R b e the p olynomial P a,b ( x ) = 2 ax 4 − bx 3 + x 2 . Since ∂ 2 P a,b 6 = 0, all lev el sets { P a,b = c } of P a,b are obv iously smo oth manifolds. 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