Area formula for regular submanifolds of low codimension in Heisenberg groups
aa r X i v : . [ m a t h . M G ] F e b AREA FORMULA FOR REGULAR SUBMANIFOLDS OF LOWCODIMENSION IN HEISENBERG GROUPS
FRANCESCA CORNI AND VALENTINO MAGNANI
Abstract.
We establish an area formula for the spherical measure of intrinsically regularsubmanifolds of low codimension in Heisenberg groups. The spherical measure is computedwith respect to an arbitrary homogeneous distance. Among the arguments of the proof,we point out the differentiability properties of intrinsic graphs and a chain rule for intrinsicdifferentiable functions.
Contents
1. Introduction 12. Definitions and preliminary results 42.1. Coordinates in the Heisenberg group 42.2. Metric structure 52.3. Differentiability and factorizations 62.4. Intrinsic derivatives 122.5. Measures and area formulas 143. Low codimensional blow-up in Heisenberg groups 164. Applications 23References 251.
Introduction
Research in the Analysis and Geometry of simply connected nilpotent Lie groups hasspread into several directions, especially in the last decade. Carnot groups, or stratifiedgroups equipped with a homogeneous left invariant distance, are an important class ofthese nilpotent groups, which are metrically different from Euclidean spaces or Riemannianmanifolds, still maintaining a rich algebraic structure. The Heisenberg group H n representsone of the prominent models.More specifically, we are interested in computing the spherical measure of submanifoldsin H n with respect to a homogeneous distance of the group. For different classes of C smooth submanifolds area formulas are available, [17]. The question has new difficulties,when we consider “intrinsic regular submanifolds” of H n , that need not be C smooth,nor Lipschitz with respect to the Euclidean distance. These submanifolds in H n and theircharacterizations have been studied in several papers, [2], [6], [7], [20]. They are called Mathematics Subject Classification.
Primary 28A75; Secondary 53C17, 22E30.
Key words and phrases.
Heisenberg group, area formula, spherical measure, centered Hausdorff measure,intrinsic differentiability, chain rule.The second author was supported by the University of Pisa, Project PRA 2018 49. H -regular surfaces of low codimension in H n (Definition 2.10) and can be seen as levelsets of continuously differentiable functions from H n to R k , 1 ≤ k ≤ n . We stress thatdifferentiability is always understood with respect to the group operation and dilations, i.e.the so-called Pansu differentiability. In the general setting of intrinsic regular submanifoldsin homogeneous groups, H -regular surfaces of low codimension correspond to ( H n , R k )-submanifolds of H n with 1 ≤ k ≤ n , [14].An implicit function theorem, proved in [6], states that any of these H -regular surfacescan be locally seen as intrinsic graphs with respect to a special semidirect factorization (Def-inition 2.5). The parametrizing mapping φ associated to the factorization acts between thetwo factors, which are a vertical and a horizontal subgroup of H n (Definition 2.2). In [2] theauthors proved that φ is uniformly intrinsic differentiable (Definition 2.8), while it is onlyH¨older continuous with respect to the Euclidean metric. Uniform intrinsic differentiabilityfor maps acting between homogeneous subgroups has been characterized in various ways,[1], [3] [4], [5], [10]. Unexpectedly, this differentiability turns out to be the natural tool toperform the blow-up of intrinsic regular submanifolds and we will apply its characterizationstated in Theorem 2.9.The main result of this work is an area formula for the spherical measure of H -regularsurfaces of low codimension. Once a homogeneous distance d on H n is fixed, the sphericalmeasure with respect to d can be introduced through the usual Carath´eodory construction(Definition 2.15). Precisely, we choose 1 ≤ k ≤ n and a parametrized H -regular surfaceΣ of codimension k (Definition 2.12). We associate to Σ a “parametrized measure” µ ,using a defining mapping f and a parametrizing map φ , according to (1). This measurehad already appeared in [6], where the authors introduced it to prove an area formula forthe centered Hausdorff measure of Σ, see [20, Theorem 4.5]. The choice of the measure µ is furthermore justified by [4, Theorem 6.1], where it is shown how µ can be rewrittenuniquely in terms of intrinsic derivatives of the group valued mapping φ .Our central result is the following theorem. Theorem 1.1 (Upper blow-up) . Let H n = W ⋊ V be equipped with a homogeneous distance d and let Σ be a parametrized H -regular surface with respect ( W , V ) and having codimension k , with ≤ k ≤ n . Let φ : U → V be its parametrization, where U ⊂ W is open and thegraph mapping Φ : U → H n of φ defines the intrinsic graph Σ = Φ( U ) . If f ∈ C h (Ω , R k ) with Σ = f − (0) and J H f ( x ) > for all x ∈ Σ , then J V f ( y ) = k∇ V f ( y ) ∧ · · · ∧ ∇ V f k ( y ) k g > for any y ∈ Σ . Let v , . . . v k ∈ H be orthonormal vectors such that V = span { v , . . . , v k } and set V = v ∧ · · · ∧ v k . Consider an orthonormal basis { w k +1 , . . . , w n , e n +1 } of W anddefine N = w k +1 ∧ · · · ∧ w n ∧ e n +1 . Let us introduce the following measure (1) µ ( B ) = k V ∧ N k g Z Φ − ( B ) J H f (Φ( n )) J V f (Φ( n )) d H n +1 − kE ( n ) for every Borel set B ⊂ H n . Then for every x ∈ Σ we have θ n +2 − k ( µ, x ) = β d ( Tan (Σ , x )) . The geometric functions θ n +2 − k ( µ, · ) and β d ( · ) are the (2 n + 2 − k )-spherical Federerdensity of µ at x (Definition 2.17) and the spherical factor (Definition 2.18), respectively.The homogeneous tangent cone Tan(Σ , x ) of Σ at x is metrically defined (Definition 2.11) REA FORMULA IN HEISENBERG GROUPS 3 and for H -regular surfaces of low codimension equals ker Df ( x ), see [6, Proposition 3.29], or[14, Theorem 1.7] in the setting of homogeneous groups. More information on the notionsinvolved in Theorem 1.1 is provided in Section 2.The terminology “upper blow-up” goes back to [16], where a Federer density was firstcomputed with applications to sets of finite perimeter in stratified groups. Indeed, theFederer density is a suitable limit superior of the ratio between the measure we wish todifferentiate and the gauge of the spherical measure with respect to a class of sets, see [15]for more information.In our higher codimensional framework, the proof of the upper blow-up involves somenew features. Three key aspects must be emphasized. First, the intrinsic differentiability ofthe parametrizing map φ (Theorem 2.7) is crucial in establishing the limit of the set (24) inthe proof of the upper blow-up. Second, we prove an “intrinsic chain rule” (Theorem 2.2)that permits us to connect the kernel of Df with the intrinsic differential of φ , accordingto (26). However, to make our chain rule work we have slightly modified the well knownnotion of intrinsic differentiability associated to a factorization (Definition 2.9). We willdeserve more attention on this differentiability and the chain rule for next investigations,since they may have an independent interest. Third, we establish a delicate algebraiclemma for computing the Jacobian of projections between vertical subgroups, that areassociated to two semidirect factorizations with the same horizontal factor (Lemma 3.1).By combining Theorem 1.1 with an abstract measure theoretic area formula, see [15,Theorem 11] or [17, Theorem 7.2], we arrive at the area formula for an H -regular surface,using µ and the spherical measure S n +2 − k with respect to any homogeneous distance d .In the assumptions of Theorem 1.1, for any Borel set B ⊂ Σ we have µ ( B ) = Z B β d (Tan(Σ , x )) d S k +2 − k ( x ) . If the factors of the semidirect product are orthogonal, then the measure µ can be writtenin terms of the intrinsic partial derivatives of the parametrization φ of Σ as follows. Theorem 1.2.
In the assumptions of Theorem 1.1, if in addition W is orthogonal to V ,then for every Borel set B ⊂ Σ we have (2) Z B β d ( Tan (Σ , x )) d S k +2 − k ( x ) = Z Φ − ( B ) J φ φ ( w ) d H n +1 − kE ( w ) where J φ φ is the natural intrinsic Jacobian of φ , introduced in Definition 2.14. If the distance d is invariant under some classes of symmetries (Definition 2.19) or it ismultiradial (Definition 2.21), then the area formula simplifies (Theorem 4.2). Precisely,in these cases the spherical factor only depends on the distance and on the fixed scalarproduct on H n , becoming a geometric constant. We may denote it by ω d (2 n + 1 − k ) andinclude it in the spherical measure by defining S n +1 − kd = ω d (2 n + 1 − k ) S n +1 − k . Thus,by the area formula (2), we have S n +2 − kd ( B ) = Z Φ − ( B ) J φ φ ( w ) d H n +1 − kE ( w )for any Borel set B ⊂ Σ. For instance, the previous formula holds for the Cygan-Kor´anyidistance and the sub-Riemannian distance.
FRANCESCA CORNI AND VALENTINO MAGNANI
Some additional applications follow from our results. By a slight modification of theproof of Theorem 1.1, a standard blow-up theorem computing the centered density of µ at any point of Σ can be established (Theorem 3.2). By combining it with the measuretheoretic area formula for the centered density [8, Theorem 3.1], we obtain an area formulafor the centered Hausdorff measure of Σ, that extends the one of [6] to any homogeneousdistance (Theorem 4.3). We finally provide the cases where spherical measure and centeredHausdorff measure do coincide (Corollary 4.4).In conclusion, we wish to point out that our scheme to establish the area formula forintrinsic regular submanifolds has actually a more general scope. It may also include moregeneral classes of stratified groups, although some of our tools are still missing. For thisreason, these questions are the object of our subsequent investigations.2. Definitions and preliminary results
The next sections will introduce notions, notations and basic tools that will be usedthroughout the paper.2.1.
Coordinates in the Heisenberg group.
The purpose of this section is to introducethe Heisenberg group, along with the special coordinates that allow us to identify H n with R n +1 . The Heisenberg group H n can be represented as a direct sum of two linear subspaces H n = H ⊕ H with dim( H )= 2 n and dim( H )= 1, endowed with a symplectic form ω on H and a fixednonvanishing element e n +1 of H . We denote by π H and π H the canonical projections on H and H , which are associated to the direct sum.We can give to H n a structure of Lie algebra by setting(3) [ p, q ] = ω ( π H ( p ) , π H ( q )) e n +1 . Then the Baker-Campbell-Hausdorff formula ensures that(4) pq = p + q + [ p, q ]2defines a Lie group operation on H n . For t >
0, the linear mapping δ t : H n → H n such that δ t ( w ) = t k w if w ∈ H k ,k = 1 ,
2, defines intrinsic dilation .Given p ∈ H n , we denote by l p the translation by p . Any left invariant vector field on H n is of the form X v ( p ) = dl p (0)( v ) for any p ∈ H n and some v ∈ H n , where we haveidentified H n with T H n . Through the Baker-Campbell-Hausdorff formula, one can checkthat the Lie algebra of left invariant vector fields Lie( H n ) is isomorphic to the given Liealgebra ( H n , [ · , · ]).We fix a symplectic basis ( e , . . . , e n ) of ( H , ω ), namely ω ( e i , e n + j ) = δ ij , ω ( e i , e j ) = ω ( e n + i , e n + j ) = 0for every i, j = 1 , . . . , n , where δ ij is the Kronecker delta. Thus, we have obtained a Heisenberg basis B = ( e , . . . , e n +1 ) , REA FORMULA IN HEISENBERG GROUPS 5 that allows us to identify H n with R n +1 . The associated linear isomorphism is defined as(5) π B : H n → R n +1 , π B ( p ) = ( x , . . . , x n +1 )for p = P n +1 j =1 x j e j . We can read the given Lie product on R n +1 as follows[( x , . . . , x n +1 ) , ( y , . . . , y n +1 )] = π B ([ n +1 X i =1 x i e i , n +1 X i =1 y i e i ])= , . . . , , n X i =1 ( x i y i + n − x i + n y i ) ! then the group product takes the following form on R n +1 (6) ( x , . . . , x n +1 )( y , . . . , y n +1 ) = x + y , . . . , x n +1 + y n +1 + n X i =1 x i y i + n − x i + n y i ! . Taking into account (6), in our coordinates we obtain the following basis of left invariantvector fields X j ( p ) = ∂ x j − x j + n ∂ x n +1 j = 1 , . . . , nY j ( p ) = ∂ x n + j + 12 x j ∂ x n +1 j = 1 , . . . , nT ( p ) = ∂ x n +1 . (7)They clearly constitute a basis ( X , . . . , X n +1 ) of Lie( H n ) such that X j (0) = e j for every j = 1 , . . . , n +1. Any linear combination of X , . . . , X n is called a left invariant horizontalvector field of H n .2.2. Metric structure.
We fix a scalar product h· , ·i that makes our Heisenberg basis B = ( e , . . . , e n +1 ) orthonormal. In the sequel, any Heisenberg basis will be understoodto be orthonormal. We denote by | · | both the Euclidean metric on R n +1 and the norminduced by h· , ·i on H n . The symmetries of the Heisenberg group are detected through thethe isometry J : H → H , that is defined by the Heisenberg basis J ( e i ) = e n + i and J ( e n + i ) = − e i for all i = 1 , . . . , n . It is then easy to check that h p, q i = ω ( p, J q ) and J = − I for all p, q ∈ H .A homogeneous distance d on H n is a function d : H n × H n → [0 , + ∞ ) such that d ( zx, zy ) = d ( x, y ) and d ( δ t ( x ) , δ t ( y )) = td ( x, y )for every x, y, z ∈ H n and t >
0. Any two homogeneous distances are bi-Lipschitz equiv-alent. We also introduce the homogeneous norm k x k = d ( x, x ∈ H n , associated to ahomogeneous distance d . Notice that this norm satisfies k xy k ≤ k x k + k y k and k δ r x k = r k x k FRANCESCA CORNI AND VALENTINO MAGNANI for x, y ∈ H n and r > T H n with H n and by left translating the fixed scalar product h· , ·i on H n we obtain a left invariant Riemannian metric g on H n . Its associated Riemannian normis denoted by k · k g . We may restrict the identification of T H n with H n to the so calledhorizontal subspace, by identifying H with H H n ⊂ T H n . Then the horizontal fiber at p ∈ H n is H p H n = dl p (0)( H H n ). The collection of allhorizontal fibers constitutes the so-called horizontal subbundle H H n . If we restrict the leftinvariant metric g to the horizontal subbundle H H n , we obtain a scalar product on eachhorizontal fiber, that is the sub-Riemannian metric. This leads in a standard way to theso-called Carnot-Carath´eodory distance , or sub-Riemannian distance , see for instance [9].This is an example of homogeneous distance.2.3.
Differentiability and factorizations.
Natural notions of differentiability are wellknown in H n and general Carnot groups, starting from the notion of Pansu differentiability,[18]. Throughout the paper Ω denotes an open subset of H n and d is a fixed homogeneousdistance. Let f : Ω → R k , x ∈ Ω and v ∈ H . If there existslim t → f ( x ( tv )) − f ( x ) t ∈ R k , then we say that it is the horizontal partial derivative at x along X v , that is the unique leftinvariant vector field such that X v (0) = v . The above limit is denoted by X v f ( x ). Noticethat X v is precisely a left invariant horizontal vector field. We say that f ∈ C h (Ω , R k ) iffor every x ∈ Ω and every horizontal vector field X ∈ Lie( H n ) the horizontal derivative Xf ( x ) exists and it is continuous with respect to x ∈ Ω.A linear mapping L : H n → R k that is homogeneous, i.e. tL ( v ) = L ( δ t v ) for all t > v ∈ H n , is an h-homomorphism , that stands for “homogeneous homomorphism”. Ifthere exists an h-homomorphism L : H n → R k that satisfies | f ( xw ) − f ( x ) − Df ( x )( w ) | = o ( d ( w, d ( w, → , then it is unique and it is called the h-differential , or Pansu differential , of f at x . We denoteit by Df ( x ). Notice that f ∈ C h (Ω , R k ) if and only if it is everywhere Pansu differentiableand x → Df ( x ) is continuous as a map from Ω to the space of h-homomorphisms, see forinstance [13, Section 3]. Definition 2.1.
Let Ω ⊂ H n be an open set and let f ∈ C h (Ω , R ). We call horizontalgradient of f at x ∈ Ω the unique vector ∇ H f ( x ) of H such that Df ( x )( z ) = h∇ H f ( x ) , z i for every z ∈ H n .When differentiability meets the factorizations of the Heisenberg group, the notion of in-trinsic differentiability comes up naturally, see [20] for more information. Now we introducesome algebraic properties of factorizations in H n in order to define intrinsic differentiabilityand its basic properties. Definition 2.2.
If a Lie subgroup of H n is closed under intrinsic dilations, we call it a homogeneous subgroup . Homogeneous subgroups of H n containing H are called verticalsubgroups . Homogeneous subgroups contained in H are called horizontal subgroups . REA FORMULA IN HEISENBERG GROUPS 7
It is easy to realize that any homogeneous subgroup of H n is either horizontal or vertical.We also notice that normal homogeneous subgroups of H n coincide with vertical subgroups. Definition 2.3.
Let M and V be a vertical subgroup and a horizontal subgroup of H n ,respectively. We say that H n is the semidirect product of M and V and write H n = M ⋊ V if H n = MV and M ∩ V = { } . Definition 2.4.
Let M , W and V be homogeneous subgroups of H n such that(8) H n = M ⋊ V = W ⋊ V . The semidirect product
W ⋊ V automatically yields the unique projections π W : H n → W and π V : H n → V such that x = π W ( x ) π V ( x ) for every x ∈ H n . If necessary, to emphasize the dependence onthe semidirect factorization we will also introduce the notation π W , VW = π W and π W , VV = π V .The same holds for M ⋊ V . We define the following restrictions π W , VW , M = π W , VW | M : M → W and π M , VM , W = π M , VM | W : W → M Remark 2.1.
The uniqueness of the factorizations (8) implies that both restrictions π W , VW , M and π M , VM , W are invertible and(9) π W , VW , M = ( π M , VM , W ) − . If H n = M ⋊ V , then by the local compactness of H n , it is immediate to observe thatthere exists a constant c ∈ (0 , M and V , such that for all m ∈ M and v ∈ V the following holds(10) c ( k m k + k v k ) ≤ k mv k ≤ k m k + k v k . Remark 2.2.
Whenever two homogeneous subgroups W and V of H n satisfy H n = WV and W ∩ V = { } , then one of them must be necessarily vertical and the other one must be horizontal.We recall now definitions and results about intrinsic graphs of functions between twohomogeneous subgroups. For more information, see [20]. Definition 2.5.
Let H n = W ⋊ V and let U ⊂ W be a set. If φ : U → V , then we defineits intrinsic graph as the set graph( φ ) = { mφ ( m ) : m ∈ U } . We denote by Φ : U → Σ, Φ( m ) = mφ ( m ). We call Φ the graph map of φ . Remark 2.3.
The notion of graph is intrinsic, since both translations and dilations sendintrinsic graphs to new intrinsic graphs, with respect to a different function.Translating intrinsic graphs requires some preliminary notions.
Definition 2.6.
Let H n = W ⋊ V be a semidirect product. Let us consider x ∈ H n . Wedefine σ x : W → W as follows σ x ( η ) = π W ( l x ( η )) = xη ( π V ( x )) − . FRANCESCA CORNI AND VALENTINO MAGNANI
Given a set U ⊂ W and a function φ : U → V , the translation of φ at x , φ x : σ x ( U ) → V is defined as(11) φ x ( η ) = π V ( x ) φ ( x − ηπ V ( x )) = π V ( x ) φ ( σ x − ( η )) . Remark 2.4.
The map σ x is invertible on W σ x − ( η ) = x − ηπ V ( x − ) − = x − ηπ V ( x ) = σ − x ( η ) . Then for η ∈ σ x ( U ) we may also write(12) φ x ( η ) = π V ( x ) φ ( σ − x ( η )) . Next we recall the content of [2, Propositions 3.6].
Proposition 2.1.
Let H n = W ⋊ V be a semidirect product. Let U ⊂ W be an open setand φ : U → V be a function. Then we have l x (graph( φ )) = { ηφ x ( η ) : η ∈ σ x ( U ) } . Definition 2.7.
Let H n = W ⋊ V be a semidirect product. Let U ⊂ W be an open setand let φ : U → V be a function. Let us take ¯ w ∈ U and define x = ¯ wφ ( ¯ w ). The function φ is intrinsic differentiable at ¯ w if there exists an h-homomorphism L : W → V such that(13) d ( L ( w ) , φ x − ( w )) = o ( k w k )as w →
0. The function L is called the intrinsic differential of φ at ¯ w , it is uniquely definedand we denote it by dφ ¯ w . Remark 2.5.
By virtue of [2, Proposition 3.23], in our setting any intrinsic linear functionis actually an h-homomorphisms. We also observe that the assumption ¯ w ∈ U implies that0 ∈ σ x − ( U ). In addition σ x − ( U ) is an open set, hence the limit (13) is entirely justified. Remark 2.6.
By [2, Proposition 3.25], condition (13) is equivalent to ask that for all w ∈ U k dφ ¯ w ( ¯ w − w ) − φ ( ¯ w ) − φ ( w ) k = o ( k φ ( ¯ w ) − ¯ w − wφ ( ¯ w ) k )as k φ ( ¯ w ) − ¯ w − wφ ( ¯ w ) k → Definition 2.8.
Let H n = W ⋊ V be a semidirect product of H n . Let U ⊂ W be an openset and let φ : U → V be a function. The map φ is uniformly intrinsic differentiable on U if for any point ¯ w ∈ U there exists an h-homomorphism dφ ¯ w : W → V such that(14) lim δ → sup < k ¯ w − w ′ k <δ sup < k w k <δ d ( dφ ¯ w ( w ) , φ Φ( w ′ ) − ( w )) k w k = 0where Φ is the graph map of φ .The following definition is a slight modification of the notion of intrinsic differentiability. Definition 2.9.
Let W be a vertical subgroup of H n , let U ⊂ W be an open set and let F : U → R k with u ∈ U . We fix any horizontal subgroup V ⊂ H n such that H n = W ⋊ V and choose v ∈ V . We define x = uv in H n and the corresponding translated function F x − ( w ) = F ( σ x ( w )) − F ( u ) REA FORMULA IN HEISENBERG GROUPS 9 for w ∈ σ x − ( U ). We say that F is extrinsically differentiable at u with respect to ( W , V , x )if there exists an h-homomorphism L : W → R k such that(15) | F x − ( w ) − L ( w ) |k w k → w → L allows us to denote it by d W , V x F .The terminology extrinsic differentiabilty arises from the fact that the subgroup V andthe point x cannot be detected from the information we have on F . They are actually ar-tifically added from outside. However, when the factor V as metric spaces replaces R k and v = F ( w ), hence in the numerator of (15) becomes d ( F x − ( w ) , L ( w )), then extrinsic differ-entiability yields intrinsic differentiability. We have introduced this notion only to makesense of the following chain rule involving intrinsic differentiability. Somehow extrinsic andintrinsic differentiability compensate each other in the following theorem. Theorem 2.2 (Chain rule) . Let H n = W ⋊ V be a semidirect product. Let us considertwo open sets U ⊂ W , Ω ⊂ H n and two functions f : Ω → R k , φ : U → V . Assume Φ( U ) ⊂ Ω where Φ is the graph function of φ . Let us consider x W ∈ U and set x = Φ( x W ) .If f and φ are h-differentiable at x and intrinsic differentiable at x W , respectively, then thecomposition F = f ◦ Φ : U → R k , given by F ( u ) = f ( uφ ( u )) for all u ∈ U , is extrinsically differentiable at x W with respect to ( W , V , x ) . For every w ∈ W the formula (16) d W , V x F ( w ) = Df ( x )( wdφ x W ( w )) holds. If in addition f ( wφ ( w )) = c for every w ∈ U and some c ∈ R , then we obtain (17) ker( Df ( x )) = graph( dφ x W ) . Proof.
Let us first show that F is extrinsically differentiable at x W with respect to ( W , V , x ).We define L ( w ) = Df ( x )( wdφ x W ( w )) = Df ( x )( w ) + Df ( x )( dφ x W ( w ))for w ∈ W , that is an h-homomorphism. For w small enough, we have | F x − ( w ) − L ( w ) |k w k = | f ( xwx − V φ ( xwx − V )) − f ( x ) − L ( w ) |k w k = | f ( xwφ x − ( w )) − f ( x ) − Df ( x )( wdφ x W ( w )) |k w k≤ | f ( xwφ x − ( w )) − f ( x ) − Df ( x )( wφ x − ( w )) |k w k + | Df ( x )( wφ x − ( w )) − Df ( x )( wdφ x W ( w )) |k w k . Let us consider the last two addends separately: | f ( xwφ x − ( w )) − f ( x ) − Df ( x )( wφ x − ( w )) |k w k = | f ( xwφ x − ( w )) − f ( x ) − Df ( x )( wφ x − ( w )) |k wφ x − ( w ) k k wφ x − ( w ) kk w k → as k w k →
0, by the Pansu differentiability of f at x and by the validity of k wφ x − ( w ) kk w k ≤ k φ x − ( w ) kk w k = 1 + (cid:13)(cid:13)(cid:13)(cid:13) dφ x W (cid:18) w k w k (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) + k dφ x W ( w ) − φ x − ( w ) kk w k ≤ C x for all w = 0 and sufficiently small. It is indeed a consequence of the intrinsic differentia-bility of φ at x W . For the second addend, the previous intrinsic differentiability yields | Df ( x )( dφ x W ( w ) − φ x − ( w )) |k w k = (cid:12)(cid:12)(cid:12)(cid:12) Df ( x ) (cid:18) dφ x W ( w ) − φ x − ( w )) k w k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) → w →
0. This complete the proof of the first claim and also establishes formula (16).Let us now assume the constancy of w → f ( wφ ( w )) on U . Since we have proved that F is extrinsically differentiable at x W with respect to ( W , V , x ), being in this case F x − identically vanishing, we obtain d W , V x F ( w ) = o ( k w k )as w →
0. Therefore, for any u ∈ W , we have k Df ( x )( δ t udφ x W ( δ t u )) k = o ( t )as t →
0. Due to the h-linearity, it follows that Df ( x )( udφ x W ( u )) = 0 . We have proved the inclusion graph( dφ x W ) ⊂ ker( Df ( x )) of homogeneous subgroups withthe same dimension, hence formula (17) is established. (cid:3) The notion of H -regular surface in H n was first given in [6]. Definition 2.10.
Let Σ ⊂ H n be a set and let 1 ≤ k ≤ n . We say that Σ is an H - regularsurface of low codimension if for every x ∈ Σ there exist an open neighbourhood Ω suchthat x ∈ Ω and a function f = ( f , . . . , f k ) ∈ C h (Ω , R k ) such that(i) Σ ∩ U = { y ∈ Ω : f ( y ) = 0 } ,(ii) ∇ H f ( y ) ∧ · · · ∧ ∇ H f k ( y ) = 0 for all y ∈ Ω . We can characterize the metric tangent cone of an H -regular surface of codimension k . Definition 2.11.
For any set A ⊂ H n , x ∈ A , the homogeneous tangent cone is given bythe set Tan( A, x ) = n ν ∈ H n : ν = lim h →∞ δ r h ( x − x h ) , r h > , x h ∈ A, x h → x o . From [6, Proposition 3.29], we have the following characterization.
Proposition 2.3. If Σ is an H -regular surface of low codimension and f ∈ C h (Ω , R k ) isas in Definition 2.10, then ker Df ( x ) = Tan (Σ , x ) for all x ∈ Σ ∩ Ω . Given an open subset Ω ⊂ H n , a function f ∈ C h (Ω , R k ) and x ∈ Ω, we define the horizontal Jacobian J H f ( x ) = k∇ H f ( x ) ∧ · · · ∧ ∇ H f k ( x ) k g , where the norm is given through our fixed left invariant metric g . REA FORMULA IN HEISENBERG GROUPS 11 If V ⊂ H is a k dimensional subspace, we set ∇ V f ( x ) as the unique vector of V suchthat Df ( x )( z ) = h∇ V f ( x ) , z i for every z ∈ V . As a consequence, we can also define the Jacobian with respect to V , namely J V f ( x ) = k∇ V f ( x ) ∧ · · · ∧ ∇ V f k ( x ) k g . The next implicit function theorem is proved in [6, Theorem 3.27]. Its general version inthe framework of homogeneous groups is given in [14, Theorem 1.3].
Theorem 2.4 (Implicit function theorem) . Let Ω ⊂ H n be an open set, let f ∈ C h (Ω , R k ) be a function and consider a point x ∈ Ω such that J H f ( x ) > . Then there exists ahorizontal subgroup V such that J V f ( x ) > . We set Σ = { x ∈ Ω : f ( x ) = f ( x ) } and wefix a homogeneous subgroup W complementary to V . Setting π W ( x ) = η and π V ( x ) = υ ,there exist an open set Ω ′ ⊂ Ω ⊂ H n , with x ∈ Ω ′ , an open set U ⊂ W with η ∈ U and aunique continuous function φ : U → V such that φ ( η ) = υ and Σ ∩ Ω ′ = { wφ ( w ) : w ∈ U } . Definition 2.12 (Parametrized H -regular surface) . Let Σ ⊂ Ω be an H -regular surfaceand assume that there exist a semidirect factorization H n = W ⋊ V , an open set U ⊂ W and a continuous mapping φ : U → V such that Σ = { uφ ( u ) ∈ H n : u ∈ U } . We saythat Σ is a parametrized H -regular surface with respect to ( W , V ). We say that φ is the parametrization of Σ. Proposition 2.5.
Let f ∈ C h (Ω , R k ) be such that f − ( f ( p )) = Σ for some p ∈ Ω , andsuppose that for some horizontal subgroup V ⊂ H n it holds J V f ( x ) > for all x ∈ Σ . If W ⊂ H n is any vertical subgroup such that H n = W⋊V , then Σ is a parametrized H -regularsurface with respect to ( W , V ) .Proof. The proof follows by exploiting the local factorization given by the implicit functiontheorem of [6, Proposition 3.13]. (cid:3)
The proof of the following proposition is a simple application of Theorem 2.2.
Proposition 2.6.
Let φ : U → V , where U ⊂ W be open, and assume that φ is everywhereintrinsic differentiable. Let Σ = { nφ ( n ) : n ∈ U } and let f : Ω → R k be everywhere h-differentiable with Σ = f − ( f ( p )) ∩ ( U V ) for some p ∈ U V . If J H f ( x ) > for all x ∈ Σ , then J V f ( x ) > for all x ∈ Σ .Proof. We consider x = wφ ( w ), so by Theorem 2.2 the function F = f ◦ Φ is extrinsicallydifferentiable at w with respect to ( W , V , x ) and0 = d W , V x F ( v ) = Df ( x )( vdφ x W ( v )) = D W f ( x )( v ) + D V f ( x )( dφ x W ( v ))where v ∈ W and D S f ( x ) = Df ( x ) | S for any homogeneous subgroup S of H n . If bycontradiction D V f ( x ) : V → V would not be a isomorphism, then its image T would havelinear dimension less than k . Then the previous equalities would imply that the image of D W f ( x ) would be contained in T , hence the same would hold for the image of Df ( x ). Thisconflicts with the fact that Df ( x ) is surjective. (cid:3) We conclude this section by pointing out that the intrinsic graph in the implicit functiontheorem is suitably differentiable.
Theorem 2.7 ([2, Theorem 4.2]) . In the assumption of Theorem 2.4, φ is uniformlyintrinsic differentiable on U . Intrinsic derivatives.
In this section we recall some results about uniform intrinsicdifferentiability in Heisenberg groups. Throughout this section, we assume that H n is asemidirect product W ⋊ V with W orthogonal to V . The following proposition ensures thatwe can always find a Heisenberg basis which is adapted to this factorization. Proposition 2.8.
Let H n = W⋊V be a semidirect product, where the horizontal subgroup V is spanned by an orthonormal basis v , . . . , v k ∈ H and W is orthogonal to V , hence k ≤ n .Then there exist v k +1 , . . . , v n , w , . . . , w n ∈ H such that ( v k +1 , . . . , v n , w , . . . , w n , e n +1 ) isan orthonormal basis of W and ( v , . . . , v n , w , . . . , w n , e n +1 ) is a Heisenberg basis of H n .Proof. Since V is commutative, an element v = J ( w ) with v, w ∈ V satisfies | v | = h v, J ( w ) i = − ω ( v, w ) = 0 , therefore V ∩ J ( V ) = { } . We set J ( v i ) = w i ∈ W for i = 1 , . . . , k and define the2 k -dimensional subspace S = V ⊕ J ( V ) ⊂ H . We notice that dim( S ⊥ ∩ H ) = 2( n − k ) . If k < n , we pick a vector v k +1 ∈ S ⊥ ∩ H ofunit norm and define w k +1 = J v k +1 . It is easily observed that both w k +1 and v k +1 areorthogonal to S , so that ( v , . . . , v k +1 , w , . . . , w k +1 , e n +1 ) is a Heisenberg basis of S ⊕ span { e n +1 } , where we have defined S = V ⊕ span { v k +1 } ⊕ J ( V ⊕ span { v k +1 } ). Indeed, the previoussubspace has the structure of a 2 k + 3-dimensional Heisenberg group. One can iterate thisprocess until a Heisenberg basis of H n is found. (cid:3) From now on, we assume that ( v , . . . , v n , w , . . . , w n , e n +1 ) is the Heisenberg basis pro-vided by Proposition 2.8. We can identify V with R k and W with R n +1 − k through thefollowing diffeomorphisms i V : V → R k , i V k X i =1 x i v i ! = ( x , . . . , x k ) ,i W : W → R n +1 − k ,i W ze n +1 + n X i = k +1 ( x i v i + y i w i ) + k X i =1 η i v i ! = ( x k +1 , . . . , x n , η , . . . , η k , y k +1 , . . . , y n , z ) . We identify any function from an open subset U ⊂ W , φ : U → V with the correspondingfunction from an open subset e U ⊂ R n +1 − k , e φ : e U → R k : e φ ( w ) = i V ( φ ( i − W ( w ))) ∀ w ∈ e U = i W ( U ) ⊂ R n +1 − k . Any h-homomorphism L : W → V can be associated as before to a map e L : R n +1 − k → R k ,and so it can be identified with a k × (2 n − k ) matrix M L with real coefficients such thatfor every w ∈ R n +1 − k e L ( w ) = M L π ( w ) t , where π : R n +1 − k → R n − k is the canonical projection on the first 2 n − k components. REA FORMULA IN HEISENBERG GROUPS 13 If U ⊂ W is an open set and φ : U → V is intrinsic differentiable at a point w ∈ U , wedenote by D φ φ ( w ) the matrix associated to dφ w and we call it intrinsic Jacobian matrix of φ at w . If U ⊂ R n +1 − k is an open set and ψ = ( ψ , . . . , ψ k ) : U → R k is a function, wedefine the family of 2 n − k vector fields: W ψj = ( i W ) ∗ ( X j + k ) j = 1 , . . . , n − k ∇ ψ j − n + k = ∂ η j − n + k + ψ j − n + k ∂ z j = n − k + 1 , . . . , n ( i W ) ∗ ( Y j + k ) j = n + 1 , . . . , n − k. Definition 2.13 (Intrinsic derivatives) . Let U ⊂ W be an open set and let ¯ w be a pointof U . Let φ : U → V be a continuous function. Let j ∈ { , . . . , n − k } , we say that φ has ∂ φ j -derivative at ¯ w if and only if there exists a vector ( α ,j , . . . , α k,j ) ∈ R k such that for all γ j : ( − δ, δ ) → e U integral curves of W e φj with γ j (0) = i W ( ¯ w ) the limitlim t → e φ ( γ j ( t )) − e φ ( ¯ w ) t exists and it is equal to ( α ,j , . . . , α k,j ) . For all j = 1 , . . . , n − k we denote it by ∂ φ j φ ( ¯ w ) = ∂ φ j φ ... ∂ φ j φ k ( ¯ w ) = α ,j ... α k,j . . Uniform intrinsic differentiability has been characterized in various ways using intrinsicderivatives and the intrinsic Jacobian matrix.
Theorem 2.9 ([4, Theorem 5.7]) . Let U ⊂ W be an open set. Let φ : U → V be a function.We define Σ = Φ( U ) where Φ is the graph mapping of φ . Then the following are equivalent: (i) φ is uniformly intrinsic differentiable on U . (ii) φ ∈ C ( U ) and for every w ∈ U there exist ∂ φ j φ ( w ) for j = 1 , . . . n − k and thefunctions ∂ φ j φ : U → R k , are continuous. (iii) φ is intrinsic differentiable on U and the map D φ φ : U → M k, n − k ( R ) is continuous. (iv) There is an open set Ω ⊂ H n and f = ( f , ..., f k ) ∈ C h (Ω , R k ) such that Σ = { x ∈ Ω : f ( x ) = 0 } , and det([ X i f j ] i,j =1 ,...,k ( x )) = 0 for all x ∈ Σ . Definition 2.14.
Let U ⊂ W be an open set. Let φ : U → V be an intrinsic differentiablefunction at ¯ w ∈ U . We define the intrinsic Jacobian of φ at ¯ w as J φ φ ( ¯ w ) = vuut k X ℓ =1 X I ∈I ℓ ( M φI ( ¯ w )) , where we have defined I ℓ as the set of multiindexes { ( i , . . . , i ℓ , j , . . . , j ℓ )) ∈ N l : 1 ≤ i < i < · · · < i ℓ ≤ n − k, ≤ j < j · · · < j ℓ ≤ k } . We have also introduced the minors M φI ( ¯ w ) = det ∂ φ i φ j . . . ∂ φ iℓ φ j . . . . . . . . .∂ φ i φ j ℓ . . . ∂ φ iℓ φ j ℓ ( ¯ w ) . Measures and area formulas. If H n is endowed with a homogeneous distance d ,we denote by B ( x, r ) = { y ∈ H n : d ( x, y ) ≤ r } and for S ⊂ H n ,diam( S ) = sup { d ( x, y ) : x, y ∈ S } . Notice that diam( B ( x, r )) = 2 r for all x ∈ H n and r > Definition 2.15 (Carath´eodory’s construction) . Let
F ⊂ P ( H n ) be a non-empty familyof closed subsets of H n , equipped with a homogeneous distance d . Let be α >
0. If δ > A ⊂ H n , we define(18) φ αδ ( A ) = inf ( ∞ X j =0 c α diam( B j ) α : A ⊂ ∞ [ j =0 B j , diam( B j ) ≤ δ, B j ∈ F ) , If F coincides with the family of closed balls, F b , with respect to the distance d , then S α ( A ) = sup δ> φ αδ ( A )is the α -spherical measure of E . If in (18) we choose c α = 2 − α , then we use the symbol S α .In the case F is the family of all closed sets and k ∈ { , . . . , n + 1 } , we define c k = L k ( (cid:8) x ∈ R k : | x | ≤ (cid:9) )2 k where L k denotes the Lebesgue measure. Then the corresponding k -dimensional Hausdorffmeasure is given by H kE ( A ) = sup δ> φ kδ ( A )where H n is equipped with the Euclidean distance induced through the identification with R n +1 . These measures are Borel regular on subsets of H n . For our purposes, it is usefulto recall a less known Hausdorff-type measure, first introduced in [19]. Given α ∈ [0 , ∞ ), δ ∈ (0 , ∞ ), we define the m − dimensional centered Hausdorff measure C α of a set A ⊂ H n as C α ( A ) = sup E ⊂ A D α ( E )where D α ( E ) = lim δ → C αδ ( E ), and, in turn, C αδ ( E ) = 0 if E = ∅ and if E = ∅ C αδ ( E ) = inf ( X i r αi : E ⊂ ∪ i B ( x i , r i ) , x i ∈ E, diam( B ( x i , r i )) ≤ δ ) . Definition 2.16.
Let α > x ∈ H n and µ be a Borel regular measure on H n . Iflim inf r → µ ( B ( x, r )) r α = lim sup r → µ ( B ( x, r )) r α we define the spherical α -centered density of µ at x as θ αc ( µ, x ) = lim r → µ ( B ( x, r )) r α . REA FORMULA IN HEISENBERG GROUPS 15
Theorem 2.10 ([8, Theorem 3.1]) . Let α > and let µ be a Borel regular measure on H n such that there exists a countable open covering of H n , whose elements have µ finitemeasure. Let B ⊂ A ⊂ H n be Borel sets. If C α ( A ) < ∞ and µ x A is absolutely continuouswith respect to C α x A , then we have that θ αc ( µ, · ) is a Borel function on A and µ ( B ) = Z B θ αc ( µ, x ) d C α ( x ) . We introduce now a crucial definition of density.
Definition 2.17.
Let F b be the family of closed balls with positive radius in H n endowedwith an homogeneous distance d . Let α > x ∈ H n and µ be a Borel regular measure on H n . We call spherical α -Federer density of µ at x the real number θ α ( µ, x ) = inf ǫ> sup (cid:26) α µ ( B )diam( B ) α : x ∈ B ∈ F b , diam( B ) < ǫ (cid:27) . This density naturally appears in representing a Borel regular measure that is absolutelycontinuous with respect to the α -dimensional spherical measure. Theorem 2.11 ([15, Theorem 11]) . Let α > and let µ be a Borel regular measure on H n such that there exists a countable open covering of H n whose elements have µ finitemeasure. If B ⊂ A ⊂ H n are Borel sets, then θ α ( µ, · ) is a Borel function on A . If inaddition S α ( A ) < ∞ and µ x A is absolutely continuous with respect to S α x A , then µ ( B ) = Z B θ α ( µ, x ) d S α ( x ) . Definition 2.18 (Spherical factor) . Let d be a homogeneous distance in H n . If W is alinear subspace of topological dimension p of H n , then we define the spherical factor of W with respect to d as β d ( W ) = max z ∈ B (0 , H pE ( W ∩ B ( z, . When we deal with a homogeneous distance d that preserves some symmetries, then thespherical factor can become a geometric constant. The following definition detects thosehomogeneous distances giving a constant spherical factor. It extends [16, Definition 6.1]to higher codimension. Definition 2.19.
Let d be a homogeneous distance on H n and let p = 1 , . . . , n + 1. If p = 1 or p = 2 n + 1, then d is called p -vertically symmetric. In the case 1 < p ≤ n , the p -vertical symmetry requires the following conditions.We refer to the fixed graded scalar product h· , ·i and we assume that there exists a family F ⊂ O ( H ) of isometries such that for any couple of p -dimensional subspaces S , S ⊂ H ,there exists L ∈ F that satisfies the condition L ( S ) = S . Taking into account that H and H are orthogonal, we introduce the class of isometries O = { T ∈ O ( H n ) : T | H = Id | H , T | H ∈ F } . Then we say that d is p -vertically symmetric if the following holds: • π H ( B (0 , B (0 , ∩ H = { h ∈ H : θ ( | π H ( h ) | ) ≤ r } for some monotonenon-decreasing function θ : [0 , + ∞ ) → [0 , + ∞ ) and r > • T ( B (0 , B (0 ,
1) for all T ∈ O . More information on p -vertically symmetric distances can be found in [11]. For instance,the sub-Riemannian distance in the Heisenberg group is vertically symmetric. Verticallysymmetric distances were already introduced in [16].The next theorem specializes to the Heisenberg group a general result from [11]. Theorem 2.12. If p = 1 , . . . , n +1 and d is a homogeneous p -vertically symmetric distanceon H n , then the spherical factor β d ( W ) is constant on every p -dimensional vertical subgroup W ⊂ H n . The previous theorem motivates the following definition.
Definition 2.20.
If we have a class of p -dimensional homogeneous subgroups F for which β d ( S ) remains constant as S ∈ F , then we denote the spherical factor by ω d ( p ), withoutindicating the special class of subgroups. Definition 2.21 ([17, Definition 8.5]) . Let d be a homogeneous distance on H n . We saythat d is multiradial if there exists a function θ : [0 , + ∞ ) → [0 , + ∞ ), which is continuousand monotone non-decreasing on each single variable, with d ( x,
0) = θ ( | π H ( x ) | , | π H ( x ) | ) . The function θ is also assumed to be coercive in the sense that θ ( x ) → + ∞ as | x | → + ∞ . Proposition 2.13. If d : H n × H n → [0 , ∞ ) is multiradial, then it is also p -verticallysymmetric for every p = 1 , . . . , n + 1 . A more general statement can be found in [17]. One may also check that both d ∞ andthe Cygan-Kor´anyi distance are multiradial.It is also possible to find conditions under which the spherical factor has a simplerrepresentation. The next theorem is established in [11]. Theorem 2.14. If p = 1 , . . . , n + 1 and d is a homogeneous distance in H n whose unitball B (0 , is convex, then for every p -dimensional vertical subgroup W we have β d ( W ) = H pE ( W ∩ B (0 , . Low codimensional blow-up in Heisenberg groups
Our main result needs the following algebraic lemma.
Lemma 3.1.
We consider two vertical subgroups M , W of H n and a k -dimensional hori-zontal subgroup V ⊂ H n such that H n = M ⋊ V = W ⋊ V . We introduce the multivectors V = v ∧ · · · ∧ v k , N = w ∧ · · · ∧ w n − k ∧ e n +1 , M = m ∧ · · · ∧ m n − k ∧ e n +1 , where ( v , . . . , v k ) , ( w , . . . w n − k , e n +1 ) and ( m , . . . , m n − k , e n +1 ) are orthonormal basesof V , W and M , respectively. Then for every Borel set B ⊂ M , we have ( π M , VM , W ) ♯ H n +1 − kE ( B ) = H n +1 − kE ( π W , VW , M ( B )) = k V ∧ M k g k V ∧ N k g H n +1 − kE ( B ) , REA FORMULA IN HEISENBERG GROUPS 17 where the projections π M , VM , W and π W , VW , M have been introduced in Definition 2.4. The norms of V ∧ M and V ∧ N are taken with respect to the Hilbert structure of Λ n +1 ( H n ) induced byour scalar product on H n .Proof. It is clearly not restrictive to relabel the bases of M and W as w k +1 , . . . , w n , e n +1 and m k +1 , . . . , m n , e n +1 . We define the isomorphisms i W : W → R n +1 − k , i W x n +1 e n +1 + n X i = k +1 x i w i ! = ( x k +1 , . . . , x n +1 )and i M : M → R n +1 − k , i M x n +1 e n +1 + n X i = k +1 x i m i ! = ( x k +1 , . . . , x n +1 )and i V : V → R k i V k X i = i x i v i ! = ( x , . . . , x k ) . We introduce Ψ : R n +1 → H n ,(19) Ψ ( x , . . . , x n +1 ) = (cid:16) x n +1 e n +1 + n X i = k +1 x i w i (cid:17)(cid:16) k X j =1 x i v i (cid:17) . We now notice that J Ψ ( x ) = k V ∧ N k g for every x = ( x , . . . , x n +1 ) ∈ R n +1 . It sufficesto observe that J Ψ = k ∂ x Ψ ∧ · · · ∂ x n +1 Ψ n +1 k g and use the explicit form of (19). We define another map Ψ : R n +1 → H n ,Ψ ( x , . . . , x n +1 ) = (cid:16) x n +1 e n +1 + n X i = k +1 x i m i (cid:17)(cid:16) k X j =1 x i v i (cid:17) , and we observe in the same way that J Ψ ( x ) = k V ∧ M k g . We introduce the embedding q : R n +1 − k → R n +1 , q ( x , . . . , x n +1 − k ) = (0 , . . . , , x , . . . , x n +1 − k )and the projection p : R n +1 → R n +1 − k , p ( x , . . . , x n +1 ) = ( x k +1 , . . . , x n +1 ) . For every z ∈ H n , we observe thatΨ − ( z ) = ( i V ◦ π V ( z ) , i W ◦ π W ( z )) . It follows that i − W ◦ p ◦ Ψ − = π W . If we take any m ∈ M , then π W ( m ) = i − W ◦ p ◦ Ψ − ◦ Ψ ◦ Ψ − ( m )= i − W ◦ p ◦ Ψ − ◦ Ψ ◦ q ◦ i M ( m )= π W , VM ( m ) . (20)We notice that Ψ − ◦ Ψ is a polynomial diffeomorphism, whose Jacobian matrix at x hasthe following form I R R L ( x ) L ( x ) 1 ∈ R (2 n +1) × (2 n +1) , where I ∈ R k × k , R ∈ R k × (2 n − k ) , R ∈ R (2 n +1 − k ) × (2 n +1 − k ) and L : R n +1 → R k , L : R n +1 → R n − k are affine functions. From the definition of q : R n +1 − k → R n +1 and p : R n +1 −→ R n +1 − k , by explicit computation, it follows that(21) J (Ψ − ◦ Ψ ) = | det R | = J ( p ◦ Ψ − ◦ Ψ ◦ q ) . As a consequence, taking into account that k V ∧ M k g k V ∧ N k g = J (Ψ − ◦ Ψ )the following equalities hold H n +1 − kE ( B ) = L n +1 − k ( j ( B ))= k V ∧ N k g k V ∧ M k g L n +1 − k ( p ( L (( L − ( q ( j ( B )))))= k V ∧ N k g k V ∧ M k g H n +1 − kE ( i − ( p ( L ( L − ( q ( j ( B )))))= k V ∧ N k g k V ∧ M k g H n +1 − kE ( π W , VW , M ( B )) . (cid:3) We are now in the position to prove our main result.
Proof of Theorem 1.1.
Let us consider x ∈ Σ. By formula (1), for any y ∈ Ω, taking t > µ ( B ( y, t )) = k V ∧ N k g Z Φ − ( B ( y,t )) J H f (Φ( n )) J V f (Φ( n )) d H n +1 − kE ( n ) . We denote by ζ ∈ U the element such that x = Φ( ζ ) = ζ φ ( ζ ) . We now perform the change of variables n = σ x (Λ t ( η )) = x (Λ t η )( π V ( x )) − = x (Λ t η )( φ ( ζ )) − , REA FORMULA IN HEISENBERG GROUPS 19 where Λ t = δ t | W . The Jacobian of Λ t is t n +2 − k . It is well known that σ x has unit Jacobian(see for instance [7, Lemma 2.20]). Setting α ( x ) = J H f ( x ) /J V f ( x ), we obtain that µ ( B ( y, t )) t n +2 − k = k V ∧ N k g Z Λ /t ( σ − x (Φ − ( B ( y,t )))) ( α ◦ Φ)( σ x (Λ t ( η )))) d H n +1 − kE ( η ) . By the general definition of Federer density we obtain that θ n +2 − k ( µ, x ) = inf r> sup y ∈ B ( x,t )0
1) such that lim p →∞ δ /t p ( x − y p ) = z. For the sake of simplicity, we use the notation M x = ker Df ( x ) . Using the projection introduced in Definition 2.4, we set S z = π W , VW , M x ( M x ∩ B ( z, ⊂ W . Claim 1:
For each ω ∈ W \ S z , there existslim p →∞ Λ /tp ( σ − x (Φ − ( B ( y p ,t p ))) ( ω ) = 0 . By contradiction, if we had a subsequence of the integers p such that( δ /t p ( y − p x )) ω (cid:18) φ x − (Λ t p ω ) t (cid:19) ∈ B (0 , , then by a slight abuse of notation, we could still call t p the sequence such that(25) ( δ /t p ( y − p x )) ωdφ ζ ( ω ) (cid:18) ( dφ ζ (Λ t p ω )) − φ x − (Λ t p ω ) t p (cid:19) ∈ B (0 , p , where we have used the homogeneity of the intrinsic differential dφ ζ of φ , seeDefinition 2.7 for the notion of intrinsic differential. Indeed, by Theorem 2.7, the function φ is in particular intrinsic differentiable at ζ . Due to the intrinsic differentiability, takinginto account (25) as p → ∞ , it follows that ωdφ ζ ( ω ) ∈ B ( z, . It is now interesting to observe that the chain rule of Theorem 2.2 yields(26) graph( dφ ζ ) = ker( Df ( x ))) = M x . As a consequence, ωdφ ζ ( ω ) ∈ B ( z, ∩ M x and then(27) ω = π W , VW , M x ( ωdφ ζ ( ω )) ∈ π W , VW , M x ( M x ∩ B ( z, S z , that is not possible by our assumption. This concludes the proof of Claim 1.Now we introduce the density function α ( t, η ) = J H f (Φ( σ x (Λ t ( η ))) J V f (Φ( σ x (Λ t ( η )))to write k V ∧ N k g Z Λ /tp ( σ − x (Φ − ( B ( y p ,t p )))) α ( t p , η ) d H n +1 − kE ( η ) = I p + J p . The sequence I p , defined in the following equality, satisfies the estimate I p = k V ∧ N k g Z S z ∩ Λ /tp ( σ − x (Φ − ( B ( y p ,t p )))) α ( t p , η ) d H n +1 − kE ( η ) ≤ k V ∧ N k g Z S z α ( t p , η ) d H n +1 − kE ( η ) . REA FORMULA IN HEISENBERG GROUPS 21
Analogously for J p , we find J p = k V ∧ N k g Z Λ /tp ( σ − x (Φ − ( B ( y p ,t p )))) \ S z α ( t p , η ) d H n +1 − kE ( η ) ≤ k V ∧ N k g Z B W (0 ,R ) \ S z Λ /tp ( σ − x ((Φ − ( B ( y p ,t p )))) ( η ) α ( t p , η ) d H n +1 − kE ( η ) . Claim 1 joined with the dominated convergence theorem prove that J p → p → ∞ ,hence I p → θ n +2 − k ( µ, x ). To study the asymptotic behavior of I p , we first observe that α ( t p , η ) → J H f ( x ) J V f ( x ) = c ( x )as p → ∞ . It follows that(28) θ n +2 − k ( µ, x ) = lim p →∞ I p ≤ k V ∧ N k g c ( x ) H n +1 − kE ( S z ) . Claim 2 . We set M x = ker( Df ( x )) and consider N x = m k +1 ∧ · · · ∧ m n ∧ e n +1 suchthat ( m k +1 , . . . , m n , e n +1 ) is an orthonormal basis of M x . We have that(29) c ( x ) = J H f ( x ) J V f ( x ) = 1 k V ∧ N x k g . Since span {∇ H f ( x ) , . . . , ∇ H f k ( x ) } is orthogonal to M x , it is a standard fact that(30) m k +1 ∧ · · · ∧ m n ∧ e n +1 = ∗ ( ∇ H f ( x ) ∧ · · · ∧ ∇ H f k ( x )) λ for some λ ∈ R , see for instance [12, Lemma 5.1]. Here we have defined the Hodge operator ∗ in H n with respect to the fixed orientation e = e ∧ . . . e n ∧ e n +1 and the fixed scalar product h· , ·i . Precisely, we are referring to an orthonormal Heisenbergbasis ( e , . . . , e n , e n +1 ), according to Sections 2.1 and 2.2. Therefore ∗ η is the unique(2 n + 1 − k )-vector such that(31) ξ ∧ ∗ η = h ξ, η i e for all k -vectors ξ . Since the Hodge operator is an isometry, we get(32) | λ | = 1 k∇ H f ( x ) ∧ . . . ∇ H f k ( x ) k g . Due to (32) and (31), we have k V ∧ N x k g = | λ |kk v ∧ · · · ∧ v k ∧ ( ∗ ( ∇ H f ( x ) ∧ · · · ∧ ∇ H f k ( x ))) k g = kh v ∧ · · · ∧ v k , ∇ H f ( x ) ∧ · · · ∧ ∇ H f k ( x ) i e k g k∇ H f ( x ) ∧ · · · ∧ ∇ H f k ( x ) k g = |h v ∧ · · · ∧ v k , ∇ H f ( x ) ∧ · · · ∧ ∇ H f k ( x ) i|k∇ H f ( x ) ∧ · · · ∧ ∇ H f k ( x ) k g = k∇ V f ( x ) ∧ · · · ∧ ∇ V f k ( x ) k g k∇ H f ( x ) ∧ · · · ∧ ∇ H f k ( x ) k g = J V f ( x ) J H f ( x ) , hence establishing Claim 2.As a result, taking into account (28), we have proved that(33) θ n +2 − k ( µ, x ) ≤ k V ∧ N k g k V ∧ N x k g H n +1 − kE ( S z ) . By Lemma 3.1, for B = M x ∩ B ( z, H n +1 − kE ( π W , VW , M x ( M x ∩ B ( z, k V ∧ N x k g k V ∧ N k g H n +1 − kE ( M x ∩ B ( z, . It follows that(35) θ n +2 − k ( µ, x ) ≤ H n +1 − kE ( M x ∩ B ( z, ≤ H n +1 − kE ( M x ∩ B ( z , , where z ∈ B (0 ,
1) is chosen such that β d ( M x ) = H n +1 − kE ( M x ∩ B ( z , y t = xδ t z ∈ B ( x, t ) and fix λ >
1. We have thatsup
0, therefore(36) lim sup t → + µ ( B ( y t , λt ))( λt ) n +2 − k ≤ θ n +2 − k ( µ, x ) . We introduce the set A t = Λ /λt ( σ − x (Φ − ( B ( y t , λt )))= (cid:26) η ∈ Λ /λt ( σ − x ( U )) : η (cid:18) φ x − (Λ λt η ) λt (cid:19) ∈ B ( δ /λ z , (cid:27) . The second equality can be deduced from (24). Then we can rewrite µ ( B ( y t , λt ))( λt ) n +2 − k = k V ∧ N k g Z A t α ( λt, η ) d H n +1 − kE ( η )= k V ∧ N k g λ n +2 − k Z δ λ A t α ( λt, δ /λ η ) d H n +1 − kE ( η )(37)The domain of integration satisfies δ λ A t = (cid:26) η ∈ Λ /t ( σ − x ( U )) : η (cid:18) φ x − (Λ t η ) t (cid:19) ∈ B ( z , λ ) (cid:27) . Due to (23) and the definition of A t , it holds δ λ A t ⊂ B W (0 , λR ) . Claim 3:
For every η ∈ π W , VW , M x ( M x ∩ B ( z , λ )), we have(38) lim t → + δ λ A t ( η ) = 1 . The intrinsic differentiability of φ at ζ shows that η (cid:18) φ x − (Λ t η ) t (cid:19) → ηdφ ζ ( η ) as t → . REA FORMULA IN HEISENBERG GROUPS 23
Taking into account (9) and (27), we get π M x , VM x , W ( η ) = ηdφ ζ ( η ) , hence our assumption on η can be written as follows d ( ηdφ ζ ( η ) , z ) < λ. We conclude that η ∈ δ λ A t for any t > k V ∧ N k g λ n +2 − k Z π W , VW , M x ( M x ∩ B ( z ,λ )) lim inf t → (cid:0) δ λ A t ( η ) α ( λt, δ /λ η ) (cid:1) d H n +1 − kE ( η ) ≤ θ n +2 − k ( µ, x ) . Claim 3 joined with (29) yield1 λ n +2 − k k V ∧ N k g k V ∧ N x k g H n +1 − kE (cid:16) π W , VW , M x ( M x ∩ B ( z , (cid:17) ≤ θ n +2 − k ( µ, x ) . Applying again (34), we obtain1 λ n +2 − k H n +1 − kE ( M x ∩ B ( z , ≤ θ n +2 − k ( µ, x ) . Taking the limit as λ → + and considering the opposite inequality (35), the proof iscomplete. (cid:3) Adding some natural modifications in the proof of the previous theorem, we can alsoobtain the following “centered blow-up”.
Theorem 3.2.
In the assumptions of Theorem 1.1, for every x ∈ Σ , we have θ n +2 − kc ( µ, x ) = H n +1 − kE ( Tan (Σ , x ) ∩ B (0 , . Applications
Combining Theorems 2.11 and Theorem 1.1 we immediately get the following result.
Theorem 4.1 (Area formula) . In the assumptions of Theorem 1.1, for any Borel set B ⊂ Σ we have (39) µ ( B ) = Z B β d ( Tan (Σ , x )) d S k +2 − k ( x ) . We are now in the position to prove Theorem 1.2.
Proof of Theorem 1.2.
Since W and V are orthogonal, by Proposition 2.8 we can fix aHeisenberg basis ( v , . . . , v k , v k +1 , . . . , v n , w , . . . w n , e n +1 ) such that V = span { v , . . . , v k } and W = span { v k +1 , . . . , v n , w i , . . . , w n , e n +1 } . Our claim follows by representing the mea-sure µ in terms of the intrinsic partial derivatives of the parametrization φ of Σ, arguingas in the proof [4, Theorem 6.1]. For the reader’s convenience we report the main pointsof the proof.Taking into account Theorem 2.7, Σ = Φ(Ω) is the graph of a uniformly intrinsic differ-entiable function φ . Arguing as in the proof of [5, Theorem 4.1] or [2, Theorem 4.2], there exist an open set Ω ′ ⊂ H n and a function g ∈ C h (Ω ′ , R k ) such that Σ ⊂ g − (0) and forevery m ∈ U the following holds(40) Dg (Φ( m )) = ∇ H g (Φ( m )) . . . ∇ H g k (Φ( m )) = (cid:2) I k − D φ φ ( m ) (cid:3) . By Theorem 4.1, for any Borel set B ⊂ Σ, µ ( B ) = Z B β d (Tan(Σ , x )) d S k +2 − k ( x ) = Z Φ − ( B ) J H g (Φ( n )) J V g (Φ( n )) d H n +1 − kE ( n ) . Notice that J V g (Φ( m )) = 1 for every m ∈ U . By Definition 2.14, taking into account theform of Dg (Φ( m )) in (40), the proof is achieved. (cid:3) We now restrict our attention to homogeneous distances with symmetries. Using bothTheorem 2.12 and Proposition 2.13 we obtained simpler versions of the area formula.
Theorem 4.2.
Let d be either a (2 n + 1 − k ) -vertically symmetric distance or a multiradialdistance of H n . Then in the assumptions of Theorem 1.1, we have that (41) µ = ω d (2 n + 1 − k ) S k +2 − k x Σ . Therefore, by defining S n +2 − kd = ω d (2 n + 1 − k ) S n +1 − k , we have (42) S n +2 − kd x Σ = k V ∧ N k g Φ ♯ (cid:18) J H fJ V f ◦ Φ (cid:19) H n +1 − kE x W . In the assumptions of the previous theorem, assuming in addition that W and V areorthogonal, equation (42) can be rewritten for any Borel set B ⊂ Σ as(43) S n +2 − kd x Σ( B ) = Z Φ − ( B ) J φ φ ( w ) d H n +1 − kE ( w ) , where J φ φ the intrinsic Jacobian of φ defined in 2.14.By Theorem 2.10 and Theorem 3.2 we have the area formula for the centered Hausdorffmeasure. It is the analogous of Theorem 4.1. Theorem 4.3.
In the assumptions of Theorem 1.1, for any Borel set B ⊂ Σ we have (44) µ ( B ) = Z B H n +1 − kE ( Tan (Σ , x ) ∩ B (0 , d C k +2 − k ( x ) . When d = d ∞ the previous theorem recovers [6, Theorem 4.1]. Corollary 4.4.
Let d be an homogeneous distance on H n such that B (0 , is convex. Inthe assumptions of Theorem 1.1, for every x ∈ Σ we obtain (45) θ n +2 − kc ( µ, x ) = θ n +2 − k ( µ, x ) and C n +2 − k x Σ = S n +2 − k x Σ . Proof.
By Theorem 2.14 and Theorem 3.2, for every x ∈ Σ we have β d (ker( Df ( x )) = H n +1 − k (ker( Df ( x )) ∩ B (0 , θ n +2 − kc ( µ, x ) . Finally, the area formulas (39) and (44) conclude the proof. (cid:3)
REA FORMULA IN HEISENBERG GROUPS 25
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Dip.to di Matematica, Universit`a di Bologna, Piazza di Porta San Donato, 5, 40126,Bologna, Italy
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