On transversal submanifolds and their measure
aa r X i v : . [ m a t h . M G ] N ov ON TRANSVERSAL SUBMANIFOLDS AND THEIR MEASURE
VALENTINO MAGNANI, JEREMY T. TYSON, AND DAVIDE VITTONE
Abstract:
We study the class of transversal submanifolds. We characterizetheir blow-ups at transversal points and prove a negligibility theorem for their“generalized characteristic set”, with respect to the Carnot-Carath´eodoryHausdorff measure. This set is made by all points of non-maximal degree.Observing that C submanifolds in Carnot groups are generically transversal,the previous results prove that the “intrinsic measure” of C submanifoldsis generically equivalent to their Carnot-Carath´eodory Hausdorff measure.As a result, the restriction of this Hausdorff measure to the submanifoldcan be replaced by a more manageable integral formula, that should be seenas a “sub-Riemannian mass”. Another consequence of these results is anexplicit formula, only depending on the embedding of the submanifold, thatcomputes the Carnot-Carath´eodory Hausdorff dimension of C transversalsubmanifolds. Contents
1. Introduction 12. Notation and preliminary results 43. Blow-up at transversal points 124. Negligibility of lower degree points in transversal submanifolds 175. Size of the characteristic set for C ,λ submanifolds 25References 271. Introduction
A stratified group G is a connected, simply connected nilpotent Lie group, whoseLie algebra G has a special grading that allows for the existence of natural dilations Date : November 8, 2018.2010
Mathematics Subject Classification.
Primary 53C17; Secondary 22E25, 28A78.
Key words and phrases.
Stratified groups, submanifolds, Hausdorff measure.The first author acknowledges the support of the European Project ERC AdG *GeMeThNES*.The second author acknowledges the support of the US National Science Foundation Grants DMS-0901620 and DMS-1201875.The third author acknowledges the support of MIUR, GNAMPA of INDAM (Italy), Universityof Padova, Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: models, analy-sis, and control-theoretic problems” and University of Padova research project “Some analytic anddifferential geometric aspects in Nonlinear Control Theory, with applications to Mechanics”. along with a homogeneous distance that respect both dilations and group operation.The first developments of Geometric Analysis in the non-Riemannian framework ofstratified groups were mainly focused on geometric properties of domains in relationwith Sobolev embeddings (see, for instance, [6], [12], [16]), problems from the calculusof variations (e.g., [5], [10]), differential geometric calculus on hypersurfaces ([9]), andthe structure of finite perimeter sets (a very incomplete list of references includes [1],[7], [8], [13], [14], and [20]). The preceding lists of references are far from exhaustive,representing only a small sample of the rapidly expanding literature in the field ofsub-Riemannian geometric analysis.The study of finite perimeter sets and domains naturally connects with the study ofhypersurfaces and their Hausdorff measure. The fact that this measure is constructedby a fixed homogeneous distance of the group is understood. An important object inthis context is the so-called G -perimeter measure. It can be defined using a volumemeasure and a smooth left invariant metric on the horizontal subbundle of the group.This measure is equivalent, in the sense of (1.2) below, to the ( Q − C smooth domains in arbitrary stratified groups,[20], where Q is the Hausdorff dimension of G .The G -perimeter measure for regular sets has a precise integral formula that replacesthe Hausdorff measure and that does not contain the homogeneous distance. In fact,it is more manageable for minimization problems. In the development of GeometricMeasure Theory on stratified groups, a natural question arises: what is the “rightmeasure” replacing the G -perimeter measure for higher codimensional sets?In [23], a general integral formula for the “intrinsic measure” of C submanifoldshas been found: let Σ be a C smooth submanifold of G and define(1.1) µ Σ ( U ) = ˆ Φ − ( U ) k ( ∂ t Φ ∧ · · · ∧ ∂ t p Φ) D, Φ( t ) k dt where Φ : A → U ⊂ Σ is a local parametrization of Σ, A is an open set of R p and D is the degree of Σ, see Subsection 2.4 for more details. This measure yields theperimeter measure in codimension one and in several cases it is equivalent to H D Σup to geometric constants, where D is the Hausdorff dimension of Σ, namely,(1.2) C − H D Σ µ Σ C H D Σ . This equivalence already appears in [23] for C , smooth submanifolds in stratifiedgroups, under the key assumption that points of degree less than D are H D -negligible.Under this assumption, the equivalence (1.2) is a consequence of a “blow-up theorem”performed at each point of degree D , see [23, Theorem 1.1]. For more details on thenotion of degree, see Subsection 2.4.The previously mentioned H D -negligibility condition holds in many cases: for C , smooth submanifolds in two step stratified groups [22], and in the Engel group [19],for C smooth non-horizontal submanifolds in all stratified groups [20, 21], and for C smooth curves in all groups [18]. In all these cases, the equivalence (1.2) holds. Infact, when Σ is C , this is a consequence of the blow-up theorem of [23], while for RANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 3 the case of C smoothness the blow-up at points of degree D is established in [21] fornon-horizontal submanifolds and in [18] for all curves.Surprisingly, for C smooth submanifolds in stratified groups the equivalence (1.2)is an intriguing open question. One of the reasons behind this new difficulty is that,in higher codimension, submanifolds may belong to different classes, namely, theymay have different Hausdorff dimensions, while keeping the same topological dimen-sion. Simple examples of this phenomenon are given by the one dimensional homoge-neous subgroups, that have different Hausdorff dimensions according to their degree.Clearly, analogous examples can be easily found for higher dimensional homogeneoussubgroups. It is instructive to compare these cases with that of codimension onesubmanifolds, whose Hausdorff dimension must equal Q − horizontal fibers , this class is formed bynon-horizontal submanifolds, for which (1.2) holds, [21]. In higher codimension, thisclass is formed by transversal submanifolds . A transversal submanifold is easily definedas a top-dimensional submanifold among all submanifolds having the same topologicaldimension p . We have a precise formula for this maximal Hausdorff dimension D ( p ),see Section 2 for precise definitions.In this paper, we prove that transversal submanifolds in arbitrary codimension haveproperties similar to those of hypersurfaces. In fact, our main result is that (1.2) holdsfor all C smooth transversal submanifolds in arbitrary stratified groups. This followsby combining two key results: a blow-up theorem and a negligibility result, that arestated below. The estimates (1.2) show in particular that the Hausdorff dimension of C smooth transversal submanifolds is equal to D ( p ). This fact should be comparedwith [17, 0.6.B], where M. Gromov provides a formula for the Hausdorff dimensionof generic submanifolds. Gromov also introduces the number D H (Σ) associated witha submanifold Σ; this number coincides with the degree d (Σ) introduced in Subsec-tion 2.4, see [22, Remark 2] for a proof of this fact.Another motivation for our study of transversal submanifolds is that C smooth sub-manifolds are generically transversal, namely, “most” C submanifolds are transversal.This suggests that these submanifolds are important in the subsequent study of highercodimensional submanifolds in Carnot groups. The fact that transversality is a genericproperty can be seen for instance as a simple consequence of our Lemma 2.11 and thenarguing as in [21, Section 4].The main results of our work are a “blow-up theorem” and an H D ( p ) -negligibilitytheorem for all C smooth transversal submanifolds. These theorems extend the blow-up theorem of [21] and the negligibility theorem of [20]. Theorem 1.1 (Blow-up theorem) . If Σ is a C smooth submanifold and x ∈ Σ istransversal, then, for every compact neighbourhood F of , we have (1.3) F ∩ δ /r ( x − Σ) → F ∩ Π Σ ( x ) as r → + where the convergence is in the sense of the Hausdorff distance between compact setsand Π Σ ( x ) is a p -dimensional normal homogeneous subgroup of G , having Hausdorff VALENTINO MAGNANI, JEREMY T. TYSON, AND DAVIDE VITTONE dimension equal to D ( p ) . Moreover, the following limit holds (1.4) lim r → + ˜ µ (cid:0) Σ ∩ B ( x, r ) (cid:1) r D ( p ) = θ dg (cid:0)(cid:0) τ Σ ( x ) (cid:1) D ( p ) ,x (cid:1) k (cid:0) τ Σ , ˜ g ( x ) (cid:1) D ( p ) ,x k . The proof of this theorem is given in Section 3. This section, along with Section 2,also contains the definitions of the relevant notions. It is worth to mention that inthe case of C , regularity the blow-up at transversal points is already contained in[23]. In our case, where Σ is only C , the approach of [23] does not apply, so we followthe method used in [18] for curves. The point here is to provide a special “weightedreparametrization” of Σ around the blow-up point, see (3.8). Our next main result isthe following generalized negligibility theorem. Theorem 1.2 (Negligibility theorem) . Let Σ ⊂ G be a p -dimensional C submanifoldand let Σ c ⊂ Σ denote the subset of points with degree less than D ( p ) . We have (1.5) H D ( p ) (Σ c ) = 0 . We refer to Section 4 for the definition of the generalized characteristic set Σ c .The proof of Theorem 1.2 relies on covering arguments and a number of technicallemmata, that aim to estimate the behaviour of the number of small balls coveringthe generalized characteristic set. The difficulty here is to properly translate theinformation on the lower degree of the points into concrete estimates on the best“local coverings” around these points, see Lemma 2.3, Lemma 4.1 and Lemma 4.2.Theorem 1.2 extends to Euclidean Lipschitz submanifolds using standard arguments,see Theorem 4.5. Arguing the same way, one also realizes that estimates (1.2) extendto all Euclidean Lipschitz transversal submanifolds.The method to prove Theorem 1.2 can also be used to establish new estimates onthe Carnot-Carath´eodory Hausdorff dimension of Σ c for C ,λ submanifolds in Carnotgroups, where 0 < λ
1. These estimates, proved in Theorem 5.3, show that theCarnot-Carath´eodory Hausdorff dimension of Σ c can be estimated from above by abound smaller than D ( p ). Both Theorems 1.2 and 5.3 generalize some results proved,in the Heisenberg group framework, in the fundamental paper [2], compare Remark 5.4.We do not know whether the estimates of Theorem 5.3 are sharp. Even in Heisenberggroups, this sharpness seems to be an interesting open question. We refer to [3] forresults and open problems akin to that of estimating the size of Σ c .The validity of (1.2) for a large class of submanifolds makes the intrinsic measure(1.1) a reasonable notion of “sub-Riemannian mass”. This should be seen for instancein the perspective of studying special classes of isoperimetric inequalities, when eitherthe filling current or the filling submanifold must be necessarily transversal, as it occursfor higher dimensional fillings in Heisenberg groups.2. Notation and preliminary results
Carnot groups and exponential coordinates.
Let us start with a brief in-troduction to stratified groups; we refer to [15] for more details on the subject.
RANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 5
Let G be a connected and simply connected Lie group with stratified Lie algebra G = V ⊕ · · · ⊕ V ι of step ι , satisfying the conditions V i +1 = [ V , V i ] for every i > V ι +1 = { } . We set(2.1) n j := dim V j and m j := n + · · · + n j , j = 1 , . . . , ι ;we will also use m := 0. The degree d j of j ∈ { , . . . , n } is defined by the condition m d j − + 1 j m d j . We denote by n the dimension of G , therefore n = m ι . We say that a basis ( X , . . . , X n )of G is adapted to the stratification , or in short adapted , if X m j − +1 , . . . , X m j is a basis of V j for any j = 1 , . . . , ι. In the sequel, we will fix a graded metric g on G , namely, a left invariant Riemannianmetric on G such that the subspaces V k are orthogonal. Definition 2.1.
An adapted basis ( X , . . . , X n ) of G that is also orthonormal withrespect to a left invariant Riemannian metric is a graded basis .Clearly, the Riemannian metric in the previous definition must be necessarily graded.When either an adapted or a graded basis is understood, we identify G with R n bythe corresponding exponential coordinates of the first kind.We use two different ways of denoting points x of G with respect to fixed exponentialcoordinates of the first kind adapted to a graded basis of G . We use both the standardnotation with “lower indices” x = ( x , . . . , x n ) ∈ R n and the one with “upper indices” x = ( x , x , . . . , x ι ) ∈ R n × R n × · · · × R n ι , where clearly x j = ( x m j − +1 , . . . , x m j ) ∈ R n j for all j = 1 , . . . , ι . By the Baker-Campbell-Hausdorff formula, the group law reads in coordinates as(2.2) x · y = x + y + Q ( x, y ) , for a suitable polynomial function Q : R n × R n → R n . Precisely, Q = ( Q , . . . , Q n )can be written in the form(2.3) Q j ( x, y ) = X k,h : d k
VALENTINO MAGNANI, JEREMY T. TYSON, AND DAVIDE VITTONE
Stratified groups as abstract vector spaces.
To emphasize some intrinsic notionson stratified groups, while preserving the ease of using a linear structure, stratifiedgroups can be also regarded as abstract vector spaces. In fact, connected and simplyconnected nilpotent Lie groups are diffeomorphic to their Lie algebra through theexponential mapping exp :
G → G , that is an analytic diffeomorphism. As a result, weequip the Lie algebra of G with a Lie group operation, given by the Baker-Campbell-Hausdorff series, that makes this Lie group isomorphic to the original G . This allowsus to consider G as an abstract linear space, equipped with a polynomial operationand a grading G = H ⊕ · · ·⊕ H ι . Under this identification a graded basis ( X , . . . , X n )becomes an orthonormal basis of G as a vector space, where ( X m j − +1 , . . . , X m j ) is anorthonormal basis of H j for all j = 1 , . . . , ι . Remark 2.2.
When a stratified group G is seen as an abstract vector space, equippedwith a graded basis X , . . . , X n , then the associated graded metric g makes this basisorthonormal. As a result, the metric g becomes the Euclidean metric with respect tothe corresponding coordinates ( x , x , . . . , x ι ).2.1.2. Dilations.
For every r >
0, a natural group automorphism δ r : G → G can bedefined as the unique algebra homomorphism such that δ r ( X ) := rX for every X ∈ V . This one parameter group of linear isomorphisms constitutes the family of the so-called dilations of G . They canonically yield a one parameter group of dilations on G andcan be denoted by the same symbol. With respect to our coordinates, we have δ r ( x , . . . , x j , . . . , x n ) = ( rx , . . . , r d j x j , . . . , r ι x n ) . Left translations.
For each element x ∈ G , the group operation of G automat-ically defines the corresponding left translation l x : G → G , with l x ( z ) = xz for all z ∈ G . Right translations r x are defined in analogous way.2.2. Metric facts.
We will say that d is a homogeneous distance on G if it is acontinuous distance on G satisfying the following conditions(2.5) d ( zx, zy ) = d ( x, y ) and d ( δ r ( x ) , δ r ( y )) = rd ( x, y ) ∀ x, y, z ∈ G , r > . Important examples of homogeneous distances are the well known Carnot-Carath´eo-dory distance and those constructed in [14]. It is easily seen that two homogeneousdistances are always equivalent. We will denote by B ( x, r ) and B E ( x, r ), respectively,the open balls of center x and radius r with respect to a (fixed) homogeneous distance d and the Euclidean distance on R n ≡ G .For r > boxes Box(0 , r ) := { y ∈ R n : | y j | < r j ∀ j = 1 , . . . , ι } = ( − r, r ) n × ( − r , r ) n × · · · × ( − r ι , r ι ) n ι Box( x, r ) := x · Box(0 , r ) , x ∈ G . RANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 7
By homogeneity, it is easy to observe for any homogeneous distance d there exists C BB = C BB ( d ) > x, r/C BB ) ⊂ B ( x, r ) ⊂ Box( x, C BB r ) . We will also use the notationBox µE (0 , r ) := { y ∈ R µ : | y j | < r ∀ j = 1 , . . . , µ } = ( − r, r ) µ . When given 0 < s < r and a linear subspace W of R µ we poseBox µW ⊥ ⊕ W (0; r, s ) := { y ∈ Box µE (0 , r ) : | π W ( y ) | < s } , where π W ( y ) is the canonical projection of y on W . If w , . . . , w H is an orthonormalbasis of W we clearly have(2.7) { y ∈ Box µE (0 , r ) : |h y, w i i| < s √ H ∀ i = 1 , . . . , H } ⊂ Box µW ⊥ ⊕ W (0; r, s ) . From now on, a homogeneous distance d is fixed. We will use several times thefollowing simple fact. Lemma 2.3.
There exists C = C ( d ) > with the following property. For any fixed r , x ∈ B E (0 , r ) and j ∈ { , . . . , ι } there exists ˜ x ∈ G such that ˜ x = · · · = ˜ x j = 0 , d ( x, ˜ x ) Cr /j (2.8) | ˜ x h − x h | Cr for any h = j + 1 , . . . , ι. (2.9) Proof.
In the case j = 1, we define˜ x = x · ( − x , , . . . ,
0) = ( x , . . . , x ι ) · ( − x , , . . . , , x + O ( r ) , . . . , x ι + O ( r )) , where the last equality follows from (2.4) and O ( · ) is understood with respect to theEuclidean norm. By (2.6) we have d ( x, ˜ x ) = d (0 , ( − x , , . . . , C BB r whence (2.8) and (2.9) follow.We now argue by induction on j >
2, assuming the existence of some ¯ x such that¯ x = · · · = ¯ x j − = 0 , d ( x, ¯ x ) Cr / ( j − and | ¯ x h − x h | Cr for any h = j, . . . , ι . Defining ˜ x = ¯ x · (0 , . . . , , − ¯ x j , , . . . , x = (¯ x , . . . , ¯ x ι ) · (0 , . . . , , − ¯ x j , , . . . , , . . . , , , ¯ x j +1 + O ( r ) , . . . , ¯ x ι + O ( r )) . Thus, by inductive hypothesis we get ˜ x = (0 , . . . , , , x j +1 + O ( r ) , . . . , x ι + O ( r )).As a result, we arrive at the following inequalities d ( x, ˜ x ) d ( x, ¯ x ) + d (¯ x, ˜ x ) Cr / ( j − + d (0 , (0 , . . . , , − ¯ x j , , . . . , Cr /j + C BB | ¯ x j | /j = Cr /j + C BB | x j + O ( r ) | /j e Cr /j . that complete the proof. (cid:3) VALENTINO MAGNANI, JEREMY T. TYSON, AND DAVIDE VITTONE
Hausdorff measures and coverings.
For the sake of completeness, we recall thedefinitions of Hausdorff measures. Let q > δ > H qδ ( E ) := inf ( ∞ X i =1 (diam E i ) q : E ⊂ ∪ i E i , diam E i < δ ) S qδ ( E ) := inf ( ∞ X i =1 (diam B i ) q : E ⊂ ∪ i B i , B i = B ( x i , r i ) balls , diam B i < δ ) . The q -dimensional Hausdorff measure of E ⊂ G is H q ( E ) := lim δ → + H qδ ( E )while the q -dimensional spherical Hausdorff measure of E is S q ( E ) := lim δ → + S qδ ( E ) . The
Hausdorff dimension of E isdim H E := inf { q : H q ( E ) = 0 } = sup { q : H q ( E ) = + ∞} . It is well-known that dim H G coincides with the homogeneous dimension Q := n +2 n + · · · + ιn ι of G . The standard Euclidean Hausdorff measure on G = R n is denotedby H q |·| . For more information on the properties of these measures, see for instance[11, 24, 27].We state without proof the following simple fact. Proposition 2.4.
Let θ > and let E ⊂ X , where X is a metric space. If for all ǫ ∈ (0 , the set E can be covered by N ǫ balls of radius ǫ β with N ǫ C ǫ − q and C > independent from ǫ , then the Hausdorff dimension of E is not greater than q/β . The following result, see e.g. [27, Theorem 3.3], will be useful in the sequel.
Theorem 2.5 (5 r -covering) . Let ( X, d ) be a separable metric space and E ⊂ X ; let r > be fixed. Then there exists a subset F ⊂ E at most countable such that E ⊂ [ x ∈ F B ( x, r ) and B ( x, r ) ∩ B ( x ′ , r ) = ∅ ∀ x, x ′ ∈ F, x = x ′ . Multi-indices, degrees and maximal dimension.
We denote by I p the set ofthose multi-indices α = ( α , . . . , α p ) ∈ { , . . . , n } p such that 1 α < · · · < α p n .We also set d ( α ) := d α + · · · + d α p . We denote by D ( p ) the maximum integer d ( α ) when α varies in I p . We call this num-ber the maximal dimension , that is uniquely defined for any given p ∈ { , , . . . , n } .Clearly, D ( n ) equals the homogeneous dimension Q of G . The maximal dimensioncan be computed in the following way. Define ℓ = ℓ ( p ) by imposing(2.10) ℓ := ι if p n ιι X j = ℓ +1 n j < p ι X j = ℓ n j otherwise. RANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 9
Clearly, ℓ depends on p and it can be equivalently defined by ℓ ( p ) := d n +1 − p , thatrepresents the lowest possible degree among tangent vectors of span { X n − p +1 , . . . , X n } ,where ( X , . . . , X n ) is an adapted basis of the stratified Lie algebra G . It is also easyto see that(2.11) D ( p ) = ι X j = ℓ ( p )+1 jn j + ℓ ( p ) (cid:16) p − ι X j = ℓ ( p )+1 n j (cid:17) , where the two summations in (2.11) have to be understood as 0 when ℓ ( p ) = ι . Wealso set(2.12) r p := p − ι X j = ℓ ( p )+1 n j > , so that(2.13) p = r p + n ℓ ( p )+1 + · · · + n ι and D ( p ) = ℓ ( p ) r p + ( ℓ ( p ) + 1) n ℓ ( p )+1 + · · · + ιn ι . It is worth noticing that D ( p ) = β + ( p ), where β + is the upper dimension comparisonfunction for G , introduced in [4].2.4. Degree of submanifolds, projections and subdilations.
By Σ ⊂ G , wedenote a p -dimensional Lipschitz submanifold of G . We define the singular set Σ ∗ := { x ∈ Σ : C x Σ is not a p -dimensional subspace of R n } , where C x Σ is the (Euclidean) tangent cone to Σ at x , i.e., C x Σ := n tv ∈ R n : t > , v = lim i →∞ x i − x | x i − x | for some sequence ( x i ) i ∈ N ⊂ Σ with x i → x o . We have the following fact.
Proposition 2.6.
For any p -dimensional Lipschitz submanifold Σ ⊂ G , we have (2.14) H D ( p ) (Σ ∗ ) = 0 . Proof.
Since H p |·| (Σ ∗ ) = 0, by [4, Proposition 3.1] our claim immediately follows. (cid:3) Given a point x ∈ Σ \ Σ ∗ , we denote by τ Σ ( x ) its tangent vector, i.e., the p -dimensional multivector associated with the p -plane C x Σ. We can write τ Σ ( x ) = X α ∈ I p c α X α , where X α := X α ∧ . . . ∧ X α p . We then define the degree d Σ ( x ) of Σ at x as d Σ ( x ) := max { d ( α ) : α ∈ I p and c α = 0 } . The degree of Σ is d (Σ) := max { d Σ ( x ) : x ∈ Σ \ Σ ∗ } . Clearly, we have d (Σ) D ( p ).The underlying metric g on G gives a natural scalar product on multivectors, whosenorm will be denoted by k · k . If ˜ g is any Riemannian metric on G , then at any x ∈ G we have a canonically defined scalar product on any Λ p S x where S x ⊂ T x G isa p -dimensional subspace. We will denote by k · k ˜ g,x its corresponding norm. Definition 2.7.
Let ˜ g be any Riemannian metric on G , let Σ be a p -dimensionalLipschitz submanifold and let x ∈ Σ \ Σ ∗ . We define the unit tangent p -vector withrespect to ˜ g as follows(2.15) τ Σ , ˜ g ( x ) = τ Σ ( x ) k τ Σ ( x ) k ˜ g,x , where τ Σ ( x ) is any tangent p -vector of Σ at x .Dilations of G canonically extend to dilations on multivectors as follows(Λ p δ r )( v ∧ . . . ∧ v p ) = ( δ r v ) ∧ . . . ∧ ( δ r v p )for all v , . . . , v p ∈ G , therefore we have(Λ p δ r )( X α ) = (Λ p δ r )( X α ∧ . . . ∧ X α p ) = r d α + ··· + d αp X α = r d ( α ) X α . Definition 2.8. A p -vector v ∈ Λ p G is homogeneous of degree l ∈ N \ { } if (Λ p δ r ) v = r l v for all r > p ( G ) allows us to introduce the following canonical projec-tions, hence homogeneous multivectors of different degrees are orthogonal. Definition 2.9.
Let p, D ∈ N be such that 1 p D D ( p ). Let us introduce thelinear subspace Λ D,p ( G ) of Λ p G made by all homogeneous p -multivectors of degree D .With respect to the scalar product of G , the following orthogonal projection π D : Λ p ( G ) → Λ D,p ( G )is uniquely defined. We say that π D is the projection of degree D . If we considera p -vector t ∈ Λ p S x with S x ⊂ T x G , a pointwise projection π D,x ( t ) is automaticallydefined, taking the left translated multivector (Λ p dl x − )( t ) ∈ Λ p ( T G ), identifying T G with G and applying π D to this translated multivector, hence π D,x ( t ) = (Λ p dl x ) ◦ π D ◦ (Λ p dl x − )( t ) . To simplify notation, both projection π D applied to v ∈ Λ p G and π D,x applied to w ∈ Λ p S x will be also denoted by ( v ) D and ( w ) D,x , respectively.
Remark 2.10.
The previous notions allow us to consider the “density function” withrespect to ˜ g , defined as Σ ∋ x → k ( τ Σ , ˜ g ( x )) D k . This function naturally appears in therepresentation of the D -dimensional “intrinsic measure” of a submanifold.The following lemma will be useful in the sequel. It can be proved repeating exactlythe same arguments of [23, Lemma 3.1]. Lemma 2.11.
Let Σ ⊂ G be a p -dimensional Lipschitz submanifold and fix x ∈ Σ \ Σ ∗ .Then we can find a graded basis X , . . . , X n of G and a basis v , . . . , v p of T x Σ such RANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 11 that, writing v j = P ni =1 C ij X i ( x ) , we have (2.16) C := ( C ij ) i =1 ,...,nj =1 ,...,p = Id α · · · ∗ · · · ∗ Id α · · ·
00 0 · · · ∗ ... ... . . . ... · · · Id α ι · · · where α k are integers satisfying α k n k and α + · · · + α ι = p . The symbols and ∗ denote null and arbitrary matrices of the proper size, respectively. We have (2.17) d Σ ( x ) = ι X k =1 kα k . Remark 2.12.
As already observed in [23, Remark 3.2], the previous lemma alongwith its proof are understood to hold also in the case where some α k possibly vanishes.In this case the α k columns of (2.16) containing Id α k and the corresponding vectors v kj are meant to be absent.The integers α , . . . , α ι of Lemma 2.11 define a “sub-grading” for a p -dimensionalsubspace of R n , so that in analogy with the integers m j defined in (2.1), we set(2.18) µ = 0 and µ k = k X l =1 α l for all k = 1 , . . . , ι. This new grading allows us to define for every j ∈ { , . . . , p } the subdegree σ j definedas follows(2.19) σ j := k if and only if µ k − < j µ k for some k ∈ { , . . . , ι } . The corresponding subdilations λ r : R p → R p are defined as follows λ r ( ξ , . . . , ξ p ) = ( r σ ξ , r σ ξ , . . . , r σ p ξ p ) for all r > . Transversal points and transversal submanifolds.
Let us fix a p -dimensionalLipschitz submanifold Σ and consider x ∈ Σ \ Σ ∗ , where Σ ∗ is its singular set. We saythat x is transversal if d Σ ( x ) equals the maximal dimension D ( p ). In this case we saythat Σ is a transversal submanifold , that is equivalent to the condition d (Σ) = D ( p ). Remark 2.13.
For hypersurfaces, transversal points coincide with noncharacteristicpoints and when p > n − n a p -dimensional submanifold of G is transversal if andonly if it is non-horizontal , according to the terminology of [21].The following corollary is an easy consequence of the fact that X i (0) = e i . Corollary 2.14.
Under the assumptions and notations of Lemma 2.11, the point x istransversal if and only if the following conditions hold (2.20) α ι = n ι , α ι − = n ι − , . . . , α ℓ +1 = n ℓ +1 , α ℓ = r p , α ℓ − = 0 , . . . , α = 0 , where ℓ = ℓ ( p ) is defined by (2.10) and r p is defined in (2.12) . If x = 0 is transversal,then the vectors v , . . . , v p in Lemma 2.11 constitute the columns of the matrix (2.21) C = ( C ij ) = ∗ · · · · · · · · · ∗ Id r p · · · · · · ∗ · · · · · · ∗ Id n ℓ +1 · · · ... . . . . . . · · · ...... . . . · · · . . . · · · · · · Id n ι . The previous corollary shows that at transversal points the associated grading givenby the integers of (2.18) and (2.19) yields µ = · · · = µ ℓ − = 0 , µ ℓ = r p and µ ℓ + j = r p + j X i =1 n ℓ + i for all j = 1 , . . . , ι − ℓ, therefore the subdegrees are the following ones(2.22) σ = ℓ, . . . , σ r p = ℓ and σ r p + s + P ji =1 n ℓ + i = σ µ ℓ + j + s = ℓ + j + 1for all s = 1 , . . . , n ℓ + j +1 and j = 0 , , . . . , ι − ℓ −
1, where the term P ji =1 n ℓ + i in theprevious formulae is meant to be zero when j = 0.3. Blow-up at transversal points
This section is devoted to the proof of Theorem 1.1 stated in the Introduction. Wehave first to recall some more notions and fix other auxiliary objects. First of all ˜ g will denote any auxiliary Riemannian metric on G . The corresponding Riemanniansurface measure induced on a C smooth submanifold Σ ⊂ G will be denoted by ˜ µ . Definition 3.1.
A graded metric g on G is fixed and we set B = { x ∈ G : d (0 , x ) < } ,where d is a homogeneous distance. The stratified group G is seen as an abstract vectorspace and S denotes one of its p -dimensional linear subspaces. We consider any simple p -vector τ associated to S . Then we define the metric factor (3.1) θ dg ( τ ) = H p |·| ( S ∩ B ) . Here | · | denotes the Euclidean metric on G with respect to a fixed graded basis( X , . . . , X n ) and the sets S and B are represented with respect to the associatedcoordinates of the first kind. Remark 3.2.
In the previous definition, any other simple p -vector λτ with λ = 0defines the same subspace S . Conversely, whenever a simple p -vector ζ is associatedto S , that is ζ = ζ ∧ · · · ∧ ζ p and ( ζ , . . . , ζ p ) is basis of S , then ζ = t τ for some t = 0. Remark 3.3.
The metric factor only depends on the Riemannian metric g and thehomogeneous distance d . In fact, under the assumptions of Definition 3.1, let usconsider another graded basis ( Y , . . . , Y n ) with associated coordinates ( y , . . . , y n ) ofthe first kind. Then the linear change of variables from these coordinates to the original RANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 13 coordinates ( x , . . . , x n ) associated to ( X , . . . , X n ) is an isometry of R n , hence thenumber (3.1) is preserved under the coordinates ( y i ).About the statement of Theorem 1.1, we wish to clarify that the Lie subgroup Π Σ ( x )appearing in (1.3) is also homogeneous in the sense that is closed under dilations.Furthermore, it is a p -dimensioanl homogeneous subgroup of G of the form Z ⊕ H ℓ +1 ⊕ · · · ⊕ H ι , where S ⊂ H ℓ is a linear space of dimension r p . The integers ℓ and r p are defined in(2.10) and (2.12). In particular, Π Σ ( x ) is also a normal subgroup. The same subgroupis more conveniently defined later in the proof of Theorem 1.1, see (3.25). Proof of Theorem 1.1.
First of all, our claim allows us to assume that there exists anopen neighbourhood U ⊂ R p of the origin such that Ψ : U → Σ is a C smoothdiffeomorphism with Ψ(0) = x . Defining the translated submanifold Σ x = l x − (Σ) ,we observe that d Σ x (0) = d Σ ( x ) = D ( p ) = d (Σ) = d (Σ x ) . We consider the translated diffeomorphism φ = l x − ◦ Ψ, with φ : U → Σ x . Taking intoaccount Corollary 2.14, we have a graded basis of left invariant vector fields X , . . . , X n and linearly independent vectors v , . . . , v p ∈ T Σ x such that the matrix C = ( C ij ),defined by v j = P nj =1 C ij X i (0) = P nj =1 C ij e i is given by (2.21). The vector fields X i in our coordinates have the form(3.2) X i = n X l =1 a li e l and their (nonconstant) coefficients satisfy (see e.g. [15])(3.3) a li = (cid:26) δ li d l d i homogeneous polynomial of degree d l − d i otherwise,where homogeneity refers to intrinsic dilations of the group. After a linear change ofvariable on φ , we can also assume that ∂ t i φ (0) = v i for all i = 1 , . . . , p , where each v i is the i -th column of (2.21). Let π : R n → R p be the projection π ( x , . . . , x n ) = ( x m ℓ − +1 , . . . , x m ℓ − + r p , x m ℓ +1 , . . . , x n ) . Thus, d ( π ◦ φ )(0) is invertible and the inverse mapping theorem provides us with newvariables y = ( y m ℓ − +1 , . . . , y m ℓ − + r p , y m ℓ +1 , . . . , y n ) such that Σ x is represented nearthe origin by γ = φ ◦ ( π ◦ φ ) − , which can be written as follows (cid:0) γ ( y ) , . . . , γ m ℓ − ( y ) , y m ℓ − +1 , . . . , y m ℓ − + r p , γ m ℓ − + r p +1 ( y ) , . . . , γ m ℓ ( y ) , y m ℓ +1 , . . . , y n (cid:1) and it is defined in some smaller neighbourhood ( − c , c ) p ⊂ U for some c > d ( π ◦ φ )(0) is the identity mapping of R p , we get(3.4) ( ∂ γ )(0) = v , ( ∂ γ )(0) = v , . . . ( ∂ p γ )(0) = v p , so that we have continuous functions C ij ( y ) with C ij (0) = C ij such that(3.5) ( ∂ j γ )( y ) = n X i =1 C ij ( y ) X i ( γ ( y )) for all j = 1 , . . . , p . Due to the structure of C given in (2.21), whenever σ j = ℓ , or equivalently when j = 1 , . . . , r p , we have(3.6) C ij ( y ) = δ i − m ℓ − ,j + o (1) for any i such that d i > ℓ . When ℓ < σ j ι , or equivalently in the case j > r p and j = µ σ j − + 1 , . . . , µ σ j , we get(3.7) C ij ( y ) = (cid:26) δ i − m σj − ,j − µ σj − + o (1) if d i = σ j or 1 i − m σ j − n σ j o (1) if d i > σ j . Let us introduce the C smooth homeomorphism η : R p → R p as follows(3.8) η ( t ) = (cid:18) | t | σ σ sgn( t ) , . . . , | t p | σ p σ p sgn( t p ) (cid:19) , where its inverse mapping is given by the formula ζ ( τ ) = (cid:18) sgn( τ ) σ p σ | τ | , . . . , sgn( τ p ) σp q σ p | τ p | (cid:19) and all σ j satisfy (2.22). We consider the C smooth reparametrization Γ( t ) = γ (cid:0) η ( t ) (cid:1) with partial derivatives(3.9) ∂ t j Γ( t ) = | t j | σ j − ( ∂ j γ )( η ( t )) = | t j | σ j − n X s,i =1 C ij ( η ( t )) a si (Γ( t )) e s for all j = 1 , . . . , p , where we have used both (3.2) and (3.5). We first observe that(3.10) Γ i ( t ) = o ( | t | d i ) for 1 d i < ℓ. In fact, we have γ (0) = 0 and η ( t ) = O ( | t | ℓ ), hence(3.11) Γ i ( t ) = γ i ( η ( t )) = O ( | η ( t ) | ) = (cid:26) o ( | t | d i ) if d i < ℓO ( | t | ℓ ) if d i = ℓ . The main point is to prove the following rates of convergence(3.12) (cid:26) Γ i ( t ) = O ( | t | ℓ ) for m ℓ − < i m ℓ − + r p Γ i ( t ) = o ( | t | ℓ ) for m ℓ − + r p < i m ℓ . Since the first equation of (3.12) is already contained in (3.11), we have nothing toprove in the case r p = n ℓ (because (3.12) does not have the second case). Thus, wewill assume that r p < n ℓ and then prove the second formula of (3.12). First of all, weapply (3.9) and compute the following partial derivatives(3.13) ∂ t j Γ i ( t ) = | t j | σ j − ( ∂ j γ i )( η ( t )) = | t j | σ j − n X k =1 C kj ( η ( t )) a ik (Γ( t ))for all j = 1 , . . . , p and all i = 1 , . . . , n . If 1 j r p , we rewrite the previous sum as | t j | σ j − (cid:18) X k : d k <ℓ C kj ( η ( t )) a ik (Γ( t )) + X k : d k > ℓ C kj ( η ( t )) a ik (Γ( t )) (cid:19) . RANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 15
As a consequence, taking into account (3.6), it follows that ∂ t j Γ i ( t ) = | t j | ℓ − (cid:18) a ij + m ℓ − (Γ( t )) + X k : d k <ℓ C kj ( η ( t )) a ik (Γ( t )) + X k : d k > ℓ o (1) a ik (Γ( t )) (cid:19) = | t j | ℓ − (cid:18) a ij + m ℓ − (Γ( t )) + o (1) + X k : d k <ℓ C kj ( η ( t )) a ik (Γ( t )) (cid:19) . Since d k < ℓ , we have that a ik is a nonconstant homogenous polynomial. It followsthat a ik ◦ Γ = o (1) and we get(3.14) ∂ t j Γ i ( t ) = | t j | ℓ − (cid:16) a ij + m ℓ − (Γ( t )) + o (1) (cid:17) Since m ℓ − + j m ℓ − + r p < i m ℓ and d m ℓ − + j = d i , formula (3.3) implies that a ij + m ℓ − is the null polynomial. It follows that(3.15) ∂ t j Γ i ( t ) = o ( | t | ℓ − ) whenever 1 j r p and m ℓ − + r p < i m ℓ . If r p < j p , then (3.13) implies that ∂ t j Γ i ( t ) = | t j | σ j − n X k =1 C kj ( η ( t )) a ik (Γ( t )) = | t j | σ j − O (1) = O ( | t | σ j − ) . Since in this case σ j > ℓ , we get in particular that(3.16) ∂ t j Γ i ( t ) = o ( | t | ℓ − ) whenever r p < j p and 1 i n . Joining (3.15) with (3.16), it follows that ∇ Γ i ( t ) = o ( | t | ℓ − ) for all i = m ℓ − + r p + 1 , . . . , m ℓ , that proves the second equation of (3.12).Now, we write explicitly the form of Γ as the composition γ ◦ η . By the previousformulae for γ and η , we getΓ( t ) = (cid:16) Γ ( t ) , . . . , Γ m ℓ − ( t ) , | t | ℓ ℓ sgn( t ) , . . . , | t r p | ℓ ℓ sgn( t r p ) , Γ m ℓ − + r p +1 ( t ) , . . .. . . , Γ m ℓ ( t ) , | t r p +1 | ℓ +1 ℓ + 1 sgn( t r p +1 ) , . . . , | t p | ι ι sgn( t p ) (cid:17) . (3.17)The new parametrization γ of Σ x around the origin yields Φ : ( − c , c ) p → Σ definedas Φ := l x ◦ γ , that is our “adapted parametrization” of Σ around x . Taking r > µ (Σ ∩ B ( x, r )) r D ( p ) = r − D ( p ) ˆ Φ − ( B ( x,r )) k ( ∂ y Φ ∧ · · · ∧ ∂ y p Φ)( y ) k ˜ g dy . where ˜ µ is the Riemannian surface measure induced by ˜ g on Σ. We perform the changeof variable y = λ r t , where λ r is the subdilation of the form(3.19) λ r ( t , . . . , t p ) = ( r ℓ t , . . . , r ℓ t r p , r ℓ +1 t r p +1 , . . . , r ℓ +1 t r p + n ℓ +1 , . . . , r ι t p ) that yields the formula(3.20) ˜ µ (Σ ∩ B ( x, r )) r D ( p ) = ˆ λ /r (Φ − ( B ( x,r ))) k ∂ y Φ( λ r t ) ∧ · · · ∧ ∂ y p Φ( λ r t ) k ˜ g dt . The point is then to study the “behaviour” of the set λ /r (cid:0) Φ − ( B ( x, r )) (cid:1) as r → + .To do this, we will use the formula (3.17) for Γ and the rates of convergence (3.12),taking into account the change of variables (3.8). Since Φ − ( B ( x, r )) = γ − ( B (0 , r )),it follows that(3.21) λ /r (cid:0) Φ − ( B ( x, r )) (cid:1) = (cid:8) t ∈ R p : δ /r (cid:0) γ ( λ r t ) (cid:1) ∈ B (cid:9) , where B = { z ∈ G : d ( z, < } . We observe that γ ( λ r t ) = Γ( ζ ( λ r t )) = Γ( r ζ ( t )) , therefore the previous rescaled set can be written as follows(3.22) λ /r (cid:0) Φ − ( B ( x, r )) (cid:1) = (cid:8) t ∈ R p : δ /r (cid:0) Γ( rζ ( t )) (cid:1) ∈ B (cid:9) . By (3.17), an element t ∈ R p of the previous set is characterized by the property that (cid:16) Γ ( rζ ( t )) r , . . . , Γ m ℓ − ( rζ ( t )) r ℓ − , t , . . . , t r p , Γ m ℓ − + r p +1 ( rζ ( t )) r ℓ , . . .. . . , Γ m ℓ ( rζ ( t )) r ℓ , t r p +1 , . . . , t p (cid:17) (3.23)belongs to B . This is a simple consequence of the equalities η ( rζ ( t )) = λ r η ( ζ ( t )) = λ r t .We now use both (3.11) and (3.12) to conclude that the element represented in (3.23)converges to(3.24) (cid:16) , . . . , , t , . . . , t r p , , . . . , , t r p +1 , . . . , t p (cid:17) as r → + , uniformly with respect to t that varies in a bounded set. By standard factson Hausdorff convergence, the previous limit implies the convergence in (1.3) where(3.25) Π Σ ( x ) := { z ∈ G : z = · · · = z m ℓ − = z m ℓ − + r p +1 = · · · = z m ℓ = 0 } . It can be easily seen that Π Σ ( x ) is the homogeneous subgroup of G associated withthe Lie subalgebraspan { X m ℓ − +1 , X m ℓ − +2 , . . . , X m ℓ − + r p , X m ℓ +1 , . . . X n } . Moreover, the convergence of the element represented in (3.23) to (3.24) also gives(3.26) lim r → + ˜ µ (Σ ∩ B ( x, r )) r D ( p ) = H p |·| ( B ∩ S ) k ( ∂ y Φ ∧ · · · ∧ ∂ y p Φ)(0) k ˜ g,x where S = { (0 , . . . , , t , . . . , t r p , , . . . , , t r p +1 , . . . , t p ) ∈ R n : t , . . . , t p ∈ R } and themetric unit ball B is represented with respect to the same coordinates. Taking intoaccount (3.4) and the matrix (2.21), we have the projection π D ( p ) , (cid:0) ( ∂ y γ ∧ · · · ∧ ∂ y p γ )(0) (cid:1) = (cid:0) X m ℓ − +1 ∧ · · · ∧ X m ℓ − + r p ∧ X m ℓ +1 ∧ · · · ∧ X n (cid:1) (0)and the formulae ∂ y j Φ( x ) = dl x (cid:0) ∂ y j γ (0) (cid:1) yield π D ( p ) ,x (cid:0) ( ∂ y Φ ∧ · · · ∧ ∂ y p Φ)(0) (cid:1) = (cid:0) X m ℓ − +1 ∧ · · · ∧ X m ℓ − + r p ∧ X m ℓ +1 ∧ · · · ∧ X n (cid:1) ( x ) . RANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 17
We have the unit tangent p -vector τ Σ , ˜ g ( x ) = ( ∂ y Φ ∧ · · · ∧ ∂ y p Φ)(0) k ( ∂ y Φ ∧ · · · ∧ ∂ y p Φ)(0) k ˜ g,x then the previous equations for projections give (cid:13)(cid:13)(cid:0) τ Σ , ˜ g ( x ) (cid:1) D ( p ) ,x (cid:13)(cid:13) = 1 k ( ∂ y Φ ∧ · · · ∧ ∂ y p Φ)(0) k ˜ g,x . As a result, in view of Definition 3.1, the limit (3.26) proves our last claim (1.4). (cid:3) Negligibility of lower degree points in transversal submanifolds
The aim of this section is to prove Theorem 1.2 for a C p -dimensional transversalsubmanifold Σ ⊂ G , where we define(4.1) Σ c := { x ∈ Σ : d Σ ( x ) < D ( p ) } . Since Σ is transversal, the subset Σ c plays the role of a generalized characteristic set of Σ. Since any left translation is a diffeomorphism, for each point x ∈ Σ there holds(4.2) T ( x − · Σ) = dl x − ( T x Σ) . Clearly, a basis for T ( x − · Σ) is given by dl x − ( v ) , . . . , dl x − ( v p ) , where the vectors v , . . . , v p are given by Lemma 2.11. If v j = P i C ij X i ( x ), by the leftinvariance of X i , we have(4.3) dl x − ( v j ) = n X i =1 C ij X i (0) for any j = 1 , . . . , p . In particular, d x − · Σ (0) = d Σ ( x ) and 0 ∈ ( x − · Σ) c if and only if x ∈ Σ c .Taking into account (2.20), we observe that Σ c can be written as the disjoint union(4.4) Σ c = Σ Ac ∪ Σ Bc , where we have definedΣ Ac := { x ∈ Σ c : ∃ ¯ > ℓ + 1 such that α ¯ < n ¯ } Σ Bc := { x ∈ Σ c : α j = n j ∀ j > ℓ + 1 and α ℓ < r p } . (4.5)The integer ℓ , depending on p , is introduced in (2.10) and the nonnegative integers α , . . . , α ι are defined in Lemma 2.11. In particular, ℓ will be used throughout thissection. We notice that in the case ℓ = 1, we must have α ℓ = r p , hence Σ Bc = ∅ .We begin by making the further assumption that Σ is of class C and such that Σ ⊂ φ ([0 , p ) for some C -regular map φ : [0 , p → G . By the uniform differentiability of φ , the boundedness of Σ and the continuity of left translations, the following statementholds: for any ǫ >
0, there exists ¯ r ǫ > |h y, w i| ǫr ∀ r ∈ (0 , ¯ r ǫ ) , ∀ x ∈ Σ , ∀ y ∈ ( x − · Σ) ∩ B E (0 , r ) , ∀ w ∈ ( T ( x − · Σ)) ⊥ , | w | = 1 , where h· , ·i denotes the Euclidean scalar product. The orthogonal space ( T ( x − · Σ)) ⊥ is understood with respect to the same product. Notice that such coordinates areassociated with the basis X , . . . , X n given by Lemma 2.11; in particular, they dependon the chosen basepoint x ∈ Σ.The proof of the negligibility stated in Theorem 1.2 stems from the following keylemmata. The proofs of these lemmata could be rather simplified; however, we presentthem in a form which will be helpful for some refinement provided in Subsection 5.
Lemma 4.1.
Let Σ be a C submanifold such that Σ ⊂ φ ([0 , p ) for some C map φ : [0 , p → G ; let θ := 1 /ℓ . Then, there exists a constant C A = C A (Σ) > such thatthe following property holds. For any x, ǫ, r satisfying (4.7) x ∈ Σ Ac , ǫ ∈ (0 , and < r min { ¯ r ǫ , ǫ ℓ } , the set ( x − · Σ) ∩ B E (0 , r ) can be covered by a family { B i : i ∈ I } of CC balls withradius r θ such that I C A ǫ r p − θD ( p ) . Proof.
From now on, the numbers C i , with i = 1 , , . . . , will denote positive constantsdepending only on Σ , p, G and the fixed homogeneous distance d . For the reader’sconvenience, we divide the proof into several steps. Step 1.
By Theorem 2.5, we get a countable family { B ( x i , r θ ) : i ∈ I } such that(4.8) x i ∈ ( x − · Σ) ∩ B E (0 , r )( x − · Σ) ∩ B E (0 , r ) ⊂ S i ∈ I B ( x i , r θ ) B ( x i , r θ / ∩ B ( x h , r θ /
5) = ∅ when i = h. We have to estimate I . By Lemma 2.3, for any i ∈ I there exists ˜ x i such that˜ x i = · · · = ˜ x ℓi = 0 , d ( x i , ˜ x i ) Cr /ℓ = Cr θ , | ˜ x hi − x hi | Cr for any h = ℓ + 1 , . . . , ι. (4.9)Therefore, taking into account (2.6), we achieve(4.10) B ( x i , r θ ) ⊂ B (˜ x i , (1 + C ) r θ ) ⊂ Box(˜ x i , C r θ ) . Let us also point out that both (4.9) and the fact that x i ∈ B E (0 , r ) give(4.11) | ˜ x i | C r . Step 2.
Let us prove that there exists C > i ∈ I , there holds(4.12) Box(˜ x i , C r θ ) ⊂ Ω , where we have setΩ := ( − C r θ , C r θ ) n × ( − C r θ , C r θ ) n × · · · × ( − C r ℓθ , C r ℓθ ) n ℓ × Box µE (0 , C r )and µ := n − m ℓ . To this aim we fix y ∈ Box(0 , C r θ ), that is(4.13) | y j | < ( C r θ ) j ∀ j = 1 , . . . , ι , RANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 19 and prove that ˜ x i · y ∈ Ω. By explicit computation˜ x i · y = (0 , . . . , , ˜ x ℓ +1 i , . . . , ˜ x ιi ) · ( y , . . . , y ι )= ( y , . . . , y ℓ , ˜ x ℓ +1 i + y ℓ +1 , ˜ x ℓ +2 i + y ℓ +2 + O ( r θ ) , . . . , ˜ x ιi + y ι + O ( r θ ))(4.14)where we have used • (2.3) for the coordinates in the layers 1 , . . . , ℓ + 1; • (2.4) for the coordinates in the layers ℓ + 2 , . . . , ι , together with (4.11) and thefact that | y | = O ( r θ ).Here and in the sequel, all the quantities O ( · ) are uniform. From (4.14) and (4.13) itfollows immediately that ˜ x i · y ∈ Ω, and (4.12) follows.
Step 3.
We have not used the fact that x ∈ Σ Ac yet. By definition, there exists¯ > ℓ + 1 such that α ¯ < n ¯ . We can also assume that ¯ is maximum, i.e., that α j = n j for any j > ¯ ; set ν := n ¯ + n ¯ +1 + · · · + n ι = n ¯ + α ¯ +1 + · · · + α ι . The last ν rows of the matrix C given by Lemma 2.11 constitute a ν × p matrix M ofthe form M = · · · Id α ¯ · · · · · · ∗ · · · ∗ · · · Id n ¯ +1 · · · · · · · · · Id n ι = · · · Id α ¯ · · · ∗ · · · Id n ¯ +1 + ··· + n ι . Since M has only α ¯ + n ¯ +1 + · · · + n ι < ν nonzero columns, there exists a vector z ∈ R ν such that | z | = 1 and z is orthogonal to any of the columns of M . Therefore, the vector w := (0 , z ) ∈ R n ≡ R n − ν × R ν is orthogonal to any of the columns of C . By (4.2) and(4.3), taking into account that X k (0) = ∂ x k , these columns generate T ( x − · Σ). As aresult, since ¯ > ℓ , we are lead to the validity of the following conditions(4.15) w ∈ ( T ( x − · Σ)) ⊥ | w | = 1 w = w = · · · = w m ℓ = 0 . Step 4.
To refine the inclusion (4.12), we will use the properties (4.15). By (4.6)one has |h x i , w i| ǫr for any i ∈ I . Define w ′ := ( w m ℓ +1 , w m ℓ +2 , . . . , w n ) ∈ R µ ,where µ = n − m ℓ > ν is the same number of Step 2. By (4.14) and (4.15), for any y ∈ Box(0 , C r θ ) we have |h ˜ x i · y, w i| = (cid:12)(cid:12)(cid:10) ( y , . . . , y ℓ , ˜ x ℓ +1 i + y ℓ +1 , ˜ x ℓ +2 i + y ℓ +2 + O ( r θ ) , . . . , ˜ x ιi + y ι + O ( r θ )) , (0 , . . . , , w ℓ +1 , . . . , w ι ) (cid:11)(cid:12)(cid:12) |h (˜ x ℓ +1 i , . . . , ˜ x ιi ) , w ′ i| + |h ( y ℓ +1 , . . . , y ι ) , w ′ i| + O ( r θ )= |h ( x ℓ +1 i , . . . , x ιi ) , w ′ i| + O ( r ( ℓ +1) θ ) + O ( r θ ) ǫr + O ( r θ ) , where the second equality is justified by (4.9) and (4.13) and the last inequality followsfrom ( ℓ + 1) θ = 1 + θ . Since all the previous O ( · )s are uniform with respect to theindex i , we get |h (˜ x i · y ) µ , w ′ i| ǫr + C r θ (1 + C ) ǫr , where (˜ x i · y ) µ is the vector made by the last µ coordinates of (˜ x i · y ) µ and we usedthe fact that, by (4.7), r θ = r /ℓ ǫ . Thus, by (2.7) and (4.12) we obtain thatBox(˜ x i , C r θ ) ⊂ e Ω, where we have set e Ω := ( − C r θ , C r θ ) n × ( − C r θ , C r θ ) n × · · ·× ( − C r ℓθ , C r ℓθ ) n ℓ × Box µw ′⊥ ⊕ span w ′ (0; C r, C ǫr ) . As a consequence, by (4.10) we get B ( x i , r θ / ⊂ Box(˜ x i , C r θ ) ⊂ e Ω for all i ∈ I . Step 5.
We are ready to estimate I . The volume of e Ω is equal to a = C ǫ r θ ( n +2 n ··· + ℓn ℓ )+ µ = C ǫ r θ ( n +2 n ··· + ℓn ℓ )+ n ℓ +1 + ··· + n ι , while each B ( x i , r θ /
5) has volume b = C r θ ( n +2 n + ··· + ιn ι ) . Taking into account thatthe CC balls B ( x i , r θ /
5) are pairwise disjoint and contained in e Ω, we have I ab = C C ǫ r n ℓ +1 + ··· + n ι − θ (( ℓ +1) n ℓ +1 + ··· + ιn ι ) (2.13) = C C ǫ r p − r p − θ ( D ( p ) − ℓr p ) = C C ǫ r p − θD ( p )+( θℓ − r p which proves the claim and concludes the proof of the lemma. (cid:3) While more subtle at certain points, the proof of Lemma 4.2 follows the same linesof the previous one. For the reader’s benefit, we will try to make the analogies betweenthe two proofs as evident as possible.
Lemma 4.2.
Under the assumptions of Lemma 4.1 and ℓ > , there exists C B = C B (Σ) > such that the following property holds. For any x, ǫ, θ, r satisfying (4.16) x ∈ Σ Bc , ǫ ∈ (0 , , ℓ < θ ℓ − and < r min { ¯ r ǫ , ǫ / ( ℓθ − } , the set ( x − · Σ) ∩ B E (0 , r ) can be covered by a family { B i : i ∈ I } of CC balls withradius r θ such that I C B ǫ H r p − θD ( p ) − ( ℓθ − n ℓ − r p ) , where H = H ( x ) := n ℓ − α ℓ and the integers α j = α j ( x ) are those given by Lemma 2.11.Proof. We follow the same convention of Lemma 4.1 about the constants C i . Step 1.
By the 5 r -covering theorem we can cover ( x − · Σ) ∩ B E (0 , r ) by a familyof CC balls { B ( x i , r θ ) : i ∈ I } such that (4.8) holds. We have once more to estimate I . By Lemma 2.3, for any i ∈ I there exists ˜ x i such that(4.17) ˜ x i = · · · = ˜ x ℓ − i = 0 , d ( x i , ˜ x i ) Cr / ( ℓ − Cr θ and | ˜ x hi − x hi | Cr for any h = ℓ, . . . , ι. Therefore B ( x i , r θ / ⊂ B ( x i , r θ ) ⊂ B (˜ x i , (1 + C ) r θ ) ⊂ Box(˜ x i , C r θ ). Again(4.18) | ˜ x i | C r . RANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 21
Step 2.
Let us prove that there exists C > i ∈ I , there holds(4.19) Box(˜ x i , C r θ ) ⊂ Ω , where now Ω := ( − C r θ , C r θ ) n × ( − C r θ , C r θ ) n × · · ·× ( − C r ( ℓ − θ , C r ( ℓ − θ ) n ℓ − × Box µE (0 , C r )(4.20)and µ := n − m ℓ − = n ℓ + · · · + n ι . As before we fix y ∈ Box(0 , C r θ ),(4.21) | y j | < ( C r θ ) j ∀ j = 1 , . . . , ι and prove that ˜ x i · y ∈ Ω. Reasoning as in Step 2 in the proof of Lemma 4.1 we get˜ x i · y = (0 , . . . , , ˜ x ℓi , . . . , ˜ x ιi ) · ( y , . . . , y ι )= ( y , . . . , y ℓ − , ˜ x ℓi + y ℓ , ˜ x ℓ +1 i + y ℓ +1 + O ( r θ ) , . . . , ˜ x ιi + y ι + O ( r θ ))(4.22)where we have used (2.3), (2.4), (4.18) and the fact that | y | = O ( r θ ). All the quantities O ( · ) are uniform. The inclusion (4.19) follows from (4.18), (4.22) and the fact that | y j | < ( C r θ ) j = C j r jθ C j r j/ℓ C j r ∀ j = ℓ, . . . , ι . Step 3.
Since x ∈ Σ Bc we have by definition α ℓ < r p and α j = n j ∀ j > ℓ + 1 . Therefore the last µ rows of the matrix C from Lemma 2.11 constitute a µ × p matrix M of the form M = · · · Id α ℓ · · · · · · ∗ · · · ∗ · · · Id n ℓ +1 · · · · · · ... ... ... . . . ...0 · · · · · · Id n ι = · · · Id α ℓ · · · ∗ · · · Id n ℓ +1 + ··· + n ι There are α ℓ + n ℓ +1 + · · · + n ι nonzero columns of M ; therefore, the columns of M span a vector subspace of R µ of dimension at most α ℓ + n ℓ +1 + · · · + n ι . Since µ − ( α ℓ + n ℓ +1 + · · · + n ι ) = n ℓ − α ℓ = H, it follows that there exist H linearly independent vectors z , . . . , z H ∈ R µ such that | z k | = 1 and z k is orthogonal to any of the columns of M for any k = 1 , . . . , H . Inparticular, the unit vectors w k := (0 , z k ) ∈ R n ≡ R n − µ × R µ , k = 1 , . . . , H are orthogonal to any of the columns of C , which form a basis of T ( x − · Σ). Setting W := span( w , . . . , w H ) we have W ⊂ T ( x − · Σ) ⊥ and dim W = H > w ∈ W is of the form(4.23) w = (0 , . . . , , w ℓ , . . . , w ι ) = (0 , w ′ ) ∈ R m ℓ − × R µ . Step 4.
Again we want to refine the inclusion (4.19). By (4.6) there holds |h x i , w i| ǫr ∀ i ∈ I, ∀ w ∈ W with | w | = 1 . Recalling (4.22) and writing w = (0 , w ′ ) ∈ R m ℓ − × R µ as in (4.23), for any y ∈ Box(0 , C r θ ) we have |h ˜ x i · y, w i| = (cid:12)(cid:12)(cid:10) ( y , . . . , y ℓ − , ˜ x ℓi + y ℓ , ˜ x ℓ +1 i + y ℓ +1 + O ( r θ ) , . . . , ˜ x ιi + y ι + O ( r θ )) , (0 , . . . , , w ℓ , . . . , w ι ) (cid:11)(cid:12)(cid:12) |h (˜ x ℓi , . . . , ˜ x ιi ) , w ′ i| + |h ( y ℓ , . . . , y ι ) , w ′ i| + O ( r θ )= |h ( x ℓi , . . . , x ιi ) , w ′ i| + O ( r ℓθ ) + O ( r θ ) ǫr + O ( r ℓθ ) + O ( r θ ) ∀ w ∈ W, | w | = 1where we used (4.17) and (4.21). Since ℓθ = ( ℓ − θ + θ θ, we have r θ r ℓθ and thus, since all the O ( · )s are uniform, |h ˜ x i · y, w i| ǫr + C r ℓθ max { ǫ, C r ℓθ − } r C ǫ r ∀ w ∈ W, | w | = 1 , the last inequality following from (4.16). Using (2.7) we can then refine (4.20) toobtain B ( x i , r θ / ⊂ Box(˜ x i , C r θ ) ⊂ e Ω ∀ i ∈ I where e Ω := ( − C r θ , C r θ ) n × ( − C r θ , C r θ ) n × · · ·× ( − C r ( ℓ − θ , C r ( ℓ − θ ) n ℓ − × Box µW ⊥ ⊕ W (0; C r, C ǫr ) . Step 5.
We can now estimate I . Since dim W = H , the volume of e Ω is a = C ǫ H r θ ( n +2 n ··· +( ℓ − n ℓ − )+ µ = C ǫ H r θ ( n +2 n ··· +( ℓ − n ℓ − )+ n ℓ + ··· + n ι , while each ball B ( x i , r θ /
5) has volume b = C r θ ( n +2 n + ··· + ιn ι ) . Since the CC balls B ( x i , r θ /
5) are pairwise disjoint and contained in e Ω, we have I ab = C C ǫ H r n ℓ + ··· + n ι − θ ( ℓn ℓ + ··· + ιn ι )(2.13) = C C ǫ H r n ℓ + p − r p − θ [ ℓ ( n ℓ − r p )+ ℓr p +( ℓ +1) n ℓ +1 + ··· + ιn ι ](2.13) = C C ǫ H r p +( n ℓ − r p ) − θ [ ℓ ( n ℓ − r p )+ D ( p )] = C C ǫ H r p − θD ( p ) − ( ℓθ − n ℓ − r p ) , as claimed. (cid:3) Lemma 4.3.
Let Σ be a C submanifold such that Σ ⊂ φ ([0 , p ) for a C map φ : [0 , p → G . Then H D ( p ) (Σ c ) = 0 . RANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 23
Proof.
Clearly, it will be enough to show that(4.24) H D ( p ) (Σ Ac ) = 0 and H D ( p ) (Σ Bc ) = 0 . Step 1.
We start by proving the first equality in (4.24); let us follow the sameconvention of Lemmata 4.1 and 4.2 about the constants C i .Let ǫ ∈ (0 ,
1) and r ∈ (0 , min { ¯ r ǫ , ǫ ℓ } ] be fixed. Since ( x, y ) → x − y is locallyLipschitz and φ is Lipschitz, both with respect to the Euclidean distance, we obtain C > z , z ∈ [0 , p and | z − z | C r, then | φ ( z ) − · φ ( z ) | < r . Let us divide [0 , p , in a standard fashion, into a family of closed subcubes of diameternot greater than C r ; in this way there will be less than C r − p such subcubes. Let( Q j ) j ∈ J be the family of those subcubes with the property that φ ( Q j ) ∩ Σ Ac = ∅ and fix x j ∈ φ ( Q j ) ∩ Σ Ac . By (4.25) we have x − j · φ ( Q j ) ⊂ ( x − j · Σ) ∩ B E (0 , r ) . Writing θ := 1 /ℓ , Lemma 4.1 ensures that x − j · φ ( Q j ) can be covered by (at most) C A ǫr p − θD ( p ) balls of radius r θ ; by left invariance, the same holds for φ ( Q j ). In particu-lar, since Σ Ac ⊂ ∪ j ∈ J φ ( Q j ) and J C r − p , we have that, for any r ∈ (0 , min { ¯ r ǫ , ǫ ℓ } ],the set Σ Ac can be covered by a family of CC balls with radius r θ of cardinality con-trolled by C A C ǫr − θD ( p ) . Therefore H D ( p )2 r /ℓ (Σ Ac ) C A C ǫr − θD ( p ) (2 r θ ) D ( p ) = 2 D ( p ) C A C ǫ whence, letting r → + , H D ( p ) (Σ Ac ) D ( p ) C A C ǫ . The first part of (4.24) follows by the arbitrarity of ǫ . Step 2.
Let us prove the second equality in (4.24). Let ǫ ∈ (0 ,
1) and r ∈ (0 , min { ¯ r ǫ , ǫ ℓ − } ] be fixed; we have ǫ = r λ for a suitable λ = λ ( r ) ∈ (0 , ℓ − ]. De-fine θ = θ ( r ) := λℓ and observe that 1 /ℓ < θ / ( ℓ − r ℓθ − = r λ = ǫ ;in particular, ǫ / ( ℓθ − = r ¯ r ǫ and the conditions in (4.16) are satisfied. As before, wedivide [0 , p into a family of (at most) C r − p closed subcubes of diameter not greaterthan C r . Let ( Q k ) k ∈ K be the family of those subcubes with the property that φ ( Q k ) ∩ Σ Bc = ∅ and fix x k ∈ φ ( Q k ) ∩ Σ Ac . By (4.25) we have again x − k · φ ( Q k ) ⊂ ( x − k · Σ) ∩ B E (0 , r )so that, by Lemma 4.2, x − k · φ ( Q k ) can be covered by no more than C B ǫ H ( x k ) r p − θD ( p ) − ( ℓθ − n ℓ − r p ) balls of radius r θ = ǫ /ℓ r /ℓ ; by left invariance, the same holds for φ ( Q k ). Notice that H ( x k ) = n ℓ − α ℓ ( x k ) > n ℓ − r p + 1 ∀ k ∈ K , i.e., φ ( Q k ) can be covered by (at most) C B ǫ n ℓ − r p +1 r p − θD ( p ) − ( ℓθ − n ℓ − r p ) balls of radius r θ . As before, this implies that K C B C ǫ n ℓ − r p +1 r − θD ( p ) − ( ℓθ − n ℓ − r p ) whence, using (4.26), H D ( p )2 r θ (Σ Bc ) C B C ǫ n ℓ − r p +1 r − θD ( p ) r − ( ℓθ − n ℓ − r p ) (2 r θ ) D ( p ) =2 D ( p ) C B C ǫ n ℓ − r p +1 ǫ − ( n ℓ − r p ) =2 D ( p ) C B C ǫ . (4.27)Observing that lim r → + r θ = lim r → + r /ℓ r λ ( r ) /ℓ = lim r → + r /ℓ ǫ /ℓ = 0we can let r → + in (4.27) to obtain H D ( p ) (Σ Bc ) D ( p ) C B C ǫ . This proves the second equality in (4.24) and completes the proof. (cid:3)
Remark 4.4.
We point out for future references the following two facts proved, respec-tively, in Step 1 and Step 2 of the proof of Lemma 4.3. Let ǫ ∈ (0 ,
1) be fixed and as-sume that Σ is a C submanifold such that Σ ⊂ φ ([0 , p ) for a C map φ : [0 , p → G ;then(4.28) for any r ∈ (0 , min { ¯ r ǫ , ǫ ℓ } ], the set Σ Ac can be covered by a family of CCballs with radius r /ℓ of cardinality at most C A C ǫr − D ( p ) /ℓ .and(4.29) for any r ∈ (0 , min { ¯ r ǫ , ǫ ℓ − } ], the set Σ Bc can be covered by a family of CCballs, with radius ǫ /ℓ r /ℓ , of cardinality at most C B C ǫ ǫ − D ( p ) /ℓ r − D ( p ) /ℓ .In (4.29), we used the fact that the cardinality of the involved family is controlled by C B C ǫ n ℓ − r p +1 r − θD ( p ) − ( ℓθ − n ℓ − r p ) = C B C ǫ n ℓ − r p +1 ǫ − D ( p ) /ℓ r − D ( p ) /ℓ ǫ − ( n ℓ − r p ) = C B C ǫ ǫ − D ( p ) /ℓ r − D ( p ) /ℓ , where we also utilized the equalities r θ = ǫ /ℓ r /ℓ and r ℓθ − = ǫ .The proof of Theorem 1.2 is now at hand. Proof of Theorem 1.2.
The theorem is an easy consequence of Lemma 4.3 and a stan-dard localization argument. (cid:3)
RANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 25
Actually, Theorem 1.2 can be generalized to Lipschitz p -dimensional submanifolds;recall that the singular set Σ ∗ was defined at the beginning of Section 2.4. Clearly, thedefinition of Σ c given at (4.1) for C submanifolds extends to Lipschitz submanifoldsconsidering the subset Σ \ Σ ∗ of regular points, since the pointwise degree is definedby the existence of the pointwise tangent space. Theorem 4.5.
Let Σ ⊂ G be a p -dimensional Lipschitz submanifold, let Σ ∗ be itssingular set and denote by Σ c be the subset of points in Σ \ Σ ∗ whose degree is lessthan D ( p ) . It follows that (4.30) H D ( p ) (Σ ∗ ∪ Σ c ) = 0 . Proof.
By definition, Σ is locally the graph of a Euclidean Lipschitz function, hencewithout loss of generality, we can assume that Σ ⊂ φ ( A ), where φ is the graph functiongiven by a Lipschitz function f : A → V , A ⊂ W is a bounded open set of W and G is seens as by W × V , where W and V are linear subspaces of dimensions p and n − p ,respectively. Let ǫ > < ǫ < L p ( A ). By the classicalWhitney’s extension theorem, there exists a C function f ǫ : A → V such that the set(4.31) E ǫ := { z ∈ A : f ǫ ( z ) = f ( z ) and ∇ f ǫ ( z ) = ∇ f ( z ) } satisfies L p ( A \ E ǫ ) < ǫ . The graph function φ ǫ associated to f ǫ defines the C subman-ifold Σ ǫ := φ ǫ ( A ), hence Theorem 1.2 implies that that its generalized characteristicset Σ ǫc := { x ∈ Σ ǫ : d Σ ǫ ( x ) < D ( p ) } is H D ( p ) -negligible. By the conditions of (4.31),we have the inclusion Σ c ∩ φ ( E ǫ ) ⊂ Σ ǫc , hence Σ c ∩ φ ( E ǫ ) is also H D ( p ) -negligible. Asa consequence of [4, Proposition 3.1], there exists a geometric constant C >
0, onlydepending on the diameter of φ ( A ) and on G , such that H D ( p ) (Σ c ) = H D ( p ) (Σ c \ φ ( E ǫ )) C H p |·| (Σ c \ φ ( E ǫ )) C L p ǫ , where L > φ . The arbitrary choice of ǫ impliesthat H D ( p ) (Σ c ) = 0 and using (2.14), the proof is accomplished. (cid:3) Size of the characteristic set for C ,λ submanifolds In this section we assume that Σ is a submanifold of class C ,λ for some λ ∈ (0 , c . We first assume that Σ ⊂ φ ([0 , p ) for some map φ ∈ C ,λ ([0 , p , G ). Under this assumption, there exists C = C (Σ) > |h y, w i| Cr λ ∀ x ∈ Σ , ∀ y ∈ ( x − · Σ) ∩ B E (0 , r ) , ∀ w ∈ ( T ( x − · Σ)) ⊥ , | w | = 1 . In other words, the number ¯ r ǫ defined by (4.6) can be chosen to be ¯ r ǫ = ( ǫ/C ) /λ .As in (4.4), we write Σ c = Σ Ac ∪ Σ Bc where, following (4.5), we defineΣ Ac = { x ∈ Σ c : ∃ ¯ > ℓ + 1 such that α ¯ < n ¯ } Σ Bc = { x ∈ Σ c : α j = n j ∀ j > ℓ + 1 and α ℓ < r p } . Again, if ℓ = 1, then Σ Bc = ∅ . Lemma 5.1.
Let Σ ⊂ G be a C ,λ submanifold such that Σ ⊂ φ ([0 , p ) for some map φ ∈ C ,λ ([0 , p , G ) . Then (5.2) dim H Σ Ac D ( p ) − if λ > /ℓ dim H Σ Ac D ( p ) − ℓλ if λ /ℓ . Proof. If λ > /ℓ we havemin { ¯ r ǫ , ǫ ℓ } = min { ( ǫ/C ) /λ , ǫ ℓ } = ǫ ℓ for any ǫ > r := ǫ ℓ and obtainthat, for any ǫ > Ac can be covered by a family of balls withradius ǫ of cardinality at most C A C ǫ ǫ − D ( p ) . By Proposition 2.4 we getdim H Σ Ac D ( p ) − . On the other hand, if λ /ℓ we havemin { ¯ r ǫ , ǫ ℓ } = min { ( ǫ/C ) /λ , ǫ ℓ } = C ǫ /λ for any ǫ > C i .Using (4.28) with r := C ǫ /λ , we get that, for any ǫ > Ac can be covered by a family of balls with radius r /ℓ = C ǫ / ( ℓλ ) of cardinality at most C ǫ ǫ − D ( p ) / ( ℓλ ) . By Proposition 2.4 we getdim H Σ Ac D ( p ) − ℓλ and this concludes the proof. (cid:3) Lemma 5.2.
Let Σ ⊂ G be a C ,λ submanifold such that Σ ⊂ φ ([0 , p ) for some map φ ∈ C ,λ ([0 , p , G ) ; assume ℓ > . Then (5.3) dim H Σ Bc D ( p ) − if λ > ℓ − dim H Σ Bc D ( p ) − ℓλ λ if λ ℓ − . Proof. If λ > / ( ℓ −
1) we havemin { ¯ r ǫ , ǫ ℓ − } = min { ( ǫ/C ) /λ , ǫ ℓ − } = ǫ ℓ − for any ǫ > r := ǫ ℓ − andobtain that, for any ǫ > Bc can be covered by a family of ballswith radius ǫ /ℓ ǫ ( ℓ − /ℓ = ǫ of cardinality at most C B C ǫ ǫ − D ( p ) /ℓ r − D ( p ) /ℓ = C B C ǫ − D ( p )+1 . By Proposition 2.4 we get dim H Σ Bc D ( p ) − . On the other hand, if λ / ( ℓ −
1) we havemin { ¯ r ǫ , ǫ ℓ − } = min { ( ǫ/C ) /λ , ǫ ℓ − } = C ǫ /λ for any ǫ > r := C ǫ /λ , we get that, for any ǫ > Bc can be covered by a family of balls with radius ǫ /ℓ r /ℓ = C ǫ ( λ +1) / ( ℓλ ) of cardinality at most C B C ǫ ǫ − D ( p ) /ℓ r − D ( p ) /ℓ = C B C ǫ ǫ − D ( p ) ℓ ǫ − D ( p ) ℓλ = C B C ǫ − λ +1 ℓλ D ( p )+1 . RANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 27
By Proposition 2.4 we get dim H Σ Bc D ( p ) − ℓλλ +1 and this concludes the proof. (cid:3) Recalling that Σ Bc = ∅ if ℓ = 1, Lemmata 5.1 and 5.2 immediately lead to thefollowing result. Theorem 5.3.
Let Σ be a p -dimensional submanifold of G of class C ,λ , λ ∈ (0 , .It follows that dim H Σ c D ( p ) − λ if ℓ = ℓ ( p ) = 1dim H Σ c D ( p ) − if ℓ > and λ > ℓ − dim H Σ c D ( p ) − ℓλ λ if ℓ > and λ ℓ − . (5.4) Remark 5.4.
It is interesting to analyze Theorem 5.3 when the Carnot group G isthe Heisenberg group H n . In this case, ℓ = ℓ ( p ) = 1 for all p = 2 , . . . , n and Theorem5.3 reads as(5.5) dim H Σ c p + 1 − λ for any p -dimensional submanifold Σ ⊂ H n of class C ,λ . These estimates coincidewith the results stated in Remark 1, page 72 of [2]. In the special case p = 1, we have ℓ = 2, hence Theorem 5.3 gives(5.6) dim H Σ c D (1) − λ λ = 21 + λ − λ , where the last inequality is strict for all λ ∈ (0 ,
1) and 2 − λ = p + 1 − λ . Thus, in thisspecial case of curves ( p = 1), the estimates (5.4) improve that of Remark 1, page 72of [2]. References [1]
L. Ambrosio , Some fine properties of sets of finite perimeter in Ahlfors regular metric measurespaces , Adv. Math., , 51-67, (2001)[2]
Z. M. Balogh , Size of characteristic sets and functions with prescribed gradients , J. ReineAngew. Math., , 63-83, (2003)[3]
Z. M. Balogh, C. Pintea & H. Rohner , Size of Tangencies to Non-Involutive Distributions ,Indiana Univ. Math. J., to appear[4]
Z. M. Balogh, J. T. Tyson & B. Warhurst , Sub-Riemannian vs. Euclidean dimensioncomparison and fractal geometry on Carnot groups , Adv. Math. (2009), no. 2, 560–619[5]
V. Barone Adesi, F. Serra Cassano, D. Vittone , The Bernstein problem for intrinsicgraphs in Heisenberg groups and calibrations , Calc. Var. Partial Differential Equations , no.1, 17-49, (2007)[6] L. Capogna, D. Danielli, N. Garofalo , The geometric Sobolev embedding for vector fieldsand the isoperimetric inequality , Comm. Anal. Geom. , n.2, 203-215, (1994)[7] L. Capogna, N. Garofalo , Ahlfors type estimates for perimeter measures in Carnot-Cara-th´eodory spaces , J. Geom. Anal. , n.3, 455-497, (2006)[8] D. Danielli, N. Garofalo, D. M. Nhieu , Non-doubling Ahlfors measures, Perimeter mea-sures, and the characterization of the trace spaces of Sobolev functions in Carnot-Carath´eodoryspaces , Mem. Amer. Math. Soc. , n.857, (2006) [9]
D. Danielli, N. Garofalo, D. M. Nhieu , Sub-Riemannian calculus on hypersurfaces inCarnot groups , Adv. Math., , n.1, 292-378, (2007)[10]
D. Danielli, N. Garofalo, D. M. Nhieu , A notable family of entire intrinsic minimal graphsin the Heisenberg group which are not perimeter minimizing , Amer. J. Math, 130 (2008) no.2,317-339[11]
H. Federer , Geometric Measure Theory , Springer, (1969)[12]
B. Franchi, S. Gallot, R. L. Wheeden , Sobolev and isoperimetric inequalities for degeneratemetrics , Math. Ann. , 557-571 (1994)[13]
B. Franchi, R. Serapioni, F. Serra Cassano , Regular hypersurfaces, intrinsic perimeterand implicit function theorem in Carnot groups , Comm. Anal. Geom., , n.5, 909-944, (2003)[14] B. Franchi, R. Serapioni & F. Serra Cassano , On the structure of finite perimeter setsin step 2 Carnot groups , J. Geom. An. , 421-466, (2003)[15] G. B. Folland & E. M. Stein , Hardy spaces on homogeneous groups , Princeton UniversityPress, 1982[16]
N. Garofalo, D. M. Nhieu , Isoperimetric and Sobolev Inequalities for Carnot-Carath´eodorySpaces and the Existence of Minimal Surfaces , Comm. Pure Appl. Math. , 1081-1144 (1996)[17] M. Gromov , Carnot-Carath´eodory spaces seen from within , in
Subriemannian Geometry ,Progress in Mathematics, , edited by A. Bellaiche and J.-J. Risler, Birkh¨auser Verlag, (1996)[18]
R. Korte, V. Magnani , Measure of curves in graded groups , Illinois J. Math., to appear[19]
E. Le Donne, V. Magnani , Measure of submanifolds in the Engel group , Rev. Mat. Iberoamer-icana, , n.1, 333-346, (2010)[20] V. Magnani , Characteristic points, rectifiability and perimeter measure on stratified groups , J.Eur. Math. Soc., vol. 8, n.4, 585-609, (2006)[21]
V. Magnani , Non-horizontal submanifolds and coarea formula , J. Anal. Math., , 95-127,(2008)[22]
V. Magnani , Blow-up estimates at horizontal points and applications , J. Geom. Anal. , n.3,705-722, (2010)[23] V. Magnani & D. Vittone , An intrinsic measure for submanifolds in stratified groups , J.Reine Angew. Math. (2008), 203–232[24]
P. Mattila , Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability ,Cambridge Studies in Advanced Mathematics, , Cambridge University Press, Cambridge,1995[25] P. Mattila , Measures with unique tangent measures in metric groups , Math. Scand. , n.2,298-308 (2005)[26] R. Monti, D. Morbidelli , Regular domains in homogeneous groups , Trans. Amer. Math. Soc. , n.8, 2975-3011, (2005)[27]
L. Simon , Lectures on geometric measure theory , Proceedings of the Centre for MathematicalAnalysis, Australian National University, 3. Australian National University, Centre for Mathe-matical Analysis, Canberra, 1983. vii+272 pp.
Dipartimento di Matematica, Universit`a di Pisa, Largo Bruno Pontecorvo 5, 56127,Pisa, Italy
E-mail address : [email protected] Department of Mathematics, University of Illinois, 1409 West Green St., Urbana,IL 61801 USA
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