REGULAR HYPERSURFACES, INTRINSIC PERIMETER AND IMPLICIT FUNCTION THEOREM IN CARNOT GROUPS

BRUNO FRANCHI, RAUL SERAPIONI, AND FRANCESCO SERRA CASSANO

1. Introduction.

In the last few years, a systematic attempt to develop geometric measure theory in metric spaces has become the object of many studies. Such a program, already suggested in Federer’s book [17], has been explicitly formulated and carried on by several authors. We only mention some of them: De Giorgi [14], [15], [16], Gromov [28], [29], Preiss and Tisěr [43], Kirchheim [33], David & Semmes [11], Cheeger [9] and Ambrosio and Kirchheim [3], [4].

In this paper we study, inside a special class of metric spaces i.e. the Carnot groups, a classical problem in Geometric Measure Theory that is the problem of defining regular hypersurfaces and different reasonable surface measures on them, and of understanding their relationships (here hypersurface means simply codimension 1 surface).

First of all a few words on the ambient space: Carnot groups, each one endowed with its Carnot-Carathéodory distance d_{c} (hereafter abbreviated as cc-distance), are particularly interesting metric spaces not only because they appear in many different mathematical theories (e.g. Several Complex Variables, Partial Differential Equations, Control Theory) but also because they provide examples of spaces that are non Euclidean at any scale yet have a rich geometric structure as families of natural translations and dilations.

We recall briefly the definition: a Carnot group is a connected, simply connected, nilpotent Lie group 𝔾 ≡ (ℝ^{n}, ·) with graded Lie algebra 𝔤 that is Lie generated by its first layer V_{1}, the so called horizontal layer,

𝔤 = V_{1} ⊕ ... ⊕ V_{k}, [V_{1}, V_{i}] = V_{i+1}, V_{k} ≠ {0}, V_{i} = {0} if i > k. (1)

Assume that X_{1}, …, X_{m} is a family of left invariant vector fields that is also an orthonormal basis of V_{1} ≡ ℝ^{m} at the origin, that is X_{1}(0) = ∂_{x_{1}}, …, X_{m}(0) = ∂_{x_{m}}. The Lie algebra of the group 𝔾 can be canonically endowed with a family of dilations, so that 𝔾 is also a homogeneous group with homogeneous dimension Q = ∑_{i=1}^{k} i dim V_{i} and k is called step of the group (see [19]).

We say that an absolutely continuous curve γ : [0, T] → 𝔾 is a sub-unit curve with respect to X_{1}, …, X_{m} if there exist real measurable functions c_{1}(s), …, c_{m}(s), s ∈ [0, T] such that ∑_{j} c_{j}^{2} ≤ 1 and

γ̇(s) = ∑_{j=1}^{m} c_{j}(s)X_{j}(γ(s)), for a.e. s ∈ [0, T].

If p, q ∈ 𝔾, their cc-distance d_{c}(p, q) is

(2) d_{c}(p, q) = inf {T > 0 : γ : [0, T] → 𝔾 is subunit, γ(0) = p, γ(T) = q} .

The fact, that under assumption (1), d_{c}(p, q) is finite for any p, q is the content of Chow theorem (see e.g. [6] or [28]). We recall that the topology induced on ℝ^{n} by d_{c} is the Euclidean topology, but from a metric point of view 𝔾 and Euclidean ℝ^{n} can be dramatically different: indeed there are no (even local) bilipschitz maps from a general non commutative group 𝔾 to Euclidean spaces. In particular d_{c} is not locally equivalent to a Riemannian distance. This fact, proved by Semmes (see [44]), relies on a Rademacher’s type theorem due to Pansu ([41] see also Vodop’yanov [46]) and on algebraic and metric properties related to the non-commutativity of 𝔾. Moreover observe that the intrinsic Hausdorff dimension of a Carnot group 𝔾 (i.e. with respect to the cc-distance d_{c}) agrees with its homogeneous dimension Q. Notice that Q is always strictly larger than n, the topological dimension of 𝔾 (see [39]).

Date: June 18, 2002.

The authors are supported by GNAMPA of INdAM, project ”Analysis in metric spaces and subelliptic equations”.

B.F. is supported by University of Bologna, Italy, funds for selected research topics.

R.S. is supported by MURST, Italy, and University of Trento, Italy.

F.S.C. is supported by MURST, Italy, and University of Trento, Italy.

Coming now to the problem of surfaces and their measures, observe that the notion of regular surface in a group is not a completely obvious one and that its full comprehension is certainly a starting point for a geometrical understanding of the group. A rich and stimulating discussion on this topic can be found in Section 2 of [28].

There is a classical definition of ‘good’ surface in a metric space that goes back at least to Federer (see [17] 3.2.14). According to it, a ‘good’ surface in a metric space is the image of an open subset of an Euclidean space via a Lipschitz map. Such a notion has been successfully used recently by Ambrosio and Kirchheim (see [3], [4]) to develop a theory of currents in metric spaces. Unfortunately it does not fit the geometry of Carnot groups: indeed, as proved by the same authors in [3], in the Heisenberg group ℍ^{n} ≡ ℝ^{2n+1}, for example, any Lipschitz image of an open subset of ℝ^{2n} would be purely unrectifiable. This fact simply means that open subsets of ℝ^{d} are not appropriate as parameter spaces of surfaces inside a group 𝔾; it is necessary to use open subsets of metric spaces more strictly related to 𝔾 (about this, see also the definition of rectifiable sets in Carnot groups given in [42]).

On the other hand there is a way of circumventing this difficulty when dealing with codimension 1 surfaces. In any Euclidean space ℝ^{d}, a C^{1} hypersurface can be equivalently viewed (locally) as the zero set of a function f : ℝ^{d} → ℝ with non-vanishing gradient and in Carnot groups it is natural to follow the same approach.

If U is an open subset of 𝔾 and f : U → ℝ we say that f belongs to C_{𝔾}^{1}(U) when f and Xf := (X_{1}f, ..., X_{m}f) are continuous functions in U. We say that S ⊂ ℝ^{n} is a 𝔾-regular hypersurface if for any p ∈ S there is an open U ∋ p and f ∈ C_{𝔾}^{1}(U) such that

S ∩ U = {q ∈ U : f(q) = 0 and Xf(q) ≠ 0}. (3)

In [28], Gromov proved that a topological (n−1)-dimensional surface in 𝔾 has intrinsic Hausdorff dimension larger than Q − 1. Here we prove that regular hypersurfaces have precisely intrinsic Hausdorff dimension Q − 1 and topological dimension n − 1 (but they might have Euclidean Hausdorff dimension larger than n − 1).

We recall also that in [22] when studying regular surfaces and surface measures in the special case of the Heisenberg groups ℍ^{n}, we defined S to be an ℍ-regular surfaces if (3) holds. This definition came out to be a good one, since we were able to prove there an implicit function theorem, yielding a local continuous parametrization of S, an integral representation of the intrinsic Hausdorff measure, and an area type formula. These results made us able to extend, to the setting of ℍ^{n}, De Giorgi’s theory on the rectifiability of the boundary of finite perimeter sets as well as De Giorgi’s generalized Gauss-Green formula.

In this paper we prove in a general Carnot group 𝔾 a corresponding implicit function theorem. It might be surprising that the statement reads as in ℍ^{n}, because it is known that, usually, passing from Heisenberg groups to general Carnot groups could make things quite different because a Carnot group can be strongly different from another one, once more due to the possible different stratifications of their Lie algebras that make their geometries not (even locally) comparable.

By the way, as for De Giorgi’s rectifiability theory in general Carnot groups, the problem is far from be fully settled, even if a positive answer is given in [23] for a large class of Carnot groups containing all step 2 groups.

Our implicit function theorem states (see Theorem 2.1) that, locally, S ∩U is the graph in the directions of the integral lines of an appropriate vector field X_{i}, where X_{i}f ≠ 0, of a continuous function ϕ. A more precise statement is as follows:

Assume, without loss of generality, that in (3) we have 0 ∈ S, X_{1}f(0) ≠ 0 and that X_{1}(0) = ∂_{x_{1}} (see also (18)), then there are Ũ, I_{δ} and ϕ, with 0 ∈ Ũ ⊂ U, I_{δ} = {ξ ∈ ℝ^{n−1} : |ξ_{i}| ≤ δ} and ϕ : I_{δ} → ℝ such that

S ∩ Ũ = {p ∈ Ũ : p = exp (ϕ(ξ)X_{1}) (0, ξ) := Φ(ξ), ξ ∈ I_{δ}}. (4)

In general ϕ is not as regular as one might wish (see Example 3 and Theorem 6.5, vi in[22]), nevertheless through ϕ we can write explicitly the surface measure of S in local coordinates

|∂E|_{𝔾}(Ũ) = ∫_{I_{δ}} \frac{\sqrt{∑_{i=1}^{m} |X_{i}f(Φ)|^{2}}}{X_{1}f(Φ)} dL^{n−1} (5)

The perimeter measure |∂E|_{𝔾} comes from considering S as the topological boundary of the set E = {p ∈ 𝔾 : p = exp(tX_{1})(ξ), t < ϕ(ξ), ξ ∈ I_{δ}}. Then S ∩ Ũ = ∂E ∩ Ũ and the perimeter measure |∂E|_{𝔾} is defined as the total variation of the characteristic function of E. The measure |∂E|_{𝔾} is supported on S. The subscript 𝔾 is somehow incorrect because the perimeter as well as the cc-distance d_{c} depend on the choice of the orthonormal family X = X_{1}, …, X_{m}: changing the base in the horizontal fiber V_{1} actually changes the perimeter, but the two remain comparable. Observe that this is the same as in a Riemannian manifold: changing the metric tensor changes the perimeter and the new one is comparable with the old one. Later on we shall fix the family X and this amounts to the choice of a privileged coordinate system in 𝔤.

On the other hand also different surface measures can be considered, all of them depending on the cc-distance. We want to compare the following ones: the (Q − 1)-Hausdorff measure H_{𝔾}^{Q−1},the (Q − 1)-spherical Hausdorff measure S_{𝔾}^{Q−1} and the Minkowski content M. We recall briefly the definitions.

H_{𝔾}^{Q−1}(S) = lim_{δ→0^{+}} H_{𝔾,δ}^{Q−1}(S); S_{𝔾}^{Q−1}(S) = lim_{δ→0^{+}} S_{𝔾,δ}^{Q−1}(S)

where respectively

H_{𝔾,δ}^{Q−1}(S) = inf \{∑_{i} (\frac{diamC_{i}}{2} )^{Q−1} : S ⊂ ⋃_{i} C_{i}; diamC_{i} < δ\} ,

S_{𝔾,δ}^{Q−1}(S) = inf \{∑_{i} (\frac{diamB_{i}}{2} )^{Q−1} : S ⊂ ⋃_{i} B_{i}; diamB_{i} < δ\}

where the infimum is taken with respect to closed sets C_{i} in the first line or closed cc-balls B_{i} in the second one.

For any open U ⊂ ℝ^{n} the Minkowski content M(S)(U) is

M(S)(U) = lim_{δ→0^{+}} \frac{1}{2δ} L^{n}(S_{δ} ∩ U)

provided the limit exists, where S_{δ} = {p ∈ 𝔾 : d_{c}(p, S) < δ} and L^{n} is the n-dimensional Lebesgue measure in ℝ^{n}, which is, by the way, the Haar measure of 𝔾.

In the same hypotheses of the implicit function theorem, with the notations of (3) and (5), we prove that (see Theorem 3.3) the perimeter and the S_{𝔾}^{Q−1} (and hence H_{𝔾}^{Q−1}) are comparable measures, i.e. there is α > 1 and a Borel function 𝔰 : S ∩ U → ℝ such that for all Ũ ⊂ U we have \frac{1}{α} ≤ 𝔰 ≤ α a.e. on S ∩ Ũ and

∫_{S∩Ũ} 𝔰 dS_{𝔾}^{Q−1} = |∂E|_{𝔾}(Ũ) (6)

Moreover in [40], if S is also an Euclidean C^{∞} surface, it is proved that

|∂E|_{𝔾}(Ũ) = M(S)(Ũ). (7)

In some special Carnot groups, as the Heisenberg groups, the function 𝔰 in (6) is a constant, and one gets the following stronger version of (6)

|∂E|_{𝔾}(Ũ) = c S_{𝔾}^{Q−1}(S ∩ Ũ). (8)

It may be interesting to compare the notion of 𝔾-regular hypersurface and that of Euclidean C^{1} hypersurface. Strictly speaking, the two classes are different. However, it is enough to remove a “negligeable” closed set from an Euclidean C^{1} hypersurface to obtain a 𝔾-regular hypersurface, whereas 𝔾-regular hypersurfaces can be drammatically irregular from an Euclidean point of view: they may be (n − 1)-topological submanifold with Euclidean dimension larger than n − 1. For more precise statements, see Subsection 3.2.

We are glad to thank L.Ambrosio for his constant support. We are indebted also with Z.Balogh and V.Magnani for showing us their still unpublished papers. We thank also the referee for his careful reading and for many valuable suggestions.

1.1. Definitions and Notations. Consider a family X of vector fields X = (X_{1}, ..., X_{m}) ∈ Lip (ℝ^{n}; ℝ^{n})^{m}. As usual we identify vector fields and differential operators. If

X_{j}(x) = ∑_{i=1}^{n} c_{i}^{j}(x)∂_{i}, j = 1, ..., m,

define the m × n matrix

C(x) = [c_{i}^{j}(x)]_{{i=1,...,n \atop j=1,...,m}} . (9)

Given the family X of Lipschitz continuous vector fields it is well known that subunit curves can be defined as we do in the Introduction for Carnot groups and conseqently the Carnot-Carathéodory distance d_{c} is well defined provided that there is a subunit curve joining each couple of points. Through the paper, whenever the Carnot-Carathéodory distance is mentioned, we are assuming implicitely that this connectivity property holds and that the distance d_{c} is continuous with respect to the Euclidean topology. We shall denote U_{c}(p, r) the open balls associated with d_{c}.

X_{j}^{∗} is the operator formally adjoint to X_{j} in L^{2}(ℝ^{n}), that is the operator which for all ϕ, ψ ∈ C_{0}^{∞}(ℝ^{n}) satisfies

∫_{ℝ^{n}} ϕ(x)X_{j}ψ(x) dx = ∫_{ℝ^{n}} ψ(x)X_{j}^{∗}ϕ(x) dx.

Moreover, if f ∈ L_{loc}^{1} is a scalar function and ϕ ∈ (L_{loc}^{1})^{m} is a m-vector valued function, we define the X-gradient and X-divergence as the following distributions:

Xf := (X_{1}f, ..., X_{m}f), div_{X} (ϕ) := − ∑_{j=1}^{m} X_{j}^{∗}ϕ_{j}.

Let us remind now the notion of functions of bounded X-variation and recall some of their properties (see [8], [20] and [25]). Let Ω ⊂ ℝ^{n} be an open set and let

F (Ω; ℝ^{m}) := {ϕ ∈ C_{0}^{1}(Ω; ℝ^{m}) : |ϕ(x)| ≤ 1 ∀x ∈ Ω}. (10)

The space BV_{X}(Ω) is the set of functions f ∈ L^{1}(Ω) such that

(11) ||Xf||(Ω) := sup_{ϕ∈F (Ω;ℝ^{m})}∫_{Ω} f(x)div_{X} (ϕ)(x) dx < ∞.

The space BV_{X,loc}(Ω) is the set of functions belonging to BV_{X}(U) for each open set U ⊂⊂ Ω. From Riesz representation theorem it follows that if f ∈ BV_{X,loc}(Ω) then the total variation ||Xf|| is a Radon measures on Ω (see [17], 2.2.5).

Theorem 1.1. (Structure of BV_{X} functions) Let BV_{X,loc}(Ω), then there exists a ||Xf||- measurable function σ_{f} : Ω → ℝ^{m} such that |σ_{f}(x)|_{ℝ^{m}} = 1, for ||Xf||-almost every x, and

∫_{Ω} f(x)div_{X} ϕ(x) dx = ∫_{Ω} ⟨ϕ(x), σ_{f}(x)⟩_{ℝ^{m}}d||Xf||,

for all ϕ ∈ F (Ω; ℝ^{m}).

In perfect analogy with the Euclidean setting, the total variation ||Xf|| is lower semicontinuous with respect to L^{1} convergence, (see [8], [20] and [25]), that is we have

Proposition 1.2. (Lower semicontinuity) Let f, f_{k} ∈ L^{1}(Ω), k ∈ ℕ. If f_{k} → f in L^{1}(Ω), then

lim inf_{k→∞} ||Xf_{k}||(Ω) ≥ ||Xf||(Ω).

Definition 1.3. (X-Caccioppoli sets) A measurable set E ⊂ ℝ^{n} is a set with locally finite X-perimeter in Ω (or is a X-Caccioppoli set) if the indicatrix function 1_{E} ∈ BV_{X,loc}(Ω). In this case the total variation measure ∥X1_{E}∥_{∞} is called perimeter measure of E and is indicated as |∂E|_{X}; hence we have

|∂E|_{X}(U) := ||X1_{E}||(U) < ∞ (12)

for every open set U ⊂⊂ Ω. Moreover, the vector function σ_{1_{E}} appearing in Theorem 1.1, is called X-generalized inner normal of E and we set

ν_{E}(x) := −σ_{χ_{E}}(x). (13)

It is important to observe that when a Caccioppoli set F has a topological boundary ∂F that is an Euclidean C^{1} submanifold of ℝ^{n}, then the perimeter measure can be represented by integration with respect to the Euclidean n − 1 Hausdorff measure. Precisely we have (see [8])

Proposition 1.4. If F is a X-Caccioppoli set with C^{1} boundary, then the X-perimeter has the following representation

|∂F |_{X}(Ω) = ∫_{∂F ∩Ω} |C(x)n_{F} (x)|dH^{n−1}, (14)

here n_{F} (x) is the Euclidean unit outward normal to F, C is the coefficient matrix of the vector fields (see (9)), and H^{n−1} is the Euclidean (n − 1)-dimensional Hausdorff measure.

It is also important to notice that the domain of applicability of formula (14) is restricted to Euclidean regular hypersurfaces. On the other hand, even in simple Carnot groups (see the definition below) the boundary of finite perimeter sets is, in general, a highly irregular set from an Euclidean point of view. Indeed, not only the Euclidean normal can fail to exist almost everywere, but even the Euclidean metric dimension of the boundary could exceed n − 1, so making the right hand side of (14) divergent for all Ω ([34]). The perimeter representation proved in (vi) of Theorem 2.1 can be viewed as a generalization of (14) to the boundary of finite perimeter sets (see Remark 3.10).

Finally, we recall the definition of Carnot group and some of its structures (see [18], [31], [40],

[32] and [41]). Let 𝔾 = (ℝ^{n}, ·) be a Lie group whose Lie algebra 𝔤 admits a stratification, i.e. there exist linear subspaces V_{1}, ..., V_{k} such that

𝔤 = V_{1} ⊕ ... ⊕ V_{k}, [V_{1}, V_{i}] = V_{i+1}, V_{k} ≠ {0}, V_{i} = {0} if i > k, (15)

where [V_{1}, V_{i}] is the subspace of 𝔤 generated by the elements [X, Y] with X ∈ V_{1} and Y ∈ V_{i}.

Via the exponential map, it is possible to induce on 𝔾, in a canonical way, a family of automorphisms of the group, called dilations, δ_{λ} : ℝ^{n} → ℝ^{n} (λ > 0) such that

δ_{λ}x ≡ δ_{λ}(x_{1}, ..., x_{n}) = (λ^{α_{1}}x_{1}, ..., λ^{α_{n}}x_{n}), (16)

where 1 = α_{1} = ... = α_{m} < α_{m+1} ≤ ... ≤ α_{n} are integers and m = dim(V_{1}) (see [19] Chapter 1).

The n-dimensional Lebesgue measure in ℝ^{n}, denoted by L^{n}, is the Haar measure of the group 𝔾. This means that if E ⊂ ℝ^{n} is measurable, then L^{n}(x · E) = L^{n}(E) for all x ∈ 𝔾. Moreover, if λ > 0 then L^{n}(δ_{λ}E) = λ^{Q}L^{n}(E).

The group law can be written in the form

x · y = x + y + Q(x, y), x, y ∈ ℝ^{n} (17)

where Q : ℝ^{n} × ℝ^{n} → ℝ^{n} has polynomial components and Q_{1} = ... = Q_{m} = 0 (see [45], Chapter 12, Section 5). Note that the inverse x^{−1} of an element x ∈ 𝔾 has the form x^{−1} = −x = (−x_{1}, …, −x_{n}).

Choose now a family X = (X_{1}, …, X_{m}) of left invariant vector fields that is also an orthonormal basis of V_{1} ≡ R^{m} at the origin, in particular we choose X so that X_{1}(0) = ∂_{x_{1}}, …, X_{m}(0) = ∂_{x_{m}}. It happens that also these vector fields X_{j} have polynomial coefficients; more precisely they have the form

X_{j}(x) = ∂_{j} + ∑_{i=m+1}^{n} a_{i}^{j}(x)∂_{i}, X_{j}(0) = ∂_{j}, j = 1, ..., m; (18)

moreover each polynomial a_{i}^{j} is homogeneous with respect to the dilations of the group, that is a_{i}^{j}(δ_{λ}(x)) = λ^{α_{i}−α_{j}}a_{i}^{j}(x) (see [19], Proposition 1.26). From this homogeneity it follows that if j belongs to the l-layer, that is if h_{l−1} := ∑_{i=1}^{l−1} dimV_{i} < j ≤ h_{l} := ∑_{i=1}^{l} dimV_{i}, then

a_{i}^{j}(x) = a_{i}^{j}(x_{1}, …, x_{h_{l−1}}). (19)

We refer to {X_{1}, …, X_{m}} as canonical generating vector fields of the group. The subbundle H𝔾 of the tangent bundle T 𝔾 with fibers

H𝔾_{x} = span {X_{1}(x), …, X_{m}(x)}, x ∈ 𝔾

is called horizontal bundle. We endow each fiber H𝔾_{x} with a scalar product ⟨·, ·⟩_{x} and a norm | · |_{x} that make the moving frame {X_{1}(x), …, X_{m}(x)} to be orthonormal. We shall drop the index x in the scalar product or in the norm, writing ⟨ψ, ϕ⟩ for ⟨ψ(x), ϕ(x)⟩_{x}, if there is no ambiguity.

We shall identify each section of H𝔾 with its canonical coordinates with respect to this moving frame. This way, a section ϕ is identified with a function ϕ = (ϕ_{1}, …, ϕ_{m}) : U ⊆ ℝ^{n} → ℝ^{m}. The spaces of smooth sections of the horizontal bundle are denoted respectively by C^{k}(U, H𝔾), C_{0}^{k}(U, H𝔾), C^{∞}(U, H𝔾) or C_{0}^{∞}(U, H𝔾).

A metric d_{c} on 𝔾 is defined, via Carnot-Carathéodory construction from the vector fields X_{1}, …, X_{m}. Notice that the set of subunit curves joining any two points in 𝔾 is never empty, by Chow’s theorem, because the rank of the Lie algebra generated by X_{1}, …, X_{m} is n; hence d_{c} is a distance on 𝔾 inducing on ℝ^{n} the Euclidean topology. Notice also that, being defined through left invariant vector fields, d_{c} enjoys the further property of being well behaved with respect to left translations and dilations. We mean that

d_{c}(z · x, z · y) = d_{c}(x, y) and d_{c}(δ_{λ}x, δ_{λ}y) = λd_{c}(x, y) (20)

for x, y, z ∈ 𝔾 and λ > 0.

The integer Q = ∑_{j=1}^{n} α_{j} = ∑_{i=1}^{k} i dim V_{i} is called the homogeneous dimension of the group; Q turns out to be the Hausdorff dimension of ℝ^{n} with respect the cc-distance d_{c} (see [39]).

Inside a group 𝔾, we depart slightly from our previous notations. Once a canonical generating family of vector fields X for 𝔾 is fixed, we write ∇_{𝔾} for X, div_{𝔾} for div_{X} , |∂E|_{𝔾} for |∂E|_{X}, and so on. In particular, we say that a continuous function f belongs to C_{𝔾}^{1}(U) if ∇_{𝔾}f is a continuous vector–valued function in U.

As observed for the distance, also the perimeter measure |∂E|_{𝔾} enjoys the further properties of being invariant under group translations and (Q − 1)-homogeneous with respect to group dilations, that is, for all open O ⊂ ℝ^{n}, x ∈ 𝔾 and λ > 0, we have

|∂E|_{𝔾}(O) = |∂(x · E)|_{𝔾}(x · O) and |∂E|_{𝔾}(O) = λ^{1−Q}|∂(δ_{λ}E)|_{𝔾}(δ_{λ}O). (21)

A remarkable property of the 𝔾-perimeter is provided by the following Q-dimensional isoperimetric inequality ([25]).

Proposition 1.5. (Isoperimetric inequality) There is a positive constant c_{I} > 0 such that for any 𝔾-Caccioppoli set E, for all x ∈ 𝔾 and r > 0,

min\{L^{n}(E ∩ U_{c}(x, r)), L^{n}(E^{c} ∩ U_{c}(x, r))\}^{\frac{Q−1}{Q}} ≤ c_{I}|∂E|_{𝔾}(U_{c}(x, r)) (22)

and

min\{L^{n}(E), L^{n}(E^{c})\}^{\frac{Q−1}{Q}} ≤ c_{I}|∂E|_{𝔾}(ℝ^{n}). (23)

We define now 𝔾-regular hypersurfaces in a Carnot group 𝔾, mimicking Definition 6.1 in [22], as non critical level sets of functions in C_{𝔾}^{1}(ℝ^{\}).

Definition 1.6. (𝔾-regular hypersurfaces) Let 𝔾 be a Carnot group. We shall say that S ⊂ 𝔾 is a 𝔾-regular hypersurface if for every x ∈ S there exist a neighborhood U of x and a function f ∈ C_{𝔾}^{1}(U) such that

S ∩ U = {y ∈ U : f(y) = 0}; (i) ∇_{𝔾}f(y) ≠ 0 for y ∈ U. (ii)

𝔾-regular surfaces have a unique tangent plane at each point. This follows from a Taylor formula for functions in C_{𝔾}^{1}(ℝ^{n}) that is basically proved in [41].

Proposition 1.7. If f ∈ C_{𝔾}^{1}(U_{c}(p, r)), then

f(x) = f(p) + ∑_{j=1}^{m}(X_{j}f)(p)(x_{j} − p_{j}) + o(d_{c}(x, p)), as x → p. (24)

If S = {x : f(x) = 0} ⊂ 𝔾 is a 𝔾-regular hypersurface, the tangent group T_{𝔾}^{g}S(x) to S at x is

T_{𝔾}^{g} S(x) := {v = (v_{1}, …, v_{n}) ∈ 𝔾 : ∑_{j=1}^{m} X_{j}f(x)v_{j} = 0}.

By (17), T_{𝔾}^{g}S(x) is a subgroup of 𝔾, that is proper, by (ii) of Definition 1.6. We can define the tangent plane to S at x as

T_{𝔾}S(x) := x · T_{𝔾}^{g}S(x).

We stress that this is a good definition. Indeed the tangent plane does not depend on the particular function f defining the surface S. This is a consequence of points (i) and (iii) of implicit function theorem below that yields

T_{𝔾}^{g}S(x) = {v ∈ 𝔾 : ⟨ν_{E}(x), π_{x}v⟩_{x} = 0}

where ν_{E} is the generalized inward unit normal defined in (13) and π_{x}(v) = ∑_{j=1}^{m} v_{j}X_{j}(x). Notice that the map v ↦ π_{x}(v), for x ∈ 𝔾 fixed,

π_{x}(v) = ∑_{j=1}^{m} v_{j}X_{j}(x). (25)

is a smooth section of H𝔾.

Notice also that, once more from (iii) of Theorem 2.1, it follows that ν_{E} is a continuous function.

If v^{0} = ∑_{i=1}^{m} v_{i}X_{i}(0) ∈ H𝔾_{0} we define the halfspaces S_{𝔾}^{±}(0, v^{0}) as

S_{𝔾}^{+}(0, v^{0}) := {x ∈ 𝔾 : ∑_{i=1}^{m} x_{i}v_{i} > 0} and S_{𝔾}^{−}(0, v^{0}) := {x ∈ 𝔾 : ∑_{i=1}^{m} x_{i}v_{i} < 0}.

Their common boundary is the vertical plane

Π(0, v^{0}) := {x : ∑_{i=1}^{m} x_{i}v_{i} = 0}.

If v = ∑_{i=1}^{m} v_{i}X_{i}(y) ∈ H𝔾_{y}, S_{𝔾}^{±}(y, v) and Π(y, v) are the translated sets

S_{𝔾}^{±}(y, v) := y · S_{𝔾}^{±}(0, v^{0}) and Π(y, v) = y · Π(0, v^{0})

where v and v^{0} have the same components v_{i} with respect to the left invariant basis X_{i}. Hence

S_{𝔾}^{±}(y, v) = {x ∈ 𝔾 : ∑_{i=1}^{m}(x_{i} − y_{i})v_{i} > 0(< 0)}. (26)

Clearly, T_{𝔾}S(x) = Π(x, ν_{E}(x)).

2. The Implicit Function Theorem.

Our main result states that a 𝔾-regular hypersurface S = {f(y) = 0}, that is boundary of the set E = {f(y) < 0}, can be locally parameterized through a function Φ : ℝ^{n−1} → ℝ^{n} so that the 𝔾-perimeter of E can be written explicitly in terms of ∇_{𝔾}f and Φ.

Theorem 2.1. (Implicit Function Theorem) Let Ω be an open set in ℝ^{n} identified with a Carnot group 𝔾, 0 ∈ Ω, and let f ∈ C_{𝔾}^{1}(Ω) be such that f(0) = 0 and X_{1}f(0) > 0. Define

E = {x ∈ Ω : f(x) < 0}, S = {x ∈ Ω : f(x) = 0},

and, for δ > 0, h > 0

I_{δ} = {ξ = (ξ_{2}, …, ξ_{n}) ∈ ℝ^{n−1}, |ξ_{j}| ≤ δ}, J_{h} = [−h, h].

If ξ = (ξ_{2}, …, ξ_{n}) ∈ ℝ^{n−1} and t ∈ J_{h}, denote now by γ(t, ξ) the integral curve of the vector field X_{1} at the time t issued from (0, ξ) = (0, ξ_{2}, …, ξ_{n}) ∈ ℝ^{n}, i.e.

γ(t, ξ) = exp(tX_{1})(0, ξ).

Then there exist δ, h > 0 such that the map (t, ξ) → γ(t, ξ) is a diffeomorphism of a neighborhood of J_{h} × I_{δ} onto an open subset of ℝ^{n}, and, if we denote by U ⊂⊂ Ω the image of Int(J_{h} × I_{δ}) through this map, we have that E ∩ U is connected and

(i) E has finite 𝔾-perimeter in U; ∂E ∩ U = S ∩ U; (ii)

ν_{E}(x) = − \frac{∇_{𝔾}f(x)}{|∇_{𝔾}f(x)|_{x}} for all x ∈ S ∩ U, (iii)

where ν_{E} is the generalized inner unit normal defined by (13), that can be identified with a section of H𝔾 with |ν(x)|_{x} = 1 for |∂E|_{𝔾}-a.e. x ∈ U. In particular, ν_{E} can be identified with a continuous function and |ν| ≡ 1. Moreover, there exists a unique function

ϕ = ϕ(ξ) : I_{δ} → J_{h}

such that the following parameterization holds: if ξ ∈ I_{δ}, put Φ(ξ) = γ(ϕ(ξ), ξ), then

S ∩ Ũ = {x ∈ Ũ : x = Φ(ξ), ξ ∈ I_{δ}}; (iv)

(v) ϕ is continuous; the 𝔾-perimeter has an integral representation:

|∂E|_{𝔾}(Ũ) = ∫_{I_{δ}} \frac{\sqrt{∑_{j=1}^{m} |X_{j}f(Φ(ξ))|^{2}}}{X_{1}f(Φ(ξ))} dL_{ξ}^{n−1}.

(vi)

Proof. The proof will be divided in several steps. Step 1. Construction of the continuous function Φ. Clearly, the map (t, ξ) → γ(t, ξ) is continuously differentiable, and then it is a diffeomorphism of a neighborhood of J_{h} × I_{δ} onto an open subset of ℝ^{n} since its Jacobian determinant at (0, 0) is = 1.

Let us choose now δ, h > 0 so that X_{1}f > 0 in Ũ, and let ρ_{ϵ} be a Friedrichs’ mollifier. If we put f_{ϵ} = f ∗ ρ_{ϵ} (0 < ϵ < dist(Ũ, ℝ^{n} \ Ω)), then f_{ϵ} → f as ϵ → 0 uniformly on Ũ, because of the continuity of f. Analogously, (X_{j}f) ∗ ρ_{ϵ} → X_{j}f as ϵ → 0 uniformly on Ū for j = 1, …, m. We can use now a regularization argument that goes back to Friedrichs ([24]) and has been used recently for cc-metrics in [26] and [21]. Notice that we did not purpusely use group mollifiers ([19]), for the group structure does not really play a role in this proof (see Theorem 2.4 bel?ow). We have

X_{j}f_{ϵ} = (X_{j}f) ∗ ρ_{ϵ} − ((X_{j}f) ∗ ρ_{ϵ} − X_{j}f_{ϵ})

for j = 1, …, m. Let us prove now that (X_{j}f) ∗ ρ_{ϵ} − X_{j}f_{ϵ} → 0 as ϵ → 0 uniformly on Ũ. In fact, if x ∈ Ũ, denoting by ∂_{ℓ}f and ∂_{ℓ}ρ the partial derivatives respectively of f and ρ with respect to their ℓ-th argument, we have

((X_{j}f) ∗ ρ_{ϵ})(x) − X_{j}f_{ϵ}(x)

= ∫_{|x−y|<ϵ} ∑_{ℓ=m+1}^{n} a_{ℓ}^{j}(y)∂_{ℓ}f(y)ρ_{ϵ}(x − y)dy − ∫_{|x−y|<ϵ} ∑_{ℓ=m+1}^{n} a_{ℓ}^{j}(x)\frac{∂}{∂x_{ℓ}} (f(y)ρ_{ϵ}(x − y)) dy

= ∫_{|x−y|<ϵ} ∑_{ℓ=m+1}^{n} a_{ℓ}^{j}(y)∂_{ℓ}f(y)ρ_{ϵ}(x − y)dy − ∫_{|x−y|<ϵ} ∑_{ℓ=m+1}^{n} a_{ℓ}^{j}(x)f(y)(∂_{ℓ}ρ_{ϵ})(x − y)dy

= ∑_{ℓ=m+1}^{n} ∫_{|x−y|<ϵ} (a_{ℓ}^{j}(y) − a_{ℓ}^{j}(x))f(y)∂_{ℓ}ρ_{ϵ}(x − y)dy

(since a_{ℓ}^{j} does not depend on the ℓ-th variable)

= ∑_{ℓ=m+1}^{n} ∫_{|x−y|<ϵ} (a_{ℓ}^{j}(y) − a_{ℓ}^{j}(x))(f(y) − f(x))∂_{ℓ}ρ_{ϵ}(x − y)dy

+ f(x) ∑_{ℓ=m+1}^{n} ∫_{|x−y|<ϵ} (a_{ℓ}^{j}(y) − a_{ℓ}^{j}(x))∂_{ℓ}ρ_{ϵ}(x − y)dy = I_{1}(x) + f(x)I_{2}(x).

Again since a_{ℓ}^{j}(y) does not depend on the ℓ-th variable, we have

I_{2}(x) = − ∑_{ℓ=m+1}^{n} ∫_{|x−y|<ϵ} (a_{ℓ}^{j}(y) − a_{ℓ}^{j}(x))\frac{∂}{∂y_{ℓ}} ρ_{ϵ}(x − y)dy

= − ∑_{ℓ=m+1}^{n} ∫_{|x−y|<ϵ} \frac{∂}{∂y_{ℓ}} ((a_{ℓ}^{j}(y) − a_{ℓ}^{j}(x))ρ_{ϵ}(x − y))dy = 0,

since the function y → (a_{ℓ}^{j}(y) − a_{ℓ}^{j}(x))ρ_{ϵ}(x − y) is supported in {y : |x − y| < ϵ}. On the other hand, if we denote by ω the modulus of continuity of f, then

|I_{1}(x)| ≤ cϵ^{n+1}ω(ϵ) ∑_{ℓ=m+1}^{n} max |∂_{ℓ}ρ_{ϵ}| → 0 for any x ∈ Ũ

as ϵ → 0. This proves that

(27) X_{j}f_{ϵ} → X_{j}f as ϵ → 0 uniformly on Ū for j = 1, …, m.

Notice now that for any ξ ∈ I_{δ}

\frac{∂}{∂s}f_{ϵ}(γ(s, ξ)) = (X_{1}f_{ϵ})(γ(s, ξ))

converges uniformly with respect to s ∈ J_{h} to (X_{1}f)(γ(s, ξ)), and hence the map

s ↦ f(γ(s, ξ))

is differentiable for all s, |s| ≤ h and

\frac{∂}{∂s}f(γ(s, ξ)) = (X_{1}f)(γ(s, ξ)) > 0 (28)

when ξ ∈ I_{δ} and s ∈ J_{h}.

Since f(0) = 0 and γ(0, 0) = 0, then f(γ(h, 0)) > 0 > f(γ(−h, 0)) and, by continuity,

f(γ(h, ξ)) > 0 > f(γ(−h, ξ))

if ξ ∈ I_{δ}, provided δ is small enough, so that the existence of ϕ such that (iv) holds can be proved by an usual continuity argument.

Let us prove now that ϕ is continuous. To this end, it will be enough to show that, if ξ_{k} ∈ I_{δ} for k ∈ ℕ, and ξ_{k} → ξ, then there exists a subsequence (ξ_{k_{j}})_{j∈ℕ} such that ϕ(ξ_{k_{j}}) → ϕ(ξ). In fact, by the compactness of J_{h}, we can extract a subsequence ϕ(ξ_{k_{j}}) converging to ϕ_{0} ∈ J_{h}. By the continuity of f

0 = f(γ(ϕ(ξ_{k_{j}}), ξ_{k_{j}})) → f(γ(ϕ_{0}, ξ)),

and then ϕ_{0} = ϕ(ξ), by (iv). This proves (v).

Step 2. U ∩ ∂E = U ∩ S. By the continuity of f, clearly ∂E ⊆ S. On the other hand, let x ∈ S ∩ U be given; then x = γ(t, ξ) for some t ∈ J_{h} and ξ ∈ I_{δ}. Notice that the first component of γ(t, ξ) equals t, since the first component of X_{1} is 1; thus necessarily t = x_{1}. As we proved above, the function s → f(γ(s, ξ)) is strictly increasing and vanishes for s = x_{1}. Then there exists a sequence of points γ(s_{k}, ξ) ∈ E converging to x, so that S ⊆ ∂E. This proves (ii).

Later on, we shall need to use the fact that L^{n} (∂E ∩ U) = 0. To this end, put

S̃_{r} = \{x ∈ U, x = γ(s, ξ), ξ ∈ I_{δ}, s ∈ J_{h}, |s − ϕ(ξ)| < \frac{1}{r}\};

we have

∂E ∩ U = ⋂_{r=1}^{∞} S̃_{r}.

Indeed, if x ∈ ⋂_{r=1}^{∞} S̃_{r}, then x = γ(ϕ(ξ), ξ), so that f(x) = 0 and hence x ∈ S. If now, as usual, J_{γ} denotes the Jacobian matrix of γ, we have

|S̃_{r}| = ∫S̃_{r} dx ≤ ∫_{I_{δ}} dξ ∫_{|s−ϕ(ξ)|<1/r} |det J_{γ}(s, ξ)| ds ≤ C_{δ} \frac{1}{r},

since

|det J_{γ}(s, ξ)| = 1 + o(1) as (s, ξ) → (0, 0).

Thus L^{n} (∂E ∩ U) = 0, since S̃_{r+1} ⊆ S̃_{r} for all r ∈ ℕ.

Step 3. E has finite 𝔾-perimeter in U. The proof of this step is based on the construction of a family of functions {h_{ϵ}} bounded in BV_{𝔾}(U) converging to 1_{E} in L^{1}(U). Let us consider again the approximations f_{ϵ} we introduced at the beginning of the proof, and consider the functions g_{ϵ}, g : J_{h} × I_{δ} → ℝ as

g_{ϵ}(ξ_{1}, ξ) := f_{ϵ}(γ(ξ_{1}, ξ)) , g(ξ_{1}, ξ) := f(γ(ξ_{1}, ξ))

As we showed above, \frac{∂g_{ϵ}}{∂ξ_{1}} (ξ_{1}, ξ) converges uniformly to X_{1}f(γ(ξ_{1}, ξ)) on J_{h} × I_{δ}. Since γ(ξ_{1}, ξ) ∈ Ū, we can assume

(X_{1}f_{ϵ})(γ(ξ_{1}, ξ)) = \frac{∂g_{ϵ}}{∂ξ_{1}} (ξ_{1}, ξ) ≥ μ > 0 on J_{h} × I_{δ} for 0 < ϵ < ϵ_{0}. (29)

Thus we can apply the classical implicit function theorem in J_{h} × I_{δ} to obtain the existence of a smooth function ϕ_{ϵ} : I_{δ} → J_{h} such that

f_{ϵ}(γ(ϕ_{ϵ}(ξ), ξ)) ≡ g_{ϵ}(ϕ_{ϵ}(ξ), ξ) ≡ 0, (30)

provided δ is small enough. We stress the fact that the choice of δ and h defining I_{δ} and J_{h} can depend on μ but it is independent of ϵ: indeed first we notice that g_{ϵ}(h, 0) ≥ hμ > 0 and g_{ϵ}(−h, 0) ≤ −hμ < 0 for all ϵ ∈ (0, ϵ_{1}). Since g_{ϵ} → g uniformly, choose now ϵ_{1} = ϵ_{1}(h, μ) such that sup |g_{ϵ} − g| < hμ/3 for ϵ < ϵ_{1}, and |g(h, ξ) − g(h, 0)| < hμ/3 for ξ ∈ I_{δ}, provided δ < δ_{0} = δ_{0}(h, μ). Then, if ξ ∈ I_{δ}, 0 < ϵ < ϵ_{1}, and 0 < δ < δ_{0}, we have

g_{ϵ}(h, ξ) ≥ g_{ϵ}(h, 0) − |g_{ϵ}(h, ξ) − g_{ϵ}(h, 0)|
≥ g_{ϵ}(h, 0) − |g_{ϵ}(h, ξ) − g(h, ξ)| − |g(h, ξ) − g(h, 0)| − |g(h, 0) − g_{ϵ}(h, 0)| > 0.

Analogously g_{ϵ}(−h, ξ) < 0 for ξ ∈ I_{δ}, 0 < ϵ < ϵ_{1}, and 0 < δ < δ_{0}. Thus, the function ϕ_{ϵ}(ξ) is well defined for ξ ∈ I_{δ}. Let us prove now that ϕ_{ϵ} → ϕ uniformly in I_{δ}: by contradiction, suppose there exist σ > 0, ϵ_{k} → 0, (ξ_{k})_{k∈ℕ} in I_{δ} such that

|ϕ_{ϵ_{k}}(ξ_{k}) − ϕ(ξ_{k})| ≥ σ.

Without loss of generality we may assume that

ξ_{k} → ξ ∈ I_{δ} and ϕ_{ϵ_{k}}(ξ_{k}) → ϕ_{0} ∈ J_{h}

as k → ∞, so that |ϕ_{0} − ϕ(ξ)| ≥ σ. On the other hand

0 = g_{ϵ_{k}}(ϕ_{ϵ_{k}}(ξ_{k}), ξ_{k}) = f_{ϵ_{k}}(γ(ϕ_{ϵ_{k}}(ξ_{k})), ξ_{k}) → f(γ(ϕ_{0}, ξ)),

since f_{ϵ_{k}} → f uniformly on Ū. By the uniqueness of ϕ, this implies ϕ_{0} = ϕ(ξ), a contradiction.

Denote by γ^{−1} : U → Int(J_{h} × I_{δ}) the inverse map of γ; because of the structure of X_{1}, we can write γ^{−1}(x) = (x_{1}, θ(x)), where θ : U → Int(I_{δ}) is a smooth map.

If now H(s) = 1 for s > 0 and H(s) = 0 for s ≤ 0, put

h_{ϵ}(x) = H(ϕ_{ϵ}(θ(x)) − x_{1}).

and

E_{ϵ} = {x ∈ U : x_{1} < ϕ_{ϵ}(θ(x))}.

We want to show that

h_{ϵ} = 1_{E_{ϵ}} → 1_{E} in L^{1}(U).

By dominated convergence theorem, we need only to prove that h_{ϵ} → 1_{E} a.e. in U. This follows because ϕ_{ϵ} → ϕ uniformly on I_{δ}.

Take now x ∈ E ∩ U, x = γ(x_{1}, ξ), ξ ∈ I_{δ}; by definition

f(γ(ϕ(ξ), ξ)) = 0 > f(x) = f(γ(x_{1}, ξ)),

and hence ϕ(ξ) > x_{1}, since the map

s ↦ f(γ(s, ξ))

is strictly increasing on [−h, h]. Since ϕ_{ϵ}(ξ) → ϕ(ξ), then ϕ_{ϵ}(ξ) > x_{1} if ϵ is close to zero, that implies that h_{ϵ}(x) = 1 → 1 = 1_{E}(x) as ϵ → 0. The same argument can be carried out if f(x) > 0, and hence a.e. since |S ∩ U| = 0.

We can prove now that E has finite 𝔾-perimeter in U; to this end we need only to show that the h_{ϵ}’s have equibounded 𝔾-variations, since the 𝔾-variation is L^{1}-lsc (see Proposition 1.2).

To this end, take ψ ∈ C_{0}^{∞}(U, H𝔾), |ψ(x)|_{x} ≤ 1 for all x ∈ U. We have:

∫_{U} h_{ϵ}div_{𝔾} ψ dx = ∫_{U∩E_{ϵ}} div_{𝔾} ψ dx = ∫_{U∩E_{ϵ}} div (^{t}Cψ) dx
= ∫_{U∩∂E_{ϵ}}⟨ψ, Cn⟩_{ℝ^{m}} dH^{n−1} ≤ ∫_{U∩∂E_{ϵ}} |Cn|_{ℝ^{m}} dH^{n−1}.

On the other hand, a parameterization of U ∩ ∂E_{ϵ} is given by

Φ_{ϵ} = Φ_{ϵ}(ξ) := γ(ϕ_{ϵ}(ξ), ξ),

with ξ ∈ I_{δ}. Indeed, let us prove that Φ_{ϵ} is injective: suppose γ(ϕ_{ϵ}(ξ), ξ) = γ(ϕ_{ϵ}(ξ′), ξ′); since the first component of γ(t, ξ) is t, this implies that ϕ_{ϵ}(ξ) = ϕ_{ϵ}(ξ′), and then that ξ = ξ′, by the uniqueness of the solution of the Cauchy problem.

Thus, from standard area formula,

∫_{U} h_{ϵ}div_{𝔾} ψ dx ≤ ∫_{I_{δ}} |C(Φ_{ϵ}) \frac{∂Φ_{ϵ}}{∂ξ_{2}} ∧ ⋯ ∧ \frac{∂Φ_{ϵ}}{∂ξ_{n}} | dξ. (31)

Notice now that the j-th component of C(Φ_{ϵ}(ξ)) \frac{∂Φ_{ϵ}}{∂ξ_{2}} (ξ) ∧ ⋯ ∧ \frac{∂Φ_{ϵ}}{∂ξ_{n}} (ξ) is

⟨X_{j}(Φ_{ϵ}(ξ)), \frac{∂Φ_{ϵ}}{∂ξ_{2}} (ξ) ∧ ⋯ ∧ \frac{∂Φ_{ϵ}}{∂ξ_{n}} ⟩(ξ).

To achieve our proof, the following identity will be crucial:

|⟨X_{k}, \frac{∂Φ_{ϵ}}{∂ξ_{2}} ∧ ⋯ ∧ \frac{∂Φ_{ϵ}}{∂ξ_{n}} ⟩| = |\frac{X_{k}f_{ϵ}}{X_{1}f_{ϵ}} | , (32)

for k = 1, …, m, where \frac{∂Φ_{ϵ}}{∂ξ_{j}} = \frac{∂Φ_{ϵ}}{∂ξ_{j}} (ξ), X_{k} = X_{k}(Φ_{ϵ}(ξ)), X_{1}f_{ϵ} = X_{1}f_{ϵ}(Φ_{ϵ}(ξ)) > 0 and X_{k}f_{ϵ} = X_{k}f_{ϵ}(Φ_{ϵ}(ξ)), ξ ∈ I_{δ}.

To prove (32), let us first notice that, if we write γ = (γ_{1}, …, γ_{n}), then, because of the structure of the a_{i}^{j}’s, we get

γ_{ℓ}(t, ξ) = ξ_{ℓ} + ∫_{0}^{t} g_{ℓ}(s, ξ_{2}, …, ξ_{ℓ−1}) ds,

for suitable functions g_{ℓ} : ℝ^{ℓ} → ℝ, ℓ = 2, …, n. Hence

\frac{∂γ_{ℓ}}{∂ξ_{j}} = 0 if j > ℓ, \frac{∂γ_{ℓ}}{∂ξ_{ℓ}} = 1. (33)

Moreover, if we write ϕ_{ϵ} = ϕ_{ϵ}(ξ), Φ_{ϵ} = (Φ_{ϵ,1}, …, Φ_{ϵ,n}),

\frac{∂}{∂ξ_{j}} Φ_{ϵ,ℓ}(ξ) = \frac{∂}{∂ξ_{j}} γ_{ℓ}(ϕ_{ϵ}, ξ) = \frac{∂γ_{ℓ}}{∂ξ_{j}} (ϕ_{ϵ}, ξ) + γ′_{ℓ}(ϕ_{ϵ}, ξ)\frac{∂ϕ_{ϵ}}{∂ξ_{j}} = \frac{∂γ_{ℓ}}{∂ξ_{j}} (ϕ_{ϵ}, ξ) + a_{ℓ}^{1} \frac{∂ϕ_{ϵ}}{∂ξ_{j}} , (34)

where a_{ℓ}^{1} = a_{ℓ}^{1}(γ(ϕ_{ϵ}, ξ)). Differentiating now (30) with respect to ξ_{ℓ}, we get

0 ≡ ∑_{ℓ=1}^{n} \frac{∂f_{ϵ}}{∂x_{ℓ}} (\frac{∂γ_{ℓ}}{∂ξ_{j}} + a_{ℓ}^{1} \frac{∂ϕ_{ϵ}}{∂ξ_{j}} ) , (35)

and hence

∑_{ℓ=1}^{n} \frac{∂f_{ϵ}}{∂x_{ℓ}} \frac{∂γ_{ℓ}}{∂ξ_{j}} = −(X_{1}f_{ϵ}) · \frac{∂ϕ_{ϵ}}{∂ξ_{j}} , j = 2, …, n, (36)

i.e.

∑_{ℓ=1}^{n} \frac{∂f_{ϵ}}{∂x_{ℓ}} ∇_{ξ}γ_{ℓ} = −(X_{1}f_{ϵ}) · ∇ϕ_{ϵ}. (37)

Suppose now 2 ≤ k ≤ m; to prove (32), we want to calculate

D := |⟨X_{k}, \frac{∂Φ_{ϵ}}{∂ξ_{2}} ∧ ⋯ ∧ \frac{∂Φ_{ϵ}}{∂ξ_{n}} ⟩| = |det [X_{k}, \frac{∂Φ_{ϵ}}{∂ξ_{2}} , …, \frac{∂Φ_{ϵ}}{∂ξ_{n}} ]|
= |det [(a_{1}^{k}, ∇_{ξ}γ_{1}(ϕ_{ϵ}, ξ)), …, (a_{n}^{k}, ∇_{ξ}γ_{n}(ϕ_{ϵ}, ξ))]| ,

where, in the first determinant, X_{k}, \frac{∂Φ_{ϵ}}{∂ξ_{2}} , …, \frac{∂Φ_{ϵ}}{∂ξ_{n}} are columns, whereas in the last determinant we must see the n-dimensional vectors (a_{1}^{k}, ∇_{ξ}γ_{1}(ϕ_{ϵ}, ξ)), …, (a_{n}^{k}, ∇_{ξ}γ_{n}(ϕ_{ϵ}, ξ)) as the rows of a matrix. By (33), γ_{1}(ϕ_{ϵ}, ξ) ≡ ϕ_{ϵ}, so that ∇_{ξ}γ_{1}(ϕ_{ϵ}, ξ) = ∇_{ξ}ϕ_{ϵ}, and hence, keeping in mind that a_{1}^{k} = 0 since k > 1, by (34) and chain rule, we get

D = |det [(0, ∇_{ξ}ϕ_{ϵ}), (a_{2}^{k}, (∇_{ξ}γ_{2}) + a_{2}^{1}∇_{ξ}ϕ_{ϵ}), …, (a_{n}^{k}, (∇_{ξ}γ_{n}) + a_{n}^{1}∇_{ξ}ϕ_{ϵ})]| , (38)

where (∇_{ξ}γ_{j}) stands for (∇_{ξ}γ_{j})(ϕ_{ϵ}(ξ), ξ).

By standard properties of the determinant, i.e. by subtracting a_{i}^{1} times the first row from the i-th row, it follows that

D = |det [(0, ∇_{ξ}ϕ_{ϵ}), (a_{2}^{k}, ∇_{ξ}γ_{2}), …, (a_{n}^{k}, ∇_{ξ}γ_{n})]| .

Subtract now the first column from the k-th column; we get

D = |det [(0, ∇_{ξ}ϕ_{ϵ}), (a_{2}^{k}, ∇_{ξ}γ_{2} − a_{2}^{k}e_{k}^{n−1}), …, (a_{n}^{k}, ∇_{ξ}γ_{n} − a_{n}^{k}e_{k}^{n−1})]| ,

where {e_{2}^{n−1}, …, e_{n}^{n−1}} is the canonical orthonormal basis of ℝ^{n−1}, so that e_{2}^{n−1} = (1, 0, …, 0), and so on.

Multiply now the i-th row by \frac{∂f_{ϵ}}{∂x_{i}} (X_{1}f_{ϵ})^{−1}, sum up when i runs from 2 to n, and then add to

the first row. By (37), denoting by ∗ a suitable real number that will turn out to be irrelevant, we get

D = |det [(∗, −\frac{X_{k}f_{ϵ}}{X_{1}f_{ϵ}} e_{k}^{n−1}), (a_{2}^{k}, ∇_{ξ}γ_{2} − a_{2}^{k}e_{k}^{n−1}), …, (a_{n}^{k}, ∇_{ξ}γ_{n} − a_{n}^{k}e_{k}^{n−1})]| .

Notice now that γ_{k}(t, ξ) = ξ_{k}, since 2 ≤ k ≤ m and X_{1} has zero k-component so that ∇_{ξ}γ_{k} = e_{k}^{n−1}, and that a_{k}^{k} = 1; hence (a_{k}^{k}, ∇_{ξ}γ_{k} − a_{k}^{k}e_{k}^{n−1}) = (1, 0). Thus, if we develop the determinant with respect to the first column, all but the k-th cofactor vanish since they contain a zero row. Again since a_{k}^{k} = 1 we obtain

D = |det [−\frac{X_{k}f_{ϵ}}{X_{1}f_{ϵ}} e_{k}^{n−1}, ∇_{ξ}γ_{2} − a_{2}^{k}e_{k}^{n−1}, …, ∇_{ξ}γ_{k−1} − a_{k−1}^{k}e_{k}^{n−1},

∇_{ξ}γ_{k+1} − a_{k−1}^{k}e_{k}^{n+1}, …, ∇_{ξ}γ_{n} − a_{n}^{k}e_{k}^{n−1}]| .

Remember now that a_{i}^{k} = 0 if i ≤ m and i ≠ k, and that ∇_{ξ}γ_{i} = e_{i}^{n−1} for 2 ≤ i ≤ m. We have

D = |\frac{X_{k}f_{ϵ}}{X_{1}f_{ϵ}} | |det[e_{k}^{n−1}, e_{2}^{n−1}, …, e_{k−1}^{n−1}, e_{k+1}^{n−1}, …, e_{m}^{n−1}, ∇_{ξ}γ_{m+1} − a_{m+1}^{k}e_{k}^{n−1}, …]|
= |\frac{X_{k}f_{ϵ}}{X_{1}f_{ϵ}} | |det[e_{2}^{n−1}, …, e_{m}^{n−1}, ∇_{ξ}γ_{m+1} − a_{m+1}^{k}e_{k}^{n−1}, …]| .

By (33), keeping in mind that k ≤ m and that hence all components of e_{k}^{n−1} vanish when their index is greater than m, the matrix [e_{2}^{n−1}, …, e_{m}^{n−1}, ∇_{ξ}γ_{m+1} − a_{m+1}^{k}e_{k}^{n−1}, …] is a lower triangular matrix, with all diagonal terms equal to 1. Hence

D = |\frac{X_{k}f_{ϵ}}{X_{1}f_{ϵ}} | .

Consider now the case k = 1. Arguing as in the proof of (38), in this case we have to evaluate

D := |⟨X_{1}, \frac{∂Φ_{ϵ}}{∂ξ_{2}} ∧ ⋯ ∧ \frac{∂Φ_{ϵ}}{∂ξ_{n}} ⟩| = |det [X_{1}, \frac{∂Φ_{ϵ}}{∂ξ_{2}} , …, \frac{∂Φ_{ϵ}}{∂ξ_{n}} ]|
= |det [(a_{1}^{1}, ∇_{ξ}γ_{1}(ϕ_{ϵ}, ξ)), …, (a_{n}^{1}, ∇_{ξ}γ_{n}(ϕ_{ϵ}, ξ))]|
= |det [(1, ∇_{ξ}ϕ_{ϵ}), (a_{2}^{1}, (∇_{ξ}γ_{2}) + a_{2}^{1}∇_{ξ}ϕ_{ϵ}), …, (a_{n}^{1}, (∇_{ξ}γ_{n}) + a_{n}^{1}∇_{ξ}ϕ_{ϵ})]| ,

since a_{1}^{1} = 1. Again as above, think of the vectors

(1, ∇_{ξ}ϕ_{ϵ}), (a_{2}^{1}, (∇_{ξ}γ_{2}) + a_{2}^{1}∇_{ξ}ϕ_{ϵ}), …, (a_{n}^{1}, (∇_{ξ}γ_{n}) + a_{n}^{1}∇_{ξ}ϕ_{ϵ})

as of the row of a matrix, and subtract from the i-th row (i ≥ 2) the first row multiplied by a_{i}^{1}. In such a way, the first column turns out to have all zero entries but the first one which is 1, and we get

D = |det [∇_{ξ}γ_{2}, …, ∇_{ξ}γ_{n}]| = 1,

since the above matrix is a lower triangular matrix with all entries in the principal diagonal equal 1, by (33).

Thus (32) is proved, and hence it follows from (31) that

∫_{U} h_{ϵ}div_{𝔾} ψ dx ≤ ∫_{I_{δ}} |C(Φ_{ϵ}) \frac{∂Φ_{ϵ}}{∂ξ_{2}} ∧ ⋯ ∧ \frac{∂Φ_{ϵ}}{∂ξ_{n}} | dξ

(39)

≤ ∫_{I_{δ}} ( ∑_{j} |X_{j}f_{ϵ}(Φ_{ϵ})|^{2})^{1/2}|X_{1}f_{ϵ}(Φ_{ϵ})|^{−1}dξ ≤ C |I_{δ}|,

since the X_{j}’s and X_{1}^{−1} are equibounded, by (27) and (29).

Thus, the functions h_{ϵ} have equibounded variations, and hence, as we pointed out above, 1_{E} ∈ BV_{𝔾}(U). Step 4. The area formula for the 𝔾-perimeter of E in U. We want now to prove the explicit formula for |∂E|_{𝔾}(U) given by (vi). Again as above, if ψ ∈ C_{0}^{∞}(U, H𝔾), |ψ(x)|_{x} ≤ 1, by dominate convergence theorem, we have:

∫_{U} 1_{E} div_{𝔾}ψ dx = lim_{ϵ→0} ∫_{U} h_{ϵ}div_{𝔾}ψ dx

= lim_{ϵ→0} ∫_{I_{δ}}⟨ψ(Φ_{ϵ}(ξ)), C(Φ_{ϵ}(ξ)) \frac{∂Φ_{ϵ}}{∂ξ_{2}} ∧ ⋯ ∧ \frac{∂Φ_{ϵ}}{∂τ} ⟩_{ℝ^{m}}dξ (40)

= ∫_{I_{δ}}⟨ψ(Φ(ξ)), ∇_{𝔾}f(Φ(ξ))⟩_{Φ(ξ)}|X_{1}f(Φ(ξ))|^{−1}dξ,

again by dominate convergence theorem. Indeed, we showed above that Φ_{ϵ}(ξ) → Φ(ξ) pointwise for ξ ∈ I_{δ}, X_{j}f_{ϵ} → X_{j}f uniformly on Ũ as ϵ → 0, and Φ_{ϵ}(ξ) ∈ Ũ when ξ ∈ I_{δ}, since, as we pointed out above, |ϕ_{ϵ}(ξ)| ≤ h uniformly with respect to ϵ ∈ (0, 1).

Taking now the supremum with respect to ψ in (40), the proof of (vi) is complete.

Let us prove now (iii). Since 1_{E} ∈ BV_{𝔾}(U), then, by [20], p. 880, ∇_{𝔾}1_{E} = (X_{1}, …, X_{m})1_{E} is a Radon measure. Moreover, for all ψ ∈ C_{0}^{0}(U, H𝔾), arguing as above, we get

−⟨∇_{𝔾}1_{E}, ψ⟩_{ℝ^{m}} = ∫_{I}⟨ψ(Φ(ξ)), ∇_{𝔾}f(Φ(ξ))⟩_{Φ(ξ)}|X_{1}f(Φ(ξ))|^{−1}dξ.

Taking the supremum on all ψ ∈ C_{0}^{0}(U, H𝔾), |ψ(x)|_{x} ≤ 1, supp ψ ⊆ O ⊆ U, where O is an open set, we obtain

|∂E|_{𝔾}(O) = ∫_{Φ^{−1}(O)} \frac{|∇_{𝔾}f|}{X_{1}f} ∘ Φ dξ. (41)

Thus, with the notations of [37], Definition 1.17, |∂E|_{𝔾} is the image of the measure

dμ = \frac{|∇_{𝔾}f|}{X_{1}f} ∘ Φ dξ

in ℝ^{n−1} under the map Φ, given that equality (41) still holds for any Borel set O because both measures are Radon measures ([37], Theorem 1.18). Hence, by [37], Theorem 1.19, for all ψ ∈ C_{0}^{0}(U, H𝔾) we have

∫_{U}⟨ψ(x), ν_{E}(x)⟩_{x}d|∂E|_{𝔾} = ∫_{I}⟨ψ ∘ Φ, ν_{E} ∘ Φ⟩_{Φ} \frac{|∇_{𝔾}f|}{X_{1}f} ∘ Φ dξ.

We notice that Theorem 1.19 in [37] would require ⟨ψ(x), ν_{E}(x)⟩_{x} to be a Borel function. Nevertheless, the result is still true since ⟨ψ(x), ν_{E}(x)⟩_{x} is |∂E|_{𝔾}-measurable, thanks to [17], 2.3.6. Thus, keeping in mind that

∫_{U}⟨ψ(x), ν_{E}(x)⟩_{x}d|∂E|_{𝔾} = − ∫_{U} 1_{E} div_{𝔾} ψ dx,

by (40) and by the arbitrariness of the choice of ψ, we obtain

ν_{E} ∘ Φ = −\frac{∇_{𝔾}f}{|∇_{𝔾}f|} ∘ Φ

a.e. in I_{δ} with respect to Lebesgue measure in ℝ^{n−1}, and hence

ν_{E} = −\frac{∇_{𝔾}f}{|∇_{𝔾}f|}

|∂E|_{𝔾}-a.e. on S ∩ U, because of (41). Thus (iii) is proved.

Remark 2.2. If the assumption X_{1}f(0) > 0 is replaced by X_{j}f(0) > 0 for some j = 2, …, m, then we can always reduce ourselves to the case previously considered provided we renumber the first m variables. Hence the above Theorem still holds under this new assumption, provided we replace everywhere the X_{1} by X_{j}.

It is clear from the proof of Theorem 2.1 that in fact the group structure does not play any role in the result but for the structure of the vector fields, even if some statement throughout the proof itself must be slightly modified when the fiber bundle structure fails to exist.

Thus the following definition is quite natural.

Definition 2.3. Let X = {X_{1}, …, X_{m}} be a family of Lipschitz continuous vector fields. We say that X is a family of Carnot type if

X_{j}(x) = ∂_{j} + ∑_{i=m+1}^{n} a_{i}^{j}(x)∂_{i}, X_{j}(0) = ∂_{j}, j = 1, ..., m, (42)

with a_{i}^{j}(x) = a_{i}^{j}(x_{1}, ..., x_{i−1}), j = 1, …, m.

Then the previous Theorem can be stated as follows.

Theorem 2.4. Let X = {X_{1}, …, X_{m}} be a family of Lipschitz continuous vector fields of Carnot type in ℝ^{n}, let Ω be an open subset of ℝ^{n}, 0 ∈ Ω, and let f be a continuous realvalued function in Ω such that X_{1}f, …, X_{m}f are continuous functions. Then the conclusions of Theorem 2.1 hold.

3. Surface Measures on Hypersurfaces in 𝔾.

3.1. 𝔾-regular hypersurfaces. We want to study 𝔾-regular hypersurfaces from an intrinsic point of view. In particular we want to compare the perimeter measure, on a 𝔾-regular hypersurface S, and the intrinsic (Q − 1)-Hausdorff measure of S. Observe that it makes sense to speak of the perimeter measure of S given that S is locally the boundary of a finite 𝔾-perimeter set (as proved in Theorem 2.1).

We begin with an easy proposition, more or less explicitely contained in Theorem 2.1, showing that S is locally the homeomorphic image of a (vertical) hyperplane in ℝ^{n}. In particular this fact implies that the topological dimension of S is n − 1.

Proposition 3.1. Assume that S is a 𝔾-regular hypersurface in a Carnot group 𝔾. Then for any y ∈ S there exists an open, connected neighborhood U_{y} of y such that

(i) there are σ > 0, δ > 0 and an homeomorphism F : (−σ, σ) × Int I_{δ} → U_{y}, such that F (0, 0) = y and

F ({0} × Int I_{δ}) = S ∩ U_{y};

(ii) the set E_{1} := F ((−σ, 0) × Int I_{δ}) has finite perimeter in U_{y} and

S ∩ U_{y} = ∂E_{1} ∩ U_{y}.

Proof. By definition of 𝔾-regular hypersurface, if r > 0 is sufficiently small, there are U_{c}(y, r) and f ∈ C_{𝔾}^{1}(U_{c}(y, r)) such that

S ∩ U_{c}(y, r) = {x ∈ U_{c}(y, r) : f(x) = 0}

and Xf ≠ 0 in U_{c}(y, r). Without loss of generality we may assume X_{1}f > 0 in U_{c}(y, r). We keep the notations of Theorem 2.1. For fixed σ > 0 and δ > 0 to be chosen sufficiently small, we consider the map F = F (t, ξ) : (−σ, σ) × IntI_{δ} → ℝ^{n} given by

F (t, ξ) = y · γ(ϕ(ξ) − t, ξ) := y · exp((ϕ(ξ) − t)X_{1})(0, ξ).

It is clear that F is an homeomorphism from (−σ, σ) × IntI_{δ} → U_{y} := F ((−σ, σ) × IntI_{δ}) ⊂ U_{c}(y, r). Indeed F is the composition of the homeomorphism (t, ξ) → (ϕ(ξ) − t, ξ) and of the diffeomorphism (s, η) → y · γ(s, η) := y · exp (sX_{1}) (0, η).

By Theorem 2.1, (iv), locally S = {γ(ϕ(ξ), ξ)}, so that F ({0} × Int I_{δ}) = S ∩ U_{y}, and

U_{y} \ S = F ((−σ, 0) × Int I_{δ}) ∪ F ((0, σ) × Int I_{δ}) := E_{1} ∪ E_{2}.

Each E_{i} is a connected open set. By (28), the map s ↦ f(γ(s, ξ)) is strictly increasing and vanishes at s = ϕ(ξ). Since ϕ(ξ) − t > ϕ(ξ) when t < 0, we conclude that f > 0 in E_{1} and that f < 0 in E_{2}. Hence E_{1} = {x ∈ U_{y} : f(x) > 0} and from (i) of Theorem 2.1 the thesis follows.

Remark 3.2. Notice that Proposition 3.1 does not follow from an Euclidean local invertibility theorem, because, in general, the Euclidean C^{1}-regularity of the local chart F fails to hold in Carnot groups. Indeed, if F were continuously differentiable in the usual sense, then also the map ϕ would be continuously differentiable, since the first component of F (ξ, t) is precisely ϕ(ξ) − t. But in [22], Example 2 after Theorem 6.5, in the setting of the Heisenberg group 𝔾 = ℍ^{1}, the authors provided an example of a 𝔾-regular hypersurface such that the parameterization ϕ is not even Euclidean Lipschitz continuous.

Our next Theorem is a mild regularity result; in it we observe that 𝔾-regular hypersurfaces do not have cusps or spikes if they are studied with respect to the intrinsic cc-distance, while they can be very irregular as Euclidean submanifolds. To make precise the former statement we recall the notion of essential boundary (or measure theoretic boundary) ∂_{∗}F of a set F ⊂ 𝔾

(43) ∂_{∗}F := \{x ∈ 𝔾 : lim sup_{r→0^{+}} min \{\frac{L^{n}(F ∩ U_{c}(x, r))}{L^{n}(U_{c}(x, r))}, \frac{L^{n}(F^{c} ∩ U_{c}(x, r))}{L^{n}(U_{c}(x, r))}\} > 0\},

where we remind that U_{c}(x, r) denotes the open cc-ball. Notice that the definition above makes sense in any metric measure space and that the essential boundary does not change if, in Definition (43), the distance d_{c} is substituted by an equivalent distance d.

Theorem 3.3. Let Ω ⊂ 𝔾 be a fixed open set, and let E be such that ∂E ∩ Ω = S ∩ Ω, where S is a 𝔾-regular hypersurface. Then

∂E ∩ Ω = ∂_{∗}E ∩ Ω. (44)

Proof. Clearly, to prove (44), we need only to show that ∂E ∩ Ω ⊆ ∂_{∗}E ∩ Ω. Fix y ∈ ∂E ∩ Ω. By definition of 𝔾-regular surface and from (i) of Theorem 2.1, there exists an open ball U_{c}(r) := U_{c}(y, r) and f ∈ C_{𝔾}^{1}(U_{c}(r)) such that

∂E ∩ U_{c}(r) = {x ∈ U_{c}(r) : f(x) = 0};
|∇_{𝔾}f|_{ℝ^{m}} > 0 in U_{c}(r);
E ∩ U_{c}(r) = {x ∈ U_{c}(r) : f(x) < 0}.

If r > 0 we define the dilated sets

E_{y,r} := y · δ_{r}(y^{−1} · E) = {x ∈ 𝔾 : y · δ_{1/r}(y^{−1} · x) ∈ E},

then we have

1_{E_{y,r}} → 1_{S_{𝔾}^{−}(y,ν_{E}(y))} L^{n}-a.e. in 𝔾 as r → 0^{+}. (45)

Indeed, for all p ∈ S_{𝔾}^{−}(y, ν_{E}(y)), (iii) of Theorem 2.1 implies that

∑_{i=1}^{m} X_{i}f(y)(p_{i} − y_{i}) < 0.

and, recalling that f(y) = 0, by Proposition 1.7, we have f(y · δ_{s}(y^{−1} · p)) = s ∑_{i=1}^{m} X_{i}f(y)(p_{i} − y_{i}) + o(s) < 0 for s → 0 , hence y · δ_{s}(y^{−1} · p) ∈ E and in turn, p ∈ E_{y,s}, for s small. Reversing the inequalities, if p ∈ S_{𝔾}^{+}(y, ν_{E}(y)) we get p ∈ E_{y,s}^{c} for s small. Since L^{n}(𝔾 \ (S_{𝔾}^{+}(y, v) ∪ S_{𝔾}^{−}(y, v))) = 0 we have proved (45).

Now put S^{±}(y) := S_{𝔾}^{±}(y, ν_{E}(y)), then, from (45) it follows that

lim inf_{r→0^{+}} min \{\frac{L^{n}(E ∩ U_{c}(r))}{L^{n}(U_{c}(r))}, \frac{L^{n}(E^{c} ∩ U_{c}(r))}{L^{n}(U_{c}(r))}\}

≥ min \{\frac{L^{n}(U_{c}(1) ∩ S^{+}(y))}{L^{n}(U_{c}(1))}, \frac{L^{n}(U_{c}(1) ∩ S^{−}(y))}{L^{n}(U_{c}(1))}\} .

(46)

Indeed, from (45) it follows that 1_{E_{r}} → 1_{S^{−}(y)} in L_{loc}^{1}(𝔾), whence

L^{n}(E_{y,r} ∩ U_{c}(1)) → L^{n}(S^{−}(y) ∩ U_{c}(1)) as r → 0^{+}. (47)

On the other hand

L^{n}(E_{y,r} ∩ U_{c}(1)) = \frac{L^{n}(E ∩ U_{c}(r))}{r^{Q}} = L^{n}(U_{c}(1))\frac{L^{n}(E ∩ U_{c}(r))}{L^{n}(U_{c}(r))},

and hence from (47) we get that

lim_{r→0^{+}} \frac{L^{n}(E ∩ U_{c}(r))}{L^{n}(U_{c}(r))} = \frac{L^{n}(S^{−}(y) ∩ U_{c}(1))}{L^{n}(U_{c}(1))}.

Analogously, since ∂E ∩ Ω = ∂E^{c} ∩ Ω and ν_{E^{c}}(0) = −ν_{E}(0), we get also

lim_{r→0^{+}} \frac{L^{n}(E^{c} ∩ U_{c}(r))}{L^{n}(U_{c}(r))} = \frac{L^{n}(S^{+}(y) ∩ U_{c}(1))}{L^{n}(U_{c}(1))}.

Thus (46) holds.

Now we notice explicitly that

min \{\frac{L^{n}(U_{c}(1) ∩ S^{+}(y))}{L^{n}(U_{c}(1))}, \frac{L^{n}(U_{c}(1) ∩ S^{−}(y))}{L^{n}(U_{c}(1))}\} > 0. (48)

Indeed, both U_{c}(0, 1) ∩ IntS_{𝔾}^{+}(y, ν_{E}(y)) and U_{c}(0, 1) ∩ IntS_{𝔾}^{−}(y, ν_{E}(y)) are open and not empty, since the ball U_{c}(0, 1) is symmetric with respect to the group inversion. This concludes the proof of (44).

Our next results are about the relations between perimeter and intrinsic Hausdorff measures. In the setting of the Heisenberg group, in [10] it is proved that the perimeter of an Euclidean C^{1,1}-hypersurface is equivalent to its (Q−1)-dimensional intrinsic Hausdorff measure, whereas in [22] it is proved that on the boundary of sets of finite intrinsic perimeter the (Q−1)-dimensional intrinsic spherical Hausdorff measure coincide – after a suitable normalization – with the perimeter measure. In the setting of general Carnot groups the problem is essentially open.

Our results here are more clearly stated in a situation slightly more general than the one considered up to now. From now on d will be a metric on 𝔾, translation invariant, homogeneous and comparable with d_{c}. That is we assume that, for all x, y, z ∈ 𝔾 and λ > 0

d(x · y, x · z) = d(y, z) and d(δ_{λ}y, δ_{λ}z) = λd(y, z), (49)

and that there is c_{d} > 1 such that

\frac{1}{c_{d}} d(y, z) ≤ d_{c}(y, z) ≤ c_{d}d(y, z), for all y, z ∈ 𝔾. (50)

We indicate as U_{d}(x, r), S_{d}^{Q−1} and H_{d}^{Q−1} respectively the open balls and the Hausdorff measures with respect to the new distance d, keeeping the notation U_{c}(x, r), S_{𝔾}^{Q−1} and H_{𝔾}^{Q−1} when working with the cc-distance.

We need the following differentiation Theorem whose proof can be found in Federer’s book (see [17], Theorems 2.10.17 and 2.10.18). Notice that Federer states this result in a much more general context, i.e. for regular measures in metric spaces.

Theorem 3.4. (Federer’s differentiation Theorem) If

(51)  lim_{r→0} \frac{|∂E|_{𝔾}(U_{d}(x, r))}{2^{−Q+1}(diam U_{d}(x, r))^{Q−1}} = 𝔰(x),  for |∂E|_{𝔾}-a.e. x ∈ 𝔾,

then

|∂E|_{𝔾} = 𝔰(x)S_{d}^{Q−1} ∂E (52)

Theorem 3.5. If d is a distance on 𝔾 satisfying (49) and (50), and if 𝔰_{d} : H𝔾_{0} \ {0} → ℝ, is the 0-homogeneous function defined as

𝔰_{d}(v) := L^{n−1} (U_{d}(0, 1) ∩ Π(0, v)) ,

then

\begin{aligned}|∂E|_{𝔾} Ω & = 𝔰_{d} ∘ ν_{E} S_{𝔾}^{Q−1} (S ∩ Ω) \\ & = L^{n−1} (U_{d}(0, 1) ∩ T_{𝔾}^{g}S(x)) S_{𝔾}^{Q−1} (S ∩ Ω). \end{aligned} (53)

Moreover, there is a constant α_{d} > 1, depending only on the distance d, such that

0 < \frac{1}{α_{d}} ≤ 𝔰_{d}(v) ≤ α_{d} < ∞.

Remark 3.6. If the distance d under consideration is invariant with respect to rotations of H𝔾_{0} ≃ ℝ^{m}, then the function 𝔰_{d} is constant and, with an appropriate choice of the normalization constant in the definition of the Hausdorff measure, (53) takes the particularly neat form

|∂E|_{𝔾} = S_{d}^{Q−1} S. (54)

We do not know how large is the class of groups whose cc-distance enjoys this property. It certainly comprises the Heisenberg groups. For the groups in this class we have

|∂E|_{𝔾} = S_{c}^{Q−1} S. (55)

Nevertheless, even if d_{c} were not rotationally invariant, it always exists another true metric invariant, homogeneous, comparable with d_{c} that is also invariant by rotations of H𝔾_{0} (for an example see the Appendix in [23]). If one computes the Hausdorff measure with respect to it, then (54) holds.

Proof of Theorem 3.5. Observe that under our hypotheses on the distance d it follows that diam U_{d}(x, r) = 2r, then, from Theorem 3.4, to prove (53) it is enough to show that

lim_{r→0} r^{1−Q}|∂E|_{𝔾}(U_{c}(x, r)) = L^{n−1} (U_{c}(0, 1) ∩ T_{𝔾}^{g}S(x)) . (56)

We are going to prove (56) by a direct computation exploiting formula (vi) of Theorem 2.1. Notice also that, without loss of generality, we can assume that x = 0.

From the homogeneity of the perimeter (see (21))

r^{1−Q}|∂E|_{𝔾}(U_{c}(0, r)) = |∂E_{0,r}|_{𝔾}(U_{c}(0, 1)). (57)

For all r > 0 that are so small that ∂E ∩ U_{c}(0, r) = {x : f(x) = 0}, with f ∈ C_{𝔾}^{1}(U_{c}(0, r)) and X_{1}f(y) > c_{1} > 0 in U_{c}(0, r), we have

∂E_{0,r} ∩ U_{c}(0, 1) = \{x ∈ U_{c}(0, 1) : \frac{1}{r}f ∘ δ_{r}(x) = 0\}.

Notice that f_{1/r} := \frac{1}{r}f ∘ δ_{r} ∈ C_{𝔾}^{1}(U_{c}(0, 1)) and that X_{1}f_{1/r}(y) > c_{1} > 0 in U_{c}(0, 1).

Given that, as shown in Theorem 2.1, the equation f(x) = 0 defines implicitely a function ϕ : I_{δ} → ℝ, it follows that the equation f_{1/r}(x) = 0 defines implicitely the function ϕ_{1/r} : I_{δ/r} → ℝ acting as ϕ_{1/r}(ξ) = \frac{1}{r}ϕ ∘ δ_{r}(ξ) and a map Φ_{1/r} : I_{δ/r} → 𝔾 acting as

Φ_{1/r}(ξ) = exp (ϕ_{1/r}X_{1}) (0, ξ).

Hence by formula (vi) we get

|∂E_{0,r}|_{𝔾}(U_{c}(0, 1)) = ∫_{Φ_{1/r}^{−1}(U_{c}(0,1))} \frac{\sqrt{∑_{j=1}^{m}(X_{j}f_{1/r})^{2}}}{X_{1}f_{1/r}} ∘ Φ_{1/r} dL^{n−1}. (58)

In order to pass to the limit as r → 0 in (58), notice that, from Taylor’s formula (24),

\begin{aligned}f_{1/r}(x) & → f_{∞}(x) := ⟨Xf(0), π_{0}x⟩_{0} \\ X_{j}f_{1/r}(x) & → X_{j}f_{∞}(x) = X_{j}f(0) for j = 1, …, m, \end{aligned} (59)

as r → 0, uniformly on U_{d}(0, 1). Hence, if we put ν = (ν_{1}, …, ν_{m}) := ν_{E}(0) = \frac{Xf(0)}{|Xf(0)|_{0}} , then from the assumption X_{1}f(y) > c_{1} > 0 it follows ν_{1} > 0 and we get

\frac{\sqrt{∑_{j=1}^{m}(X_{j}f_{1/r})^{2}}}{X_{1}f_{1/r}} → \frac{1}{ν_{1}} (60)

as r → 0, uniformly on U_{d}(0, 1). Finally, for all ξ ∈ Π(0, e_{1}) ≡ ℝ^{n−1}, since f_{∞}(Φ_{∞}(0, ξ)) = 0, we get

lim_{r→0} ϕ_{1/r}(ξ) = ϕ_{∞}(ξ) = −\frac{∑_{j=2}^{m} ν_{j}ξ_{j}}{ν_{1}}
lim_{r→0} Φ_{1/r}(ξ) = Φ_{∞}(ξ) = exp (ϕ_{∞}(ξ)X_{1}) (0, ξ).

(61)

Notice that Φ_{∞} is a map from Π(0, e_{1}) to T_{𝔾}^{g}S(0) ⊂ 𝔾 and the norm of its Jacobian is precisely \frac{1}{ν_{1}} . Indeed, writing explicitely Φ_{∞}(ξ) = (Φ_{∞,1}(ξ), …, Φ_{∞,n}(ξ)), we have

Φ_{∞}(ξ) := exp (ϕ_{∞}(ξ)X_{1}) (0, ξ) = (−\frac{∑_{j=2}^{m} ν_{j}ξ_{j}}{ν_{1}} , ξ_{2}, …, ξ_{m}, Φ_{∞,m+1}(ξ), …, Φ_{∞,n}(ξ)) (62)

and from(19) we know that the last n − m components Φ_{∞,j} do not depend on all the variables ξ_{j}, but, precisely, if h_{l−1} < j ≤ h_{l}, then Φ_{∞,j}(ξ) = Φ_{∞,j}(ξ_{2}, …, ξ_{h_{l−1}}). Hence the Jacobian matrix J_{Φ_{∞}} is the n × (n − 1) matrix

(63) J_{Φ_{∞}}(ξ) = [\begin{matrix}−\frac{ν_{2}}{ν_{1}} & −\frac{ν_{3}}{ν_{1}} & … & −\frac{ν_{m}}{ν_{1}} & 0 & … & 0 \\ 1 & 0 & … & 0 & 0 & … & 0 \\ 0 & 1 & … & 0 & 0 & … & 0 \\ ⋮ & & ... & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & 1 & 0 & … & 0 \\ \frac{∂Φ_{∞,m+1}}{∂ξ_{2}} & \frac{∂Φ_{∞,m+1}}{∂ξ_{3}} & … & \frac{∂Φ_{∞,m+1}}{∂ξ_{m}} & 1 & … & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ & ... & ⋮ \\ \frac{∂Φ_{∞,n}}{∂ξ_{2}} & \frac{∂Φ_{∞,n}}{∂ξ_{3}} & … & \frac{∂Φ_{∞,n}}{∂ξ_{m}} & \frac{∂Φ_{∞,n}}{∂ξ_{m+1}} & … & 1 \end{matrix}] .

Then it is immediate to compute

∥J_{Φ_{∞}}(ξ)∥ := ∥\frac{∂Φ_{∞}}{∂ξ_{2}} ∧ ⋯ ∧ \frac{∂Φ_{∞}}{∂ξ_{n}} ∥ = \sqrt{1 + ∑_{i=2}^{m} (\frac{ν_{i}}{ν_{1}} )^{2}} = \frac{1}{ν_{1}} .

The preceeding computations yield

\begin{aligned}lim_{r→0} r^{1−Q}|∂E|_{𝔾}(U_{d}(0, r)) & =  ∫_{Φ_{∞}^{−1}(U_{d}(0,1)∩T_{𝔾}^{g}S(0))} \frac{1}{ν_{1}} dL^{n−1} \\ & =  ∫_{U_{d}(0,1)∩T_{𝔾}^{g}S(0)} dL^{n−1} \\ & = L^{n−1} (U_{d}(0, 1) ∩ T_{𝔾}^{g}S(0)) . \end{aligned} (64)

Corollary 3.7. If S is a 𝔾-regular hypersurface then the Hausdorff dimension of S, with respect to the cc-metric d_{c} or any other metric d comparable with it, is Q − 1.

3.2. Euclidean regular surfaces in 𝔾. Even if 𝔾-regular surfaces are the natural regular surfaces inside a group 𝔾, it may also be of some interest to study Euclidean C^{1} surfaces in ℝ^{n} = 𝔾.

Strictly speaking, an Euclidean regular surface S may be not 𝔾-regular. Indeed, even if S is locally the zero set of a function f ∈ C^{1}(ℝ^{n}) ⊂ C_{𝔾}^{1}(ℝ^{n}), the transversality condition ∇_{𝔾}f(x) ≠ 0, ∀x ∈ S may fail to hold. Points of S where the transversality condition fails are usually called characteristic points. More precisely the characteristic set C(S) of an Euclidean regular surface S inside a Carnot group is

C(S) = {x ∈ S : H𝔾_{x} ⊆ T_{S}(x)}

where T_{S}(x) denotes the Euclidean tangent space to S at x ∈ S.

It follows from the non integrability of the vector fields X_{1}, …, X_{m} (assumptions (15)), that C(S) is small inside S. There are many results in this line, under various regularity hypotheses on the surfaces and using different surface measures (Euclidean versus intrinsic) to estimate the smallness. Balogh (see [5]) was the first one to prove that, in the Heisenberg groups, the intrinsic (Q−1)-Hausdorff measure of the characteristic set of an Euclidean C^{1} surface vanishes. He obtained also many other related optimal estimates. Very recently, Balogh’s estimate has been extended to step 2 Carnot groups (see [23]). Precisely we have

H_{𝔾}^{Q−1}(C(S)) = 0 (65)

if S is an Euclidean C^{1} hypersurface in a Carnot group 𝔾 of step 2. Since a C^{1} hypersurface S in a Carnot group can be written as S = C(S) ∪ (S \ C(S)) and S \ C(S) is a 𝔾-regular hypersurface, then, by combining (53) and (65) we get

|∂E|_{𝔾} (Ω \ C(S)) = 𝔰S_{𝔾}^{Q−1} (S ∩ Ω).

On the other hand, the representation formula (14) for the 𝔾-perimeter of C^{1} manifolds yields

|∂E|_{𝔾} Ω = |Cn_{E}|H^{n−1} (S ∩ Ω) = 𝔰S_{𝔾}^{Q−1} (S ∩ Ω) = |∂E|_{𝔾} (Ω \ C(S)) ,

where n_{E} is the Euclidean outward normal vector field to S, since Cn_{E} = 0 on C(S). Thus the following corollary holds

Corollary 3.8. Let F be an open subset of 𝔾 with boundary S. If S is an Euclidean C^{1} hypersurface and 𝔾 is step 2 group, then

(66)  |∂F |_{𝔾} Ω = |Cn_{E}| H^{n−1} (S ∩ Ω)
= 𝔰 S_{𝔾}^{Q−1} (S ∩ Ω);

here 𝔰 is given by Theorem 3.3 and H^{n−1} is the (n−1)-dimensional Euclidean Hausdorff measure on 𝔾 ≡ ℝ^{n}.

Remark 3.9. Recently, in a private communication, Magnani informed us that he is able to extend (65) to a general Carnot group.

Remark 3.10. In order to make more evident the relationships among different integral rappresentations of the 𝔾-perimeter, we observe the following: with the notations of Proposition 1.4 and Theorem 2.1, assume that ∂F ∩ Ω is an Euclidean C^{1} hypersurface with no characteristic points, formula (14) yields

|∂F |_{𝔾} Ω = |Cn_{F} |_{ℝ^{m}} H^{n−1} (∂F ∩ Ω). (67)

On the other hand, if Ω ∩ ∂F ⊂ Φ(Int I_{δ}), (vi) of Theorem 2.1 reads as

|∂F |_{𝔾} Ω = Φ_{#} (\frac{|∇_{𝔾}f|}{X_{1}f} ∘ Φ L^{n−1}) . (68)

Notice also that

|Cn_{F} |_{ℝ^{m}} = \frac{|C∇f|_{ℝ^{m}}}{|∇f|_{ℝ^{n}}} = \frac{|∇_{𝔾}f|}{|∇f|_{ℝ^{n}}} ,

and that, by the Euclidean area formula and the non trivial computations in the proof of Theorem 2.1,

Φ_{#} (\frac{|∇f|}{X_{1}f} ∘ Φ L^{n−1}) = H^{n−1} (∂F ∩ Ω).

Thus, (67) and (68) are also formally equivalent, but the right hand side of (67) becomes meaningless if the Euclidean regularity of ∂F fails, unlike (68) that turns out to be an intrinsic generalization of (67) for 𝔾-regular hypersurfaces.

References

[1] L.Ambrosio, Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces, Adv. in Math., 159, (2001), 51–67.

[2] L.Ambrosio, N. Fusco & D. Pallara,Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Oxford University Press, (2000).

[3] L.Ambrosio & B.Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000), 527-555.

[4] L.Ambrosio & B.Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), 1-80.

[5] Z.Balogh, Size of characteristic sets and functions with prescribed gradient, Preprint,(2000).

[6] A.Bellaïche, The tangent space in subriemannian geometry, in Subriemannian Geometry, Progress in Mathematics, 144. ed. by A.Bellaiche and J.Risler, Birkhauser Verlag, Basel (1996).

[7] M.Biroli & U.Mosco, Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser. Rend. Lincei, Mat. Appl., 6, (1995), 37–44.

[8] L.Capogna, D.Danielli & N.Garofalo, The geometric Sobolev embedding for vector fields and the isoperimetric inequality, Comm. Anal. Geom. 12,(1994),203–215.

[9] J.Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geometry and Functional Analysis, 9,(1999),428–517.

[10] D.Danielli & N.Garofalo & D.M.Nhieu, Traces inequalities for Carnot–Carathèodory spaces and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 27,(1998), 195–252.

[11] G.David & S.Semmes, Fractured Fractals and Broken Dreams. Self-Similar Geometry through Metric and Measure, Oxford University Press, (1997).

[12] E.De Giorgi, Su una teoria generale della misura (r − 1)-dimensionale in uno spazio ad r dimensioni Ann.Mat.Pura Appl.(4), 36, (1954), 191–213.

[13] E.De Giorgi, Nuovi teoremi relativi alle misure (r−1)-dimensionali in uno spazio ad r dimensioni, Ricerche Mat. 4, (1955), 95–113.

[14] E.De Giorgi, Problema di Plateau generale e funzionali geodetici, Atti Sem. Mat. Fis. Univ. Modena, 43, (1995), 285–292.

[15] E.De Giorgi, Un progetto di teoria unitaria delle correnti, forme differenziali, varietà ambientate in spazi metrici, funzioni a variazione limitata, Manuscript, (1995).

[16] E.De Giorgi, Un progetto di teoria delle correnti, forme differenziali e varietà non orientate in spazi metrici, in Variational Methods, Non Linear Analysys and Differential Equations in Honour of J.P. Cecconi, (Genova 1993), M.Chicco et al. Eds. ECIG, Genova, 67–71.

[17] H.Federer, Geometric Measure Theory, Springer, 1969.

[18] G.B.Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161–207.

[19] G.B.Folland & E.M.Stein, Hardy spaces on homogeneous groups, Princeton University Press, 1982.

[20] B.Franchi, R.Serapioni & F.Serra Cassano, Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields, Houston Journal of Mathematics, 22, 4, (1996), 859–889.

[21] B.Franchi, R.Serapioni & F.Serra Cassano, Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields, Boll. Un. Mat. Ital.,(7) 11-B (1997), 83–117.

[22] B.Franchi, R.Serapioni & F.Serra Cassano, Rectifiability and perimeter in the Heisenberg group, Math. Ann., 321, (2001), 479–531.

[23] B.Franchi, R.Serapioni & F.Serra Cassano, Rectifiability and perimeter in step 2 groups, preprint, (2002).

[24] K.O. Friedrichs, The identity of weak and strong extensions of differential operators, Trans. Amer. Math. Soc. 55 (1944), 132–151.

[25] N.Garofalo & D.M.Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math., 49 (1996), 1081–1144.

[26] N.Garofalo & D.M.Nhieu, Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot Carathéodory spaces, J. Anal. Math., 74 (1998), 67–97.

[27] D.Gilbarg & N.S.Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed. (1983), Springer-Verlag, New York-Berlin-Heidelberg.

[28] M.Gromov, Carnot-Carathéodory spaces seen from within, in Subriemannian Geometry, Progress in Mathematics, 144. ed. by A.Bellaiche and J.Risler, Birkhauser Verlag, Basel (1996).

[29] M.Gromov, Metric structures for Riemannian and Non Riemannian Spaces, Progress in Mathematics, 152, Birkhauser Verlag, Boston (1999).

[30] P.Hajłasz & P.Koskela, Sobolev met Poincare, Memoirs of the American Mathematical Society, 688 (2000).

[31] D.Jerison, The Poincaré inequality for vector fields satisfying Hörmander condition, Duke Math. J., 53 (1986), 503-523.

[32] J.Heinonen, Calculus on Carnot groups, Ber. Univ. Jyväskylä Math. Inst., 68 (1995), 1–31.

[33] B.Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. AMS, 121 (1994), 113–123.

[34] B.Kirchheim & F. Serra Cassano, in preparation.

[35] V.Magnani On a general coarea inequality and applications, Ann. Acad. Sci. Fenn. Math., 27 (2002), 121– 140.

[36] V.Magnani A Blow-up Theorem for regular hypersurfaces on nilpotent groups Preprint (2001), to appear on Manuscripta Math.

[37] P.Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge U.P., 1995.

[38] P.Mattila, Hausdorff m-regular and rectifiable sets in n-spaces, Trans. Amer. Math. Soc., 205 (1975), 263–274.

[39] J.Mitchell, On Carnot-Carathèodory metrics, J.Differ. Geom. 21 (1985), 35–45.

[40] R.Monti & F.Serra Cassano, Surface measures in Carnot-Carathéodory spaces, Calc. Var. and Pde, to appear.

[41] P.Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Annals of Mathematics, 129 (1989), 1–60.

[42] S.D.Pauls, A notion of rectifiability modelled on Carnot groups, Preliminary announcement,(2000).

[43] D.Preiss & J.Tišer, On Besicovitch 1/2-problem, J. London Math. Soc., 45 (1992), 279–287.

[44] S.Semmes, On the non existence of bilipschitz parametrization and geometric problems about A_{∞} weights, Revista Matematica Iberoamericana. 12,(1996), 337–410.

[45] E.M.Stein, Harmonic Analysis, Princeton University Press, 1993.

[46] S.K.Vodop’yanov, P-differentiability on Carnot groups in different topologies and related topics, Proc.on Analysis and Geometry,603–670, Sobolev Institute Press, Novosibirsk, 2000.

Bruno Franchi: Dipartimento di Matematica, Università di Bologna, Piazza di porta San Donato,5, 40126 Bologna, Italy

   E-mail address: franchib@dm.unibo.it

Raul Serapioni: Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38050, Povo (Trento) Italy

   E-mail address: serapion@science.unitn.it

Francesco Serra Cassano: Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38050, Povo (Trento) Italy

   E-mail address: cassano@science.unitn.it