The area and co-area formulas are fundamental tools in classical geometric measure theory. These integral formulas have sub-Riemannian analogs and indeed can be generalized to much more general metric spaces. To begin, we remind the reader of the classical area formula (see [95] Section 3.2.1):
118 Chapter 6. Geometric Measure Theory and Geometric Function Theory
Theorem 6.1 (Euclidean area formula). Let A ⊂ Rn be a measurable set, let f : A→ Rmbe a Lipschitz map, and let m≥ n. Then,
A
J (f )(x) dHn(x) =
Rm
N (f, A, y) dHm(y)
where J (f ) is the Jacobian of f ,Hn is the n-dimensional Hausdorff measure, and N (f, A, y) = card{x ∈ A : f(x) = y}.
To generalize Theorem 6.1 to the setting of Carnot groups, we require a notion of horizontal Jacobian for Lipschitz maps between Carnot groups. We note that while the co-area formula holds for arbitrary Carnot groups, the reader focused on understanding the Heisenberg group may wish to assume thatGi∈ {H, Rn} in what follows. We begin with a definition of a metric notion of the Jacobian.
Definition 6.2. LetG1,G2 be Carnot groups equipped with sub-Riemannian met-rics d1, d2 respectively. Let A ⊂ G1 be an HQ measurable set, where Q is the homogeneous dimension of G1, and let f : A→ G2 be Lipschitz with respect to the sub-Riemannian metrics on G1 andG2. The metric Jacobian of f at x∈ A is defined to be
Jfm(x) = lim inf
r→0
HdQ2(f (BA(x, r))) HQd1(BA(x, r)) where BA(x, r) ={y ∈ A|d1(x, y)≤ r}.
With this notion we easily recover the following standard integration formula:
A
Jfm(x)dHdQ1(x) =HQd2(f (A)). (6.1) We note that (6.1) trivially yields a property analogous to the Sard theorem, namely, if Jfm= 0 a.e. in A, thenHQd2(f (A)) = 0.
A key ingredient of the proof of the sub-Riemannian area formula is an analysis of the set A0 ={x ∈ A : Jfm(x) = 0}. To better understand this set, we introduce Pansu’s sub-Riemannian analog for the differential.
Definition 6.3. Let (G1, d1) and (G2, d2) be Carnot groups with homogeneous structures determined by the dilations δsi, i = 1, 2. Suppose that f :G1→ G2 is a measurable map. Then the Pansu differential of f at x is the map
D0f (x) :G1→ G2
defined by
D0f (x)(y) = lim
s→0δ1/s2 f (x)−1f (xδ1sy) whenever the limit exists.
6.1. Area and co-area formulas 119
For a Carnot group G with sub-Riemannian metric dG, we may recognize the metric tangent space TxG at a point x ∈ G as the pointed Gromov–Hausdorff limit of the sequence of metric spaces (x−1G, λdG) as λ→ ∞. As G is equipped with a family of homotheties (δs),
(x−1G, sdG) = (x−1G, dG◦ (δs× δs)),
whence (x−1G, dG) and (x−1G, sdG) are isometric via the map δs. Thus the se-quence above is constant and TxG is isometric to G. From this point of view, we can think of D0f as a map between tangent spaces. Alternatively, by conjugating with the exponential map, we can view D0f as a map between the Lie algebras:
(D0f )∗:g1→ g2.
Pansu’s extension of the Rademacher differentiability theorem to Carnot groups reads as follows:
Theorem 6.4 (Pansu–Rademacher differentiation theorem). LetG1,G2be Carnot groups and let A ⊂ G1 be a measurable set. Let f : A ⊂ G1 → G2 be Lipschitz with respect to the sub-Riemannian metrics on G1 andG2. Then, for a.e. x∈ A, D0f (x) exists and is a horizontal linear map (i.e., a graded homogeneous group homomorphism, see Definition 2.1) betweenG1 andG2.
In Section 6.2, we prove a special case of this theorem when G1 = H and G2=R.
Example 6.5. Let G1 =G2 =H. The Pansu differential D0f of a Lipschitz map f = (f1, f2, f3) :H → H, acting on the Lie algebra h and expressed in terms of the standard basis X1, X2, X3, takes the form
⎛
⎝X1f1 X1f2 0 X2f1 X2f2 0
0 0 X1f1X2f2− X1f2X2f1
⎞
⎠ .
The next lemma establishes a link between D0f and Jfm.
Lemma 6.6. Let G1 andG2 be Carnot groups with homogeneous dimensions Q = Q1 and Q = Q2 respectively, with Q≥ Q, and let f : A ⊂ G1→ G2be Lipschitz.
If D0f (x) is not injective for some x∈ A, then Jfm(x) = 0.
Proof. We sketch the main ideas in the proof in the case A = G1. Let P = D0f (x)(G1) be the image of the entire group under the differential mapping. Since D0f is not injective at x, the Hausdorff dimension of P is less than or equal to Q− 1. Moreover, as the Pansu differential exists, we have
d2(D0f (x)(y), δ21/sf (x)−1f (xδs1y)) = o(1)
as s→ 0. Using left invariance of the metric and the dilation property (2.20), we find
d2(D0f (x)(y), δ1/s2 f (x)−1f (xδ1sy)) = 1
sd2(f (x)δs2D0f (x)(y), f (xδ1sy))
120 Chapter 6. Geometric Measure Theory and Geometric Function Theory
so
d2(f (x)δs2D0f (x)(y), f (xδ1sy)) = o(s).
Since D0f (x)(δ1sy) = δs2D0f (x)(y), we may rewrite this as d2(f (x)D0f (x)(y), f (xy)) = o(d1(x, y)).
Consequently, if B(x, r) ⊂ G1 is the metric ball of radius r centered at x, then its image under f lies in an o(r) neighborhood N of f (x)D0f (x)(B(o, r)) where o∈ G1 is the identity element. Since f is Lipschitz,
HQd2(f (A))≤ (Lip f)QHQd1(A) (6.2) for all measurable A⊂ G1, where
Lip f (x) = lim sup
r→0
sup{d2(f (x), f (y)) : d1(x, y)≤ r}
r
denotes the pointwise Lipschitz constant of f . Combining (6.2) with the observa-tion that f (x)D0f (x)(B(o, r)) has Hausdorff dimension less than or equal to Q−1, it follows from a covering argument that
HQd2(f (B(x, r)))≤ HQd2(N ) = o(rQ) = o(HQd1(B(x, r))).
The result follows from the definition of the metric Jacobian. We omit the proof of the following lemma, which is an adaptation of a classical argument (see 3.2.2 in [95]). For further details, see the notes to this chapter.
Lemma 6.7. Suppose that f : A ⊂ G1 → G2 is a Lipschitz map and let λ > 1.
Let ˜A be the set of points of density of A where D0f exists and is injective. Then there exist Borel sets {Ei} partitioning ˜A so that for each i,
• f|Ei is injective,
• there exist injective horizontal linear maps Li so that 1
λd2(Li(z), o)≤ d2(D0f (x)(z), o)≤ λd2(Li(z), o) (6.3) for x∈ Ei and z∈ G1, where o∈ G2 is the identity, and
• we have
Lip(f|Ei◦ Li|−1Ci)≤ λ, Lip(Li|Ei◦ f|−1Ei)≤ λ. (6.4) Using this setup and notion of metric Jacobian, we can prove a version of the area formula for Carnot groups.
6.1. Area and co-area formulas 121 proof of the area formula as it makes no contribution to either side of the equation.
By Lemma 6.6, D0f (x) is injective for x∈ A \ A0. Using Lemma 6.7, we find sets {Ci} partitioning A \ A0, with fi= f|Ci injective. Let Jim(x) := Jfm
i(x). Since the closed balls in a Carnot group form a Vitali relation, we know that
lim
r→0
HQd1(BA∩Ci(x, r))
HQd1(BA(x, r)) = 1 (6.5) for all i and all points of density x ∈ Ci. A quick calculation using (6.5) shows that Jfm= Jim at such points:
Using (6.1) and the fact that the fi are injective, we have
122 Chapter 6. Geometric Measure Theory and Geometric Function Theory
Summing over all i yields
A
Jfm(x) dHQd1(x) =
A\A0
Jfm(x) dHdQ1(x) =
G2
N (f, C, y) dHQd2(y).
To complete our discussion of the area formula, we note that a more geometric notion of the Jacobian is equivalent to the metric Jacobian at points of Pansu dif-ferentiability. Recall the notion of a horizontal linear map between Carnot groups (Definition 2.1).
Definition 6.9. The horizontal Jacobian of a horizontal linear map φ :G1→ G2 is
JH(φ) = HQd2(φ(A)) HQd1(A) ,
where A is any measurable subset ofG1of positive finite measure.
We note that a simple covering argument, combined with the homogeneity and left invariance of the Hausdorff measure, shows that the value of JH(φ) is independent of the choice of A.
Proposition 6.10. Let f : A⊂ G1→ G2 be a Lipschitz map. Then Jfm= JH(D0f ) at point of Pansu differentiability in A.
Proof. Let x be a point of Pansu differentiability for f in A. By Lemma 6.6, we may assume that D0f (x) is injective; else both sides are zero. Using Lemma 6.7 with a sequence λn → 1, we find sets En containing x as points of density and horizontal linear maps Ln satisfying (6.3) and (6.4) with λ = λn. Thus
λ−Qn JH(Ln) = lim
r→0λ−Qn HdQ2(Ln(BA(x, r))) HdQ1(BA(x, r))
= lim
r→0λ−Qn HdQ2(Ln(BA∩En(x, r))) HdQ1(BA(x, r)) , since x is a point of density of En and Ln is a horizontal linear map
≤ lim sup
r→0
HQd2(f (BA∩En(x, r))) HdQ1(BA(x, r)) , by (6.3)
≤ lim sup
r→0
HQd2(f (BA(x, r)))
HdQ1(BA(x, r)) = Jfm(x).