6.5 Mostow’s rigidity theorem for H C 2
6.5.1 Quasiconformal mappings on H
Quasiconformal maps onH play a crucial role in the above argument. Classically, quasiconformal maps may be defined in a variety of ways: metrically, geometrically or analytically. Let us review some of the basic definitions. Let f : U → U be a homeomorphism between domains in H.
(i) We say that f is metric quasiconformal if (6.40) holds for some constant H < ∞. This is an infinitesimal condition of uniformly bounded relative distortion of metric spheres.
(ii) We say that f is (locally) quasisymmetric if there exists an increasing home-omorphism η : [0,∞) → [0, ∞) so that for each Whitney ball B ⊂ U,2
d(f (x), f (y)) d(f (x), f (z)) ≤ η
d(x, y) d(x, z)
(6.42) for all x, y, z ∈ B, x = z. This is a local (but not infinitesimal) condition of uniformly bounded relative distortion of metric triples, the validity of (6.42) for some η clearly implies the validity of (6.40) with H = η(1).
(iii) We say that f is analytic quasiconformal if its coordinate functions belong to the local horizontal Sobolev class Sloc1,4, f is Pansu differentiable (in par-ticular, a contact map), and the Pansu differential (D0f )∗ : h → h verifies the pointwise distortion inequality
||(D0f )(x)∗||4≤ K det(D0G)(x)∗ for a.e. x∈ U.
Theorem 6.33 (Kor´anyi–Reimann). Conditions (i), (ii) and (iii) are quantitatively equivalent, for homeomorphisms of domains in H. Moreover, condition (i) with H = 1 implies condition (iii) with K = 1, and any such conformal map is a composition of maps of the following types:
• left translations,
• dilations,
• rotations about the x3-axis,
• the Kor´anyi inversion
jH(z, x3) =
−z
|z|2+ 4i x3
, −x3
|z|4+ 16x23
(see (2.14)).
All such conformal maps correspond, under generalized stereographic projection, with the boundary maps associated with the action of the isometry group SU (1, 2) on HC2.
2B is a Whitney ball in U if 2B⊂ U.
140 Chapter 6. Geometric Measure Theory and Geometric Function Theory
Kor´anyi and Reimann also conclusively demonstrated the existence of an extensive supply of nontrivial (e.g., nonconformal) quasiconformal mappings ofH by characterizing the infinitesimal generators of one-parameter flows of smooth quasiconformal maps.
Theorem 6.34 (Kor´anyi–Reimann). Let p ∈ C∞(H) and assume that X1X1p− X2X2p, X1X2p + X2X1p∈ L∞(H). Then the vector field
V = pX3+ (X2p)X1− (X1p)X2
generates a one-parameter family of smooth quasiconformal maps fs : H → H, s > 0, as solutions to the Cauchy problem
d
dsfs= V (fs), f0= id .
6.6 Notes
The canonical treatment of geometric measure theory remains the comprehensive tome of Federer [95]; much contemporary work in sub-Riemannian and general metric space-valued geometric measure theory involves the detailed development and elaboration of ideas and programs stemming from [95].
Notes for Section 6.1. In this section we closely follow the development of the area formula by Magnani [190] (but see also [189], and [222] for an equivalent but different treatment). Magnani’s proof is a simplified version of the proof of an area formula in a much more general class of metric spaces (see, for example, [161] or [9]). One can also give a proof of the area formula in Carnot groups analogous to the classical proof. For the details see [190, pp. 99–100]. Extensive work on the sub-Riemannian co-area formula has been done by Magnani [190–192]. The sub-Riemannian analog for the differential introduced in Definition 6.3 bears the name of Pierre Pansu, who introduced it and proved Theorem 6.4 in his important work [220]. The Pansu differential and Pansu–Rademacher differentiation theorem are foundational tools in the modern theory of geometric analysis in the Carnot–
Carath´eodory environment.
Notes for Section 6.2. The proof of Theorem 6.13 which we give is due to Calder´on;
see also Heinonen [136, Chapter 6]. Pansu’s original paper on the a.e. differentia-bility of Lipschitz functions on Carnot groups is [220]. For a recent far-reaching generalization to metric spaces, see Cheeger [60]. Arcozzi and Morbidelli [15], [16]
have given an analytic characterization for bi-Lipschitz self-maps of the Heisen-berg group using the Pansu derivative, as well as a related stability theorem for Heisenberg isometries in the spirit of F. John [157]
The eikonal equation for the CC metric was proved by Monti inH [204], and by Monti and Serra-Cassano in a wide class of CC spaces [211]. Example 6.15 is also taken from [211].
6.6. Notes 141
Notes for Section 6.3. Proposition is taken from Monti and Serra-Cassano [211], who work in a much more general setting of Carnot–Carath´eodory spaces. For an even more general perspective, see [7]. Proposition 6.18 is due to Monti [204].
Notes for Section 6.4. Our derivation of the noncharacteristic first variation of the perimeter is taken from Bonk–Capogna [37]. Alternative derivations of first variation formulas can be found in [63], [78], [221], [37], [199], [232] and [239].
The general form of the first variation is a result of Ritor´e and Rosales [231].
An essential novelty of our discussion, however, is the formulation of an explicit first variation formula for parameteric surfaces. Such formulas can be used, for instance, to compute the mean curvature of surfaces represented as graphs over spheres or other closed manifolds.
Notes for Section 6.5. Mostow’s rigidity theorem inaugurated the study of quasi-conformal maps in Carnot groups, general Carnot–Carath´eodory spaces, and most recently in metric measure spaces. While we have stated Mostow’s theorem only in the case of HC2, where the boundary quasiconformal analysis resides on the (one-point compactification of) the Heisenberg group, the original result [212] was formulated for general rank 1 symmetric spaces. Recall that a complete list of non-compact, negatively curved, rank 1 symmetric spaces consists of the real, complex and quaternionic hyperbolic spaces of dimension at least 2:
HRn, HCn, HKn, n≥ 2
(K denotes the division algebra of quaternions), and the Cayley hyperbolic plane HO2.
Mostow’s theorem 6.27 holds in all of these cases except for HR2; the essential ob-struction in the proof involves the absolute continuity in measure of the boundary quasiconformal (more properly, quasisymmetric) maps, which fails in the case of maps of S1 = ∂∞HR2. Pansu [220] obtained a stronger rigidity statement in the quaternionic and Cayley situations. See also [135], especially Section 6.
Kor´anyi and Reimann developed the full theory of quasiconformal maps on the Heisenberg groups Hn in [168], [171]. The case K = 1 in Theorem 6.33 fol-lows from their work through a regularity theorem for nonlinear subelliptic PDE proved in [246] and [50]. Alternative methods to construct quasiconformal maps in the Heisenberg groups can be found in [57]. The existence of a rich theory of quasiconformal maps in this specific non-Riemannian setting motivated further study of quasiconformal function theory in the setting of general metric measure spaces. The seminal work in this arena is due to Heinonen and Koskela [140], [141], who further studied the equivalence of definitions of quasiconformality in Theo-rem 6.33 (in more general Carnot groups and abstract metric measure spaces) and extended much of the ensuing Euclidean theory to this setting. Further work on the equivalence of definitions of quasiconformality, including the geometric def-inition of quasiconformality (which we have not touched on here) was done by
142 Chapter 6. Geometric Measure Theory and Geometric Function Theory
Tyson [251], [252]. For the most recent summary of these developments, we refer to Heinonen et al. [143]. Note that the concept of quasisymmetry, for mappings of metric spaces, was already introduced by Tukia and V¨ais¨al¨a [250] in 1980.
Quasiregular maps are a generalization of quasiconformal maps where the assumption of injectivity is relaxed. Heinonen and Holopainen [138] developed nonlinear potential theory and quasiregular maps on Carnot groups.
Chapter 7
The Isoperimetric Inequality in H
The isoperimetric inequality in H with respect to the horizontal perimeter was first proved by Pansu. We first state it in the setting of C1 sets.
Theorem 7.1 (Pansu’s isoperimetric theorem inH). There exists a constant C > 0 so that
|E|3/4≤ CPH(E) (7.1)
for any bounded open set E⊂ H with C1 boundary.
In this chapter, we present two very different proofs for this theorem. The first proof is based on the geometric Sobolev embedding S1,1⊂ L4/3 from Chap-ter 6. The second proof follows Pansu’s original approach and rests on Santal´o’s formula from integral geometry as used by Croke; it gives an explicit (nonsharp) value for C.
7.1 Equivalence of the isoperimetric and geometric Sobolev inequalities
We give a quick sketch of the argument which shows that the isoperimetric in-equality (7.1) is equivalent with the geometric Sobolev inin-equality (5.20). Suppose that E ⊂ H is a bounded, open, C1 set, and let R > 0 such that E ⊂ B(o, R).
Choose δ > 0 such that 2δ < dist( ¯E, ∂B(o, R)), where dist(·, ¯E) represents the Euclidean distance from ¯E. Define the (Euclidean) Lipschitz function
fδ(x) =
1−dist(x, ¯E) δ
+
.
144 Chapter 7. The Isoperimetric Inequality inH
Applying Proposition 5.17 to fδ we obtain
|E|34 ≤{x∈ B(o, R) : fδ(x) > t}34
≤C t
B(o,R)
|∇0fδ(y)| dy (7.2)
for each t < 1. Let Aδ be the intersection of B(o, R) with a tubular neighborhood of E of radius δ. From (7.2) and the co-area formula (see for instance [95, Theorem 3.2.3]), we obtain
|E|3/4≤ C tδ
Aδ
|∇0dist(y, ¯E)| dy;
letting t→ 1 yields
|E|3/4≤ C δ
δ 0
{y∈B(o,R):dist(y, ¯E)=s}
|∇0dist(·, ¯E)|
|∇ dist(·, ¯E)| dHn−1ds,
where dHn−1denotes the n−1-dimensional Hausdorff measure with respect to the background Euclidean metric. The proof of (7.1) is concluded once we let δ→ 0 and recall Corollary 5.8.