**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 S*_{loc}^{1,4}*, f is Pansu diﬀerentiable (in *
*par-ticular, a contact map), and the Pansu diﬀerential (D*_{0}*f )** _{∗}* :

*h → h veriﬁes*the pointwise distortion inequality

*||(D*0*f )(x)*_{∗}*||*^{4}*≤ K det(D*0*G)(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 x*3*-axis,*

*• the Kor´anyi inversion*

*j*_{H}*(z, x*3) =

*−z*

*|z|*^{2}+ 4* i x*3

*,* *−x*3

*|z|*^{4}*+ 16x*^{2}_{3}

*(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 H*_{C}^{2}*.*

2*B 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 X*1

*X*

_{1}

*p−*

*X*

_{2}

*X*

_{2}

*p, X*

_{1}

*X*

_{2}

*p + X*

_{2}

*X*

_{1}

*p∈ L*

*(*

^{∞}*H). Then the vector ﬁeld*

*V = pX*3*+ (X*2*p)X*1*− (X*1*p)X*2

*generates a one-parameter family of smooth quasiconformal maps f** _{s}* :

*H → H,*

*s > 0, as solutions to the Cauchy problem*

*d*

*dsf*_{s}*= V (f*_{s}*),* *f*_{0}*= 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 H*

_{C}

^{2}, 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:

*H*_{R}^{n}*,* *H*_{C}^{n}*,* *H*_{K}^{n}*,* *n≥ 2*

(K denotes the division algebra of quaternions), and the Cayley hyperbolic plane
*H*_{O}^{2}*.*

*Mostow’s theorem 6.27 holds in all of these cases except for H*_{R}^{2}; 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 S^{1} *= ∂*_{∞}*H*_{R}^{2}. 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 H^{n}*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 C*^{1} sets.

**Theorem 7.1 (Pansu’s isoperimetric theorem in****H). There exists a constant C > 0***so that*

*|E|*^{3/4}*≤ CP*H*(E)* (7.1)

*for any bounded open set E⊂ H with C*^{1} *boundary.*

In this chapter, we present two very diﬀerent proofs for this theorem. The
*ﬁrst proof is based on the geometric Sobolev embedding S*^{1,1}*⊂ L** ^{4/3}* 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, C*^{1} *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*

*f*_{δ}*(x) =*

1*−dist(x, ¯E)*
*δ*

+

*.*

144 Chapter 7. The Isoperimetric Inequality inH

*Applying Proposition 5.17 to f**δ* we obtain

*|E|*^{3}^{4} *≤{x∈ B(o, R) : f**δ**(x) > t}*^{3}^{4}

*≤C*
*t*

*B(o,R)*

*|∇*0*f**δ**(y)| dy* (7.2)

*for each t < 1. Let A**δ* *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*

*A*_{δ}

*|∇*0*dist(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)|* *dH*^{n}^{−1}*ds,*

*where dH*^{n}^{−1}*denotes 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.