7.3 Pansu’s proof of the isoperimetric inequality in H
In this section, we illustrate how Pansu adapts Croke’s argument to the Heisenberg group. Let Ω⊂ H be a smooth open set and denote by νHthe horizontal normal to
∂Ω as defined in (4.9). For all x∈ ∂Ω and θ ∈ [0, 2π) denote by ei θthe horizontal vector field cos θX1(x) + sin θX2(x), and define the first exit time
r(x, θ) = sup{r > 0| exp(tei θ)∈ Ω}.
The first step in Pansu’s argument consists in proving the following Santal´o-type formula:
2π|Ω| =
∂Ω
2π 0
r(x, θ)ei θ, νH1dθdµ(x), (7.10) where µ is the perimeter measure defined in Corollary 5.8.
For any point x ∈ H we define its canonical section Σx to be the set of horizontal lines through x. In other words, if sx:R2→ H denotes the section
sx(a, b) = x exp(aX1+ bY1),
then Σx= sx(R2). In our presentation of the Heisenberg group, Σxis a plane with (Euclidean) normal vector in the direction (−a/2, b/2, −1) where x = (a, b, c).
Choose polar coordinates (r, θ) in R2 centered at the point z, the projection of x ontoR2. Then we can pull back the perimeter measure dµ through sxto a measure in R2 given by
s∗x(dµ) = 1
2r2dr∧ dθ. (7.11)
Proposition 7.2. Let D ⊂ R2 be a smooth, bounded open set and let V ⊂ H be a smooth surface with boundary so(∂D). Then
µ(V )≥ µ(so(D)).
Proof. The set so(D) is foliated by horizontal lines, hence has mean curvature zero at all points. Consequently it is a sub-Riemannian minimal surface (see Section 4.3 for more details) and hence is a critical point for the perimeter variation. A calibration argument shows that so(D) is a true minimizer.
Define the open subset of Σx which is visible from the point x:
Px={x exp(ρei θ)| x exp(tei θ)∈ Ω for all 0 < t < ρ }.
We remark that Σx divides ∂Ω in two connected components whose boundaries coincide with ∂Px.
148 Chapter 7. The Isoperimetric Inequality inH
We call these components V1and V2, so that
µ(∂Ω) = µ(V1) + µ(V2). (7.12) Lemma 7.3. In the notation established above,
Proof. Denote by (ρ, φ) the coordinates in Σx ⊂ H induced by the polar coordi-nates (r, θ)∈ R2. From the definition of Px and (7.11) we have
An integration over ∂Ω with respect to µ completes the proof. Lemma 7.4. In the notation established above,
Px
ρ−2dµ≤ (3π2)1/3µ(Px)1/3. (7.13) Proof. To simplify the derivation, we pull back to the plane via the horizontal section map sx:R2→ Σx. Then (7.13) reads
Let B(o, R) be a disc with area|D|. To obtain (7.14), we estimate
by two applications of Lemma 7.5. This completes the proof of Lemma 7.4.
7.3. Pansu’s proof of the isoperimetric inequality in H 149
If h is decreasing then the inequality is reversed.
Proof. Since|B(o, R) \ D| = |D \ B(o, R)|, we have
Lemma 7.6. In the notation established above,
∂Ω
µ(Px)1/3dµ≤ 2−1/3µ(∂Ω)4/3.
Proof. From Proposition 7.2 and from (7.12) we see that 2µ(Px)≤ µ(V1) + µ(V2) = µ(∂Ω).
An integration over ∂Ω with respect to µ completes the proof. A proof of the isoperimetric inequality inH can now be obtained from (7.10) and Lemmata 7.3–7.6 via the following string of estimates:
2π|Ω| =
150 Chapter 7. The Isoperimetric Inequality inH
7.4 Notes
The approach to the isoperimetric inequality via the geometric Sobolev inequal-ity goes back to the work of Fleming and Rishel [97], see also Maz’ya [197]. The sub-Riemannian case was proved independently in [53], [104] for H¨ormander vec-tor fields and more general structures, and [35] in the setting of Dirichlet forms.
Pansu’s proof of the isoperimetric inequality appeared in his 1982 paper [217], see also [219]. For the argument of Croke, see [74]; Santal´o’s formula (7.4) can be found in his textbook on integral geometry [237]. An extension of Santal´o’s formula (7.10) to all Carnot groups can be found in [200]. A different sharp isoperimetric inequality in Hadamard manifolds (comparing with model space forms) has been given by Kleiner [163].
Chapter 8
The Isoperimetric Profile of H
This chapter is the core of this survey. We recall the definition of isoperimetric profile ofH and Pansu’s 1982 conjecture. Next we present a proof of the existence of an isoperimetric profile and describe some of the existing literature on the isoperimetric problem. Our aim is to reveal the main ideas and outlines of the proofs of various partial results and sketch some further techniques and methods which may lead to a solution, in order to guide the reader through the literature and to give a sense of the larger ideas that are in play.
8.1 Pansu’s conjecture
The isoperimetric constant of the Heisenberg group is the best constant Ciso(H) for which the isoperimetric inequality
min{|Ω|3/4,|H \ Ω|3/4} ≤ Ciso(H)PH(Ω) (8.1) holds. In other words,
Ciso(H) = sup
Ω
min{|Ω|3/4,|H \ Ω|3/4}
PH(Ω) , (8.2)
where the supremum is taken on all Caccioppoli subsets of the Heisenberg group.
In a dual manner we could define the isoperimetric constant ofH as the value
inf{PH(E) : E⊂ H is a bounded Caccioppoli set and |E| = 1}
−1
. (8.3) We also define the isoperimetric profile.
Definition 8.1. An isoperimetric profile for H consists of a family of bounded Caccioppoli sets Ωprofile= Ωprofile(V ), V > 0, with|Ωprofile(V )| = V and
|Ωprofile|3/4= Ciso(H)PH(Ωprofile).
152 Chapter 8. The Isoperimetric Profile ofH
The invariance and scaling properties of the Haar measure and perimeter measure clearly imply that the sets comprising the isoperimetric profile are closed under the operations of left translation and group dilation. In [219], Pansu con-jectured that any set in the isoperimetric profile of H is, up to translation and dilation, a bubble setB(o, R). Recall from Section 2.3 that B(o, R) is obtained by rotating around the x3-axis a geodesic joining two points at height±πR2/2. More precisely, these cylindrically symmetric surfaces have profile curve
x3= fR(r) =±1 4(r
R2− r2+ R2arccos r/R).
Setting u(x) = fR(|z|) − x3, x = (z, x3), we easily compute
|B(o, R)| = 4π
R 0
rf (r) dr = 3
16π2R4 (8.4)
and
PH(B(o, R)) = 2
B(o,R)
|∇0u| = 4π
R 0
r
f(r)2+ r2/4 dr = 1 2π2R3, yielding the following conjecture for the value of the Heisenberg isoperimetric constant and the isoperimetric profile.
Conjecture 8.2 (Pansu).
Ciso(H) = |B(o, R)|3/4
PH(B(o, R)) = 33/4 4√
π (8.5)
for any R, and equality is obtained if and only if Ω is a bubble set.1
We take this opportunity to reiterate the fact that Pansu’s conjecture is still unsolved in this generality, although numerous partial results and special cases have been established over the years.
The isoperimetric problem for Minkowski content may be formulated as fol-lows: determine the value of
min{M3(∂E) : E⊂ H is bounded, |E| = 1}. (8.6) Proposition 6.17 shows that the minima in (8.6) and in (8.3) coincide when the class of competitors is restricted to sets with C2 boundary. (In Section 8.5 we sketch an argument verifying Pansu’s conjecture in this category.) As we shall see, it is not currently known if the isoperimetric profile sets in H have C2 boundary.
However, the expression
inf{M3(∂E) : E⊂ H is bounded, |E| = 1} (8.7) is equal to the isoperimetric constant of H.
1Compare the conjectured value for Ciso(H) with the (nonsharp) value obtained in (7.15).
8.1. Pansu’s conjecture 153
Overview of the chapter. The first portion of this chapter is devoted to describing evidence supporting Pansu’s conjecture culminating, in Sections 8.5 and 8.6, with affirmative answers in the C2, respectively convex, category. It is important to note that none of the machinery necessary to analyze variational problems of this sort existed at the time when the conjecture was proposed. The development of such machinery beginning in the late 1990s was instrumental in laying the groundwork needed for potential approaches to, and partial results for, this and related conjectures.
In Section 8.2, we present an important result of Leonardi and Rigot asserting the existence of an isoperimetric profile in any Carnot group, and demonstrating weak regularity (i.e., Ahlfors-type regularity and interior and exterior corkscrew condition, see [176, Definition 2.10]) for the constituent sets. As we saw in Chapter 6, the study of sub-Riemannian geometric measure theory is still in its infancy and, in particular, there is not sufficient infrastructure to bootstrap further regularity properties of the solution. This motivates the introduction of a substantial, yet useful, restriction to the class of C2 surfaces. Under this smoothness assumption, we show, in Section 8.3, that the isoperimetric sets have constant horizontal mean curvature and, as a consequence, have particularly nice parametrizations. This allows for an analysis of the isoperimetric in this class via two different methods.
First, one may use the link to sub-Riemannian constant mean curvature surfaces to introduce the methods of geometric partial differential equations. Second, the techniques of Riemannian geometric analysis of constant mean curvature surfaces may be adapted.
In Section 8.4, we present a result of Danielli, Garofalo and Nhieu showing the validity of Pansu’s conjecture under an extra symmetry and C1 smoothness assumption. In Section 8.5, we sketch the proof of a recent ground-breaking result of Ritor´e and Rosales (Theorem 8.23) which allows for the removal of the symme-try assumption, showing that the isoperimeteric profile in the class of closed C2 surfaces is indeed given by the class of bubble sets. The results in Section 8.5 rest heavily on the work of Cheng, Hwang, Malchiodi and Yang presented in Sections 4.4.1 and 4.4.2. In Section 8.6 we present a very recent result of Monti and Rickly, where Pansu’s conjecture is proved with no smoothness assumptions but in the class of (Euclidean) convex sets. This sequence of results provides strong evidence for the validity of Pansu’s original conjecture.
In Section 8.7, we present three other possible approaches to the isoperimet-ric problem which either fail or are in some way incomplete. In Section 8.7.1, we present an approach based on Riemannian approximation. Using work of Tomter which classifies cylindrically symmetric constant mean curvature surfaces in the Riemannian Heisenberg group, one can analyze the evolution of these surfaces in (R3, gL) as L → ∞. While one recovers the bubble sets as L tends to infinity, the method is incomplete as it requires finer information concerning the limit-ing process than is currently available. In Section 8.7.2, we present an analog to the well-known approach to isoperimetry through the Brunn–Minkowski theorem,
154 Chapter 8. The Isoperimetric Profile ofH
and discuss the obstruction to implementing such a scheme in the sub-Riemannian case. Last, in Section 8.7.3, we discuss motion by horizontal mean curvature flow inH which could potentially be used to understand the isoperimetric profile. The concluding Section 8.8 discusses two related results: Monti and Morbidelli’s com-putation of the isoperimetric profile of the Grushin plane, and Ritor´e and Rosales’
classification of C2rotationally symmetric constant mean curvature surfaces inHn.