**8.8 Further results**

**8.8.2 The classiﬁcation of symmetric CMC surfaces in H n**

In this section, we present Ritor´e and Rosales’ [232] classiﬁcation of all
cylin-drically symmetric constant mean curvature surfaces in the higher-dimensional
*Heisenberg groups. Assume that S is a C*^{2}hypersurface in H* ^{n}* which is invariant
under the group of rotations in R

^{2n+1}*about the x*

*2n+1*-axis. Any such surface can be generated by rotating a curve in the

*{x*1

*≥ 0} half-plane of the x*1

*x*

_{2n+1}*-plane, g(t) = (x(t), f (t)), where t varies over an interval I in the x*

_{1}-axis, around

*the x*

*-axis. We may parametrically realize such a surface as follows. Letting*

_{2n+1}*B = I× S*

^{n}*, the map*

^{−1}*φ(t, ω) = (x(t)ω, f (t))*

*parameterizes the rotationally invariant surface formed by rotating the curve g*
*about the x** _{2n+1}*-axis. Computing the unit Riemannian normal yields

*((x(t)x*^{}*(t)ω**n+k**−f*^{}*(t)ω**k**) X**k*+ (*−x(t)x*^{}*(t)ω**k**−f*^{}*(t)ω**n+k**) X**k+1**+ x*^{}*(t) X**2n+1*)

*|g*^{}*(t)|*^{2}*+ x(t)*^{2}*x*^{}*(t)*^{2} *.*

From this, one can compute the horizontal mean curvature of such a surface, yielding,

*H =* 1
*2n*

*x*^{3}*(x*^{}*f*^{}*− x*^{}*f*^{}*) + (2n− 1)(f** ^{}*)

^{3}

*+ 2(n− 1)x*

^{2}

*(x*

*)*

^{}^{2}

*f*

^{}*x(x*

^{2}

*(x*

*)*

^{}^{2}

*+ (f*

*)*

^{}^{2})

^{3}2

*Denoting by σ(t) the angle between g** ^{}* and the vertical direction,

_{∂x}

^{∂}*2n+1*, the
formula for the mean curvature yields that if the surface of rotation is of constant

186 Chapter 8. The Isoperimetric Proﬁle ofH

*mean curvature, then g(t) satisﬁes the following system of ordinary diﬀerential*
equations:

*x*^{}*= sin(σ),*
*f*^{}*= cos(σ),*

*σ*^{}*= (2n− 1)*cos^{3}*(σ)*

*x*^{3} *+ 2(n− 1)*sin^{2}*(σ) cos(σ)*
*x*

*− 2nH(x*^{2}sin^{2}*(σ) + cos*^{2}*(σ))*^{3}2

*x*^{2} *,*

(8.68)

*whenever x > 0. Using Noether’s theorem, the authors compute the ﬁrst integral*
of this system, showing that

*E =* *x*^{2n}^{−1}*cos(σ)*

*x*^{2}sin^{2}*(σ) + cos*^{2}*(σ)*

*− Hx*^{2n}

*is constant along any solution to the system. E is called the energy of the *
sys-tem. Using certain geometric properties of the solutions, Ritor´e and Rosales then
classify all cylindrically symmetric constant mean curvature surfaces.

**Theorem 8.35. Let g(s) be a complete solution to (8.68) with energy E. Then the***surface, S* *⊂ H*^{n}*, generated by rotation about the x*_{2n+1}*-axis, is one of the ﬁve*
*following types:*

*1. If H = 0 and E = 0, then g(s) is a straight line orthogonal to the x*_{2n+1}*-axis*
*and S is a Euclidean hyperplane.*

*2. If H = 0 and E* *= 0, then S is an embedded surface of catenoidal type.*

*3. If H* *= 0 and E = 0, then S is a compact hypersurface homeomorphic to the*
*sphere.*^{3}

*4. If EH > 0, then g(s) is a periodic graph over the x*_{2n+1}*-axis. S is a cylinder*
*or an embedded hypersurface of unduloid type.*

*5. If EH < 0, then g(s) is a locally convex curve and S is a nodoid type *
*hyper-surface with self-intersections.*

**8.9** **Notes**

In the last decade, there has been an explosion of research on analogs of the mini-mal and constant mean curvature equations and associated variational problems in the setting of Carnot–Carath´eodory spaces. While some of the work most closely related to the isoperimetric problem is covered in this chapter, we point out that there is a wealth of other material that is beyond the scope of this discussion. For the study of minimal surfaces in the Heisenberg groups, see [28, 80, 82, 83, 117, 221, 223, 231, 232]. For the roto-translation group, see [68, 146]. For three-dimensional

3*For n = 1, these are precisely the bubble sets**B(o, R).*

8.9. Notes 187

pseudo-hermitian manifolds (which include both the Heisenberg groups and the roto-translation group), see [63, 64]. For general Carnot groups, see [56, 78]. For general sub-Riemannian spaces, see [116, 144, 147].

* Notes for Section 8.1. Pansu’s conjecture was ﬁrst posed in [217] and [219]. The *
ob-servation regarding the equivalence with the isoperimetric problem for Minkowski
content is due to Monti and Serra-Cassano [211].

* Notes for Section 8.2. Theorem 8.3 is due to Leonardi and Rigot [176], who *
estab-lished existence results in the general class of Carnot groups. Their proof is based
on Garofalo and Nhieu’s Theorem 8.4 and on several results established in [116].

In that paper the setting is Carnot–Carath´eodory metrics generated by special
systems of Lipschitz vector ﬁelds. We have presented a simpliﬁed proof valid in
*the Heisenberg group. Section 8.2 contains the concentration-compactness *
argu-ment of Leonardi and Rigot (Lemma 8.5) as well as their ingenious method for
demonstrating the (essential) boundedness of the isoperimetric sets. The relation
between perimeter and rate of change of the volume used in (8.11) and (8.12) was
proved by Ambrosio in [7, Lemma 3.5].

We note that Leonardi and Rigot also investigate some properties of
isoperi-metric sets Ω, showing that such sets are Ahlfors regular and satisfy a synthetic
regularity condition known as Condition B. Moreover, in the setting of the
*Heisen-berg group, such sets are also domains of isoperimetry, that is, a relative *
isoperi-metric inequality of the form

min*{|S|*^{3/4}*,|Ω \ S|*^{3/4}*} ≤ CP*H*(S, Ω)*

*holds for all sets S* *⊂ Ω and a suitable constant C < ∞. As a consequence,*
isoperimetric sets are connected. As discussion of these facts would take us away
from the main points of this survey, we refer the interested reader to the original
paper [176].

* Notes for Section 8.3. The results in this section were independently proved by*
many authors. Our presentation loosely follows the one in [231].

If, in addition to the hypotheses in Proposition 8.6, we also assume that
*H*0*∈ L*^{1}*(S, dσ) with respect to the surface measure, then we rule out the possibility*
that*H*0*is a distribution with mass supported on Σ(S). In this case, we may easily*
deduce that

*S*

*u(div*_{S}*ν*_{H}*) dσ =*

*S**\Σ(S)*

*u(div*_{S}*ν*_{H}*) dσ =H*0

*S*

*u dσ = 0* (8.69)

*for all volume-preserving C*^{1} *vector ﬁelds U with compact support on S, where*
*u =U, ν*11.

* Notes for Subsection 8.3.1. The derivation in this section is an original *
contribu-tion of this survey and represents a generalizacontribu-tion of techniques in [117], where

188 Chapter 8. The Isoperimetric Proﬁle ofH

the minimal surface case was considered. Formulas (8.18), (8.21) and (8.24) can be found in [117]. Lemma 8.7 and Proposition 8.9 are proved in [231], where the corollary is pointed out as well. Both [37], and [231, 232] contain (diﬀerent) proofs of Theorem 8.6.

* Notes for Section 8.4. Theorems 8.11 and 8.36 are proved by Danielli, Garofalo*
and Nhieu in [81]. These results continue to hold, appropriately reformulated, in
any Heisenberg groupH

*.*

^{n}One immediate consequence of Theorem 8.11 is the following isoperimetric inequality.

**Theorem 8.36. Let**E be as in Section 8.4, and denote by ˜E the class of sets of the*form L**y**δ**λ**(E) for some E∈ E, λ > 0 and y ∈ H. Then*

*|E|*^{3/4}*≤ C*iso(*H)P*H*(E)* (8.70)
*for all E* *∈ ˜E, where C*iso(H) = 3^{3/4}*/(4√*

*π), with equality if and only if E =*
*L*_{y}*B(o, R) for some R > 0 and y ∈ H.*

In the interesting work [177], Leonardi and Masnou show, among other things,
*that such u** _{o}* is a critical point (but not the unique minimizer) of the horizontal

*perimeter, when the class of competitors is restricted to C*

^{2}domains with deﬁning

*function x*

_{3}=

*±f(|z|). The same result has been also noted in [232]. We also want*to point out related results in a recent preprint by Ritor´e [230], which considers the analog of the bubble sets in higher-dimensional Heisenberg groups and proves a sharp isoperimetric inequality yielding the isoperimetric proﬁle ofH

*within the*

^{n}*class of C*

^{1}sets contained in a cylinder with axis along the center of the group.

Theorem 4.48, which plays a role in the derivation in this section, is a result of
Balogh, see Theorem 3.1 in [20]. It is worthwhile noting that the result of Theorem
*4.48 fails if C*_{loc}^{1,1}*is replaced by C*_{loc}^{1,α}*for any α < 1; examples to this eﬀect are also*
given in [20].

Unfortunately, eﬀective symmetrization procedures in the Heisenberg group
(and other Carnot groups) are noticeably lacking. An approach to symmetrization
via polarization has been developed in the classical space forms, see Baernstein
[18] or Brock–Solynin [44]. Simply put, this program seeks to realize certain
well-studied symmetrization procedures (such as Steiner symmetrization) as limits of
sequences of polarizations, i.e., reﬂective symmetrizations in hyperplanes.
Prelim-inary attempts to generalize this program to the Heisenberg setting encounter
signiﬁcant obstructions; polarizations in vertical hyperplanes (the obvious
candi-dates for producing cylindrical symmetry) are not well behaved. Ultimately, this
stems from the fact that reﬂections in such planes are not isometries of the CC
metric. (Compare the discussion following the deﬁnition of reﬂection in (8.29).)
* Notes for Sections 8.5 and 8.6. The brief sketches of Theorems 8.23 and 8.29 are*
based on the more complete arguments given by Ritor´e and Rosales [231] and
Monti and Rickly [210], respectively. In comparing the proof of Theorem 8.23