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 C2hypersurface in Hn which is invariant under the group of rotations in R2n+1 about the x2n+1-axis. Any such surface can be generated by rotating a curve in the{x1 ≥ 0} half-plane of the x1x2n+1 -plane, g(t) = (x(t), f (t)), where t varies over an interval I in the x1-axis, around the x2n+1-axis. We may parametrically realize such a surface as follows. Letting B = I× Sn−1, the map
φ(t, ω) = (x(t)ω, f (t))
parameterizes the rotationally invariant surface formed by rotating the curve g about the x2n+1-axis. Computing the unit Riemannian normal yields
((x(t)x(t)ωn+k−f(t)ωk) Xk+ (−x(t)x(t)ωk−f(t)ωn+k) Xk+1+ x(t) X2n+1)
|g(t)|2+ x(t)2x(t)2 .
From this, one can compute the horizontal mean curvature of such a surface, yielding,
H = 1 2n
x3(xf− xf) + (2n− 1)(f)3+ 2(n− 1)x2(x)2f x(x2(x)2+ (f)2)32
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)cos3(σ)
x3 + 2(n− 1)sin2(σ) cos(σ) x
− 2nH(x2sin2(σ) + cos2(σ))32
x2 ,
(8.68)
whenever x > 0. Using Noether’s theorem, the authors compute the first integral of this system, showing that
E = x2n−1cos(σ)
x2sin2(σ) + cos2(σ)
− Hx2n
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 ⊂ Hn, generated by rotation about the x2n+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 x2n+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 x2n+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
3For n = 1, these are precisely the bubble setsB(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} ≤ CPH(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 H0∈ L1(S, dσ) with respect to the surface measure, then we rule out the possibility thatH0is a distribution with mass supported on Σ(S). In this case, we may easily deduce that
S
u(divSνH) dσ =
S\Σ(S)
u(divSνH) dσ =H0
S
u dσ = 0 (8.69)
for all volume-preserving C1 vector fields U with compact support on S, where u =U, ν11.
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 groupHn.
One immediate consequence of Theorem 8.11 is the following isoperimetric inequality.
Theorem 8.36. LetE be as in Section 8.4, and denote by ˜E the class of sets of the form Lyδλ(E) for some E∈ E, λ > 0 and y ∈ H. Then
|E|3/4≤ Ciso(H)PH(E) (8.70) for all E ∈ ˜E, where Ciso(H) = 33/4/(4√
π), with equality if and only if E = LyB(o, R) for some R > 0 and y ∈ H.
In the interesting work [177], Leonardi and Masnou show, among other things, that such uo is a critical point (but not the unique minimizer) of the horizontal perimeter, when the class of competitors is restricted to C2domains with defining function x3=±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 ofHnwithin the class of C1 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 Cloc1,1 is replaced by Cloc1,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