B((0,0),R)
hu(z, u(z),∇u(z)) φ(z) (8.43)
+ ∇ph(z, u(z),∇u(z)), ∇φ(z)
dx1dx2
=
BR
∇zu + z⊥/2,∇zφ
|∇zu + z⊥/2| + λ φ
dx1dx2.
At this point, using an algebraic agument, it is possible to show the existence of a global minimizer ofJ . Such global minimizer is indeed provided by the spherically symmetric function uR in (8.41).
Proposition 8.22. Given V > 0, let R = R(V ) > 0 be as in Corollary 8.17. The functional J in (8.37) is convex on D(R). As a consequence, the function uR in (8.41) is a global minimizer of J on D(R).
Finally, the proof of Theorem 8.11 is complete once one shows that uois the unique minimizer of the variational problem (8.35). This, in turn, follows from the fact that for every function φ, not identically zero, which is D(R)-admissible at uR, the strict inequality
J [uR+ φ] > J [uR] holds.
8.5 The C
2isoperimetric profile in H
In this section we sketch an argument showing that the bubble sets are the isoperi-metric minimizers in the category of C2surfaces.
Theorem 8.23 (Ritor´e–Rosales). If Ω is an isoperimetric region in H which is bounded by a C2 smooth surface S, then S is congruent (i.e., equivalent by the composition of a Heisenberg isometry and a dilation) to the boundary of a bub-ble set.
In contrast to the previous sections, which are quite analytic or measure theoretic in spirit, the proof of Theorem 8.23 relies heavily on techniques of differ-ential geometry. We give a step-by-step outline of the proof and present some of the main ideas. Note that in the results presented below, only the bare minimum is shown to illustrate the ideas of the proof (see the notes at the end of the chapter for detailed references).
8.5. The C2 isoperimetric profile inH 169
Step 1. The starting point is the study of the characteristic set of a C2 isoperi-metric profile.
Proposition 8.24. Let S⊂ H be an oriented volume-preserving perimeter station-ary C2 compact surface enclosing a region Ω. If the characteristic locus Σ(S) contains a C1 curve C, then the rules of the Legendrian foliation of S meet C orthogonally.
Proof. Recall that by Proposition 8.6, S is CMC. In view of Section 8.3, CMC surfaces are ruled by horizontal curves of fixed curvature. To prove the proposition assume that C is a curve in Σ(S). By Lemma 4.32, C is C1. Let B⊂ S be such that B\ C is the union of two open connected sets B±. Let n+ be the inward pointing normal to B+. Finally, let u : B → R be a mean zero function with compact support in B so that u|C is supported on C∩ B. Using this u in (6.33) and taking into account the assumption that S is perimeter stationary, we have
0 =
B\Σ(S)
u divSνH dσ−
B\Σ(S)
divS(u(νH)tang) dσ
=−
B\Σ(S)
divS(u(νH)tang) dσ
(since divSνH is constant in S\ Σ(S))
=−
B+\Σ(S)
divS(u(νH)tang) dσ−
B−\Σ(S)
divS(u(νH)tang) dσ
=
C
un+, νH+1dσ−
C
un+, ν−H1dσ
(by the divergence theorem)
= 2
C
un+, νH+1dσ,
where νH±(q) = limp∈B±,p→qνH(p) for q∈ C, and νH+(q) =−νH−(q) by Proposition 4.34. Since we may choose u so that u|Cis arbitrary, we conclude thatn+, νH+1= 0 on all of C∩ B. Hence νH+ is tangent to C in B, so the rules approaching C meet
C orthogonally.
Step 2. The second step of the proof is to show improved regularity of curves in the characteristic locus.
Proposition 8.25. Let S⊂ H be an oriented perimeter stationary C2 compact sur-face. If the characteristic locus Σ(S) contains a C1curve C, then C is in fact C2 regular.
170 Chapter 8. The Isoperimetric Profile ofH
This result follows from Lemma 4.32 and an analysis of the surface using the seed curve/height function parametrization of Section 8.3.
The regularity of characteristic curves leads to a crucial property of the character-istic locus of oriented volume-preserving perimeter stationary compact surfaces.
Theorem 8.26. Let S be a complete, oriented C2 immersed volume-preserving perimeter stationary surface in H with nonvanishing mean curvature. Then any connected curve in Σ(S) is a geodesic.
This theorem follows from a careful study of the behavior of the rules of the surface emanating from a characteristic curve to determine when they return to the characteristic locus. With the observation that the rules meet the characteristic locus orthogonally, Theorem 8.26 may also be derived from the work in Section 8.3. We sketch the proof and indicate some of the main computations.
Sketch of the proof. By (8.23), we know that, for a given seed curve/height func-tion pair (γ, h0), the characteristic locus arises when Σ(s, r) = 0. By readjusting our choice of seed curve and height function, we may assume that Σ(s, 0) = 0, i.e., a portion of the characteristic locus occurs at r = 0. The cautious reader will note that the computations in Section 8.3 are performed away from the characteristic locus. However, in equation (8.23) we show the behavior of the characteristic locus if we extend the parametrization to all values of r. It is not hard to see that (8.23) and the seed curve/height function representation may be used in this context as well. Note that here we must invoke Proposition 8.25 and assert that curves in the characteristic locus are C2. In fact, our seed curves must be at least C2. Inspection of (8.23) leads to the following condition on h0:
h0(s) = 1
2γ, (γ)⊥(s).
Consider the case where Σ(S) has another component that contains a C2 smooth curve. There exists r = r(s) so that this curve is given by (F (s, r(s)), h(s, r(s))), i.e., Σ(s, r(s)) = 0. Since h0(S) = 12γ, (γ)⊥(s), we have
Σ(s, r) =−sin(ρr)
ρ +1− cos(ρr)
ρ2 κ(s) (8.44)
and hence r(s) is a solution of Σ(s, r) = 0. Next, we wish to use the fact that the rules meet Σ orthogonally. We first note that
0 =Fs, Fr = sin(ρr(s)) − κ(s) 1− cos(ρr(s)) ρ
!
using (8.19) and (8.22). Hence, for r = r(s0), Fs(s0, r(s0)) is perpendicular to Fr(s0, r(s0)). Since
∂sF (s, r(s)) = Fs(s, r(s)) + Fr(s, r(s))r(s)
8.5. The C2 isoperimetric profile inH 171
we conclude that the rules meet the characteristic locus orthogonally if and only if r(s) is constant. Inspection of (8.44) shows that this may only happen if κ(s) is also constant. Hence the seed curve is a portion of a straight line or a circle and, as we have arranged that it lifts to a horizontal curve, we conclude that
(F (s, 0), h0(s)) is a Heisenberg geodesic.
Step 3. By the previous steps, any compact C2 solution to the isoperimetric prob-lem cannot have a nontrivial curve in its characteristic locus, as any such curve would be a geodesic which would leave any bounded domain in a finite time. Thus, the characteristic locus may contain only isolated points. The third and final step of the proof is to show that if the characteristic locus of the surface contains an isolated point, then it is congruent to the boundary of a bubble set.
Theorem 8.27. Let S be a complete, connected C2 oriented immersed surface in H with nonvanishing constant mean curvature. If Σ(S) contains an isolated point, then S is congruent to the boundary of a bubble set.
Sketch of the proof. The nucleus of the argument is contained in Theorem 4.38, which indicates that, at each isolated characteristic point p, there are values r > 0 and ρ∈ R so that
{γp,v,ρ(s) : v∈ HpS,|v| = 1, s ∈ [0, r)}
is a proper subset of S. Here, γp,v,ρdenotes the sub-Riemannian geodesic passing through p with tangent v and curvature ρ. Direct computation shows that if r = π/|ρ|, then the resulting surface is congruent with the boundary of a bubble set.
The theorem follows by analyzing the cases r < π/|ρ| and r > π/|ρ| and using the structure of the geodesics of the Heisenberg group. These facts impose significant constraints on the class of C2CMC surfaces:
Theorem 8.28. Let S be a compact, connected C2immersed volume-preserving per-imeter stationary surface inH. Then S is congruent to the boundary of a bubble set.
Sketch of the proof. From the Minkowski formula (Proposition 8.9) we conclude that the horizontal mean curvature of S computed with respect to the inner normal is strictly positive, hence nonvanishing. Next, by compactness we deduce that S contains at least one characteristic point. If Σ(S) contains a curve C, then Theorem 8.26 implies that C is a complete geodesic; as such geodesics leave any bounded set in finite time, this would violate the compactness assumption. Thus Σ(S) may contain only isolated points. By Theorem 8.27, S is congruent to the boundary of
a bubble set.
Coupled with Theorem 8.3 and the characterization of isoperimetric sets as CMC surfaces, Theorem 8.28 gives a proof of Theorem 8.23.
172 Chapter 8. The Isoperimetric Profile ofH