In this section we present a proof of Pansu’s conjecture within the category of Euclidean convex sets.
Theorem 8.29 (Monti–Rickly). If Ω is a (Euclidean) convex region in H which is an isoperimetric set, then Ω is congruent to a bubble set.
From the point of view of regularity, the assumption in Theorem 8.29 (Eu-clidean convexity) is weaker than that in Theorem 8.23 (Eu(Eu-clidean Lipschitz vs.
C2 regularity). The general strategy in the proof of Theorem 8.29 is to show that patches of ∂Ω can be parameterized by lifts of circles, and then use the convexity hypothesis to conclude the argument.
For the remainder of this section, convexity always refers to convexity in the Euclidean sense.
Convex domains in H. Let Ω ⊂ H be a bounded convex open set. A supporting plane for Ω at p ∈ ∂Ω for Ω is simply a plane Π through p which has empty intersection with Ω. The notion of supporting plane generalizes the concept of tangent plane to the setting of non-smooth convex sets.
The characteristic set of a convex domain (with no further regularity assump-tions) may be defined as the set of p∈ ∂Ω for which the horizontal plane H(p) at p is a supporting plane for Ω.
We sketch the proof of Theorem 8.29 in a series of steps.
Step 1. In the first stage, we study the structure of the characteristic locus for convex domains, proving the following results.
1. Convex and bounded C1 sets have at least one characteristic point, strictly convex bounded sets have at most two.
2. In general, if Ω is a convex domain containing the origin, then Σ(∂Ω) can be written as the union of two disjoint components Σ±, separated by the hori-zontal plane at the origin, each of which is a (possibly degenerate) horihori-zontal segment.
3. If two geodesic (hence horizontal) arcs with the same curvature H > 0 are contained in ∂Ω and intersect at p ∈ ∂Ω, then they must coincide in a neighborhood of p.
Step 2. Assume now that Ω is a convex isoperimetric set in H. Decompose ∂Ω in four patches, which are graphs over convex planar domains A ⊂ R2 of the form x3 = f (x1, x2)(for the “top” and “bottom” portions), and x1 = g(x2, x3) (for the “lateral” portion), with f, g : A→ R convex (hence Lipschitz continuous with BV derivatives). Recalling (4.27), we denote by Sf the singular set of points (x1, x2)∈ A such that (x1, x2, f (x1, x2))∈ Σ(∂Ω). In a similar fashion, we define the singular set Sg.
8.6. The convex isoperimetric profile ofH 173
Step 3. Use a variational argument similar to the ones described in the previous sections to derive curvature equations for convex isoperimetric profiles. Additional difficulties arise in this non-smooth setting, as one needs first to show that for any compact set K disjoint from Sf or Sg, there exists δ > 0 such that
|∇0(x3− f(x1, x2))|, |∇0(x1− g(x2, x3))| > δ (8.45) a.e. in K. With such an estimate at hand, and in view of Remark 5.10, we may represent the perimeter of the graph of f in integral form:
respec-tively, and consider the portion of ∂Ω arising as the graph of f , we find that f must satisfy the PDE
in the distributional sense in A\ Sf. On the other hand, again in view of Remark 5.10, we may represent the perimeter of the graph of g in integral form:
If we consider the portion of ∂Ω arising as the graph of g, we find that g must satisfy the PDE in the distributional sense in A\ Sg. We will refer to the value 3P/4V as the curvature of the isoperimetric profile Ω.
Step 4. Next, one realizes that the distributional interpretation of the PDEs (8.46) and (8.47) is not sufficient to make progress towards the final result. What is needed is an extra measure of regularity for the solutions, namely, one wants the PDEs to hold in the weak Sobolev sense rather than in the distributional sense.
Such an improvement in regularity is achieved through the following proposition.
Proposition 8.30. Let A⊂ R2 be a bounded open set and let u = (u1, u2) : A→ R2 be a vector field whose components u1, u2 are BV functions. Assume that
(i) there exists δ > 0 such that |u| > δ a.e. in A, (ii) div u⊥∈ L1(A),
(iii) div(u/|u|) ∈ L1(A).
Then u/|u| ∈ W1,1(A,R2).
174 Chapter 8. The Isoperimetric Profile ofH
We apply this proposition to the vector field u = ∇0(x3 − f(x1, x2)) =
−∇f + 1/2(−x2, x1) in any compact set which avoids Sf. Observe that (i) follows from (8.45). A direct computation yields div u⊥ = −1, hence (ii) is satisfied.
Assumption (iii) follows from (8.46). We deduce that
∇0(x3− f(x1, x2))
|∇0(x3− f(x1, x2))| = −∇f +12(−x2, x1)
| − ∇f +12(−x2, x1)| ∈ W1,1(A\ Sf, R2).
A similar result holds for the patch given by x1= g(x2, x3):
∇0(x1− g(x2, x3))
|∇0(x1− g(x2, x3))| ∈ W1,1(A\ Sg,R2). (8.48) Step 5. The crucial step in the proof of Theorem 8.29 is the study of the Legendrian foliation of a convex isoperimetric profile. Observe that for any convex function f : A⊂ R2→ R, the vector fields u = −∇f + 1/2(−x2, x1) and v = 1/2(x1, x2)− (−∂x2f, ∂x1f ) are in BVloc(A,R2)∩L∞(A). Moreover div v =−1 ∈ L∞(A). These facts suffice to apply Ambrosio’s extension [8] of results of DiPerna–Lions [87] on flows generated by Sobolev vector fields: if v ∈ BVloc(R2,R2)∩ L∞(R2,R2) has bounded divergence, then for any compact set K ⊂ R2 and ρ > 0 there exists a Lagrangian flow φ : K× [−ρ, ρ] → R2 starting from K and relative to v, i.e., s → φ(q, s) solves φ(q, s) = q +s
0 v(φ(q, t)) dt for all s ∈ [−ρ, ρ]. The resulting flow is stable with respect to smooth approximations of v in the L1norm.
We apply these results to the convex functions f, g in the graphical patch representation of a convex isoperimetric profile to obtain:
Theorem 8.31. Let Ω⊂ H be a convex isoperimetric profile with mean curvature H. If we locally represent a portion of ∂Ω as a convex graph x3= f (x1, x2) over a convex region A⊂ R2, then for all compact sets K⊂ A\Sf and open neighborhoods of K, K ⊂ O ⊂ A \ Sf, there exists a sufficiently small ρ > 0 and a Lagrangian flow φ : K× [−ρ, ρ] → O relative to v = 1/2(x1, x2)− (−∂x2f, ∂x1f ) such that for a.e. z ∈ K, the curve s → φ(z, s) is an arc of a circle (oriented clockwise) of radius 1/H.
The flow φ is regular in the following sense: there exists a constant λ ≥ 1 such that for all measurable sets A⊂ K and s ∈ [−ρ, ρ] one has
1
λ|A| ≤ |φ(A, s)| ≤ λ|A|. (8.49)
Note that for each z∈ K, the vector v(z) is the projection to C of the horizontal vector
v1X1+ v2X2= 1
2x1+ ∂x2f,1
2x2− ∂x1f,−1
2 [x1∂x1f + x2∂x2f ]
!
which is a.e. tangent to ∂Ω in the graphical patch determined by f . Consequently the Lagrangian flow φ lifts to a geodesic foliation of the coordinate patch, away from characteristic points.
8.6. The convex isoperimetric profile ofH 175
Aside: sketch of the proof of Theorem 8.31. The main problem is to compute the second derivatives of an integral curve of v and interpret the PDE (8.46) pointwise.
This is not trivial because while v admits a regular Lagrangian flow (in view of the prior discussion), it is only in BVloc. On the other hand, we might be tempted to use integral lines of the normalized vector field v/|v| which is in W1,1, and so such curves would be twice differentiable a.e. However one cannot directly define a Lagrangian flow of v/|v| since its divergence is only in L1and not in L∞.
To resolve this problem one first observes that, since v and its normalization are parallel, one can find an integral curve for one by reparametrizing integral curves of the other. In standard fashion, consider a suitable reparametrization of the flow φ,
γ(s) = φ(z, τ (s)) (8.50)
defined so that γ is an integral curve of the vector field v/λ and λ : A → R is a measurable function (to be chosen later) satisfying 0 < c1 ≤ λ ≤ c2, for some constants c1, c2> 0.
At this point, we observe that in compact sets outside of Sf we may set λ = |v|, and recall that the normalized vector field w = v/|v| is in W1,1. To conclude that the integral curves γ of w defined above through a reparametrization have second derivatives a.e., we need to use a special chain rule for the composition of W1,1 vector fields w and curves γ defined as in (8.50), namely:
w◦ γ ∈ W1,1
and d
ds( w◦ γ)(s) = (∇ w ◦ γ)γ(s) a.e.
Because of our choice of w and in view of equation (8.46) we immediately obtain γ=−H(γ) a.e.
From this ODE we immediately deduce the smoothness of γ and the desired result.
An analogous result holds for the graphical patches determined by g.
Step 6. Thanks to the above results and in view of basic extension and uniqueness arguments, every convex isoperimetric profile Ω⊂ H with curvature H has bound-ary ∂Ω foliated by geodesics, which are lifts of circles of radius 1/H and which have one endpoint in Σ+and the other in Σ−. The only thing left to show is that Σ± consist each of a single point (the poles of the bubble set). To prove this we argue by contradiction and assume without loss of generality that Σ− contains a horizontal segment of the form{(0, x2, 0) :|x2| < µ} for some µ > 0.
Because of convexity, and from the definition of characteristic points, one must have
Ω⊂ H 0,µ
2, 0
!!+
∩ H 0,−µ
2, 0
!!+
, (8.51)
176 Chapter 8. The Isoperimetric Profile ofH
where we denote by H(x)+the component ofH\H(x) containing {(0, 0, x3) : x3>
R} for some sufficiently large R > 0. The right-hand side of (8.51) forms a wedge, whose edge is in ∂Ω and contains the origin. An elementary computation shows that any smooth geodesic arc emanating from the origin o must necessarily have a horizontal tangent at o; the continuity of the tangent then contradicts (8.51).
Step 7. The two points Σ± must both lie on the x3 axis, since otherwise there would be only one geodesic arc joining them and we know that any curve in the geodesic foliation joins these two points.
Step 8. In conclusion, ∂Ω is foliated by geodesics, lifts of circles with radius 1/H touching the origin o, with two isolated characteristic points Σ±. It follows that Ω is congruent with a bubble set.