Large isoperimetric regions in the product of a compact manifold with Euclidean space

Given a compact Riemannian manifold $M$ without boundary, we show that large isoperimetric regions in $M\times\mathbb{R}^k$ are tubular neighborhoods of $M\times\{x\}$, with $x\in\mathbb{R}^k$.


INTRODUCTION
We consider the isoperimetric problem of minimizing perimeter under a given volume constraint in N = M × k , where k is the k-dimensional Euclidean space and M is an m-dimensional compact Riemannian manifold without boundary. Our main result is the following: This result, in case k = 1, was first proven by Duzaar and Steffen [4,Prop. 2.11]. As observed by Morgan, an alternative proof for k = 1 can be given using the monotonicity formula and properties of the isoperimetric profile of M × (see [20,Cor. 4.12] for a proof when M is a convex body). Gonzalo considered the general problem in his Ph.D. Thesis [9]. In 1 × k , the result follows from the classification of isoperimetric regions by Pedrosa and Ritoré [19]. Large isoperimetric regions in asymptotically flat manifolds have been recently characterized by Eichmair and Metzger [5]. It is worth mentioning that W.-T. Hsiang and W.-Y. Hsiang [12] completely solved the isoperimetric problem in products of Euclidean and hyperbolic spaces. Morgan [16], after Barthé [1], using results by Ros [22], provides a lower bound of the isoperimetric profile of a Riemannian product in terms of concave lower bounds of the isoperimetric profiles of the factors.
In our proof we use symmetrization and show in Corollary 2.2 that anisotropic scaling of symmetrized isoperimetric regions of large volume L 1 -converge to a tubular neighborhood of M × {0}. This convergence is improved in Lemma 2.4 to Hausdorff convergence of the boundaries using the density estimates on tubes from Lemma 2.3, similar to the ones obtained by Ritoré and Vernadakis [21]. Results of White [23] and Grosse-Brauckmann [11] on stable submanifolds then imply that the scaled boundaries are cylinders, see Theorem 3.2. For small dimensions, it is also possible to use a result by Morgan and Ros [18] to get the same conclusion only using L 1 -convergence. Once it is shown that the symmetrized set is a tube, it is not difficult to prove that the original isoperimetric region is also a tube.
After the distribution of this manuscript, Gonzalo informed us that he had obtained a proof of Theorem 1.1 in [10]. His techniques are different from ours and similar to the ones used in [9].
Given a measurable set E ⊂ N , their perimeter and volume will be denoted by P(E) and |E|, respectively. We refer the reader to Maggi's book [14] for background on finite perimeter sets. The r-dimensional Hausdorff measure of a set E will be denoted by H r (E).
On M × k we shall consider the anisotropic dilation of ratio t > 0 defined by Since the Jacobian of the map ϕ t is t k , we have At a regular point p ∈ Σ, the unit normal ξ can be decomposed as ξ = av + bw, with a 2 + b 2 = 1, v tangent to M and w tangent to k . Then the Jacobian of ϕ t |Σ is equal to t k−1 (t 2 a 2 + b 2 ) 1/2 . For t 1 we get and the reversed inequalities when t 1. Similar properties hold for the perimeter. Equality holds in the right hand side of (1.2) if and only if a = 0, or equivalently if and only if ξ is tangent to k .
An open ball in k of radius r > 0 and center x will be denoted by D(x, r). If it is centered at the origin, we set D(r) = D(0, r). We shall also denote by T (x, r) the set M × D(x, r), and by T (r) the set M × D(r). Observe that ϕ t (T (x, r)) = T (t x, t r) and that T (x, r) is the tubular neighborhood of radius r > 0 of M × {x}.
Given any set E ⊂ N of finite perimeter, we can replace it by a normalized set sym E by requiring sym (1) |sym E| = |E|, (2) P(sym E) P(E).
The proof of Theorem 1.2 is similar to the one of symmetrization in n = m × k with respect to one of the factors, see Burago and Zalgaller [2, § 9] (or Maggi [14] for the case m = 1). The main ingredients are a corresponding inequality for the Minkowski content and approximation of finite perimeter sets by sets with smooth boundary.
Given E ⊂ N , we denote by E * its orthogonal projection onto M . If E is normalized, and u : E * → + measures the radius of the disk obtained projecting E ∩ ({p} × k ) to k , we get, assuming enough regularity on u, that where ω k = H k (D(1)), and kω k = H k−1 ( k−1 ). The above formulas imply The isoperimetric profile of M × k is the function I : (0, +∞) → [0, +∞) defined by An isoperimetric region is a set E ⊂ M × k satisfying I (|E|) = P(E). Existence of isoperimetric regions in M × k is guaranteed by a result of Morgan [17, p. 129], since the quotient of M × k by its isometry group is compact. From his arguments, it also follows that isoperimetric regions are bounded in M × k (see also [7]). From (1.3) we get for any v > 0. The regularity of isoperimetric regions in Riemannian manifolds is wellknown, see Morgan [15] and Gonzales-Massari-Tamanini [8]. The boundary is regular except for a singular set of vanishing H n−7 measure. The following properties of the isoperimetric profile hold Proposition 1.3. The isoperimetric profile I of M × k is non-decreasing and continuous.
This shows that I is non-decreasing. Let us prove now the right-continuity of I at v. Consider an isoperimetric region E of volume v. Take a smooth vector field Z with support in the regular part of the bound- Using the Inverse Function Theorem we obtain a smooth family by the monotonicity of I .
To prove the left-continuity of I at v we take a sequence of isoperimetric regions E i with v i = |E i | ↑ v and we consider balls by the monotonicity of I , and the left-continuity follows by taking limits since lim i→∞ P(B i ) = 0.
We shall also use the following well-known isoperimetric inequalities in M and M × k Lemma 1.5 follows from the facts that I (v) is strictly positive for v > 0 and asymptotic to the Euclidean isoperimetric profile when v approaches 0.

LARGE ISOPERIMETRIC REGIONS IN M × k
In this Section we shall prove that normalized isoperimetric regions of large volume, when scaled down to have constant volume v 0 , have their boundaries uniformly close to the boundary of the normalized tube of volume v 0 .
If E ⊂ N is any finite perimeter set and T (E) is the tube with the same volume as E, Let t > 0, and Ω = ϕ t (E). Since ϕ t (E + ) = Ω + , (1.1) implies A similar equality holds replacing E + by E − .
Now we claim that To prove (2.4) we argue by contradiction.
, a contradiction that proves the claim. Hence there exists w ∈ (0, H m (M )) so that and we obtain This last step to go from the particular r i to every s r i is easy to check as, for any The above arguments imply, replacing the original sequence by a subsequence, that Let a = a(w) be the constant in Lemma 1.4. For the elements of the subsequence satisfying (2.7) we have thus proving the result. In the previous inequalities we have used the coarea formula for the distance function to Then Ω i → T in the L 1 -topology, where T is the tube of volume v 0 .
Proof. Regularity results for isoperimetric regions imply that P(E i ) = H n−1 (∂ E i ), choosing as representative of every isoperimetric set the closure of the set of density one points. If Ω i does not converge to T in the L 1 -topology then, using (2.2) in Proposition 2.1 and (1.4), we get for a subsequence, thus yielding a contradiction by letting i → ∞ since |E i | → ∞.
Using density estimates, we shall show now that the L 1 convergence of the scaled isoperimetric regions can be improved to Hausdorff convergence.
In a similar way to Leonardi and Rigot [13, p. 18] (see also [21] and David and Semmes [3]), given E ⊂ N , we define a function h : k × (0, +∞) → + by for x ∈ k and R > 0. We remark that the quantity h(x, R) is not homogeneous in the sense of being invariant by scaling since h(x, R) 1 2 (kω k H m (M )) R k−n , which goes to infinity when R goes to 0. When the set E should be explicitly mentioned, we shall write h(E, x, R) = h(x, R). Define

Lemma 2.3. Let E ⊂ N be an isoperimetric region of volume
The function m(r) is non-decreasing and, for r R 1, we get By the coarea formula, when m ′ (r) exists, we get T (x, r)).
We improve now the L 1 -convergence of normalized isoperimetric regions obtained in Corollary 2.2 to Hausdorff convergence of their boundaries Then for every r > 0, ∂ Ω i ⊂ (∂ T ) r , for large enough i ∈ , where T is the tube of volume v 0 .
As Ω i → T in L 1 (N ) by Corollary 2.2, we can choose a sequence r i → 0 so that Now fix some 0 < r < 1. We reason by contradiction assuming that, for some subsequence, there exist We distinguish two cases. First case: x i ∈ N \ T , for a subsequence. Choosing i large enough, (2.17) implies T (x i , r i ) ∩ T = and (2.16) yields By Lemma 2.3, we conclude that |Ω i ∩ T (x i , r i /2)| = 0, a contradiction. Second case: x i ∈ T . Choosing i large enough, (2.17) implies T (x i , r i ) ⊂ T and so Then, by (2.16), we get So, for i large enough, we get By Lemma 2.3, we conclude that |T (x i , r i /2) \ Ω i | = 0, and we get again contradiction that proves the Lemma.

STRICT O(k)-STABILITY OF TUBES WITH LARGE RADIUS
In this Section we consider the orthogonal group O(k) acting on the product M × k through the second factor.
Let Σ ⊂ M × k be a compact hypersurface with constant mean curvature. It is wellknown that Σ is a critical point of the area functional under volume-preserving deformations, and that Σ is a second order minimum of the area under volume-preserving variations if and only if for any smooth function u : Σ → with mean zero on Σ. In the above formula ∇ is the gradient on Σ and q is the function Ric(ξ, ξ) + |σ| 2 , where |σ| 2 is the sum of the squared principal curvatures in Σ, ξ is a unit vector field normal to Σ, and Ric is the Ricci curvature on N .
A hypersurface satisfying (3.1) is usually called stable and condition (3.1) is referred to as stability condition. In case Σ is O(k)-invariant we can consider an equivariant stability condition: we shall say that Σ is strictly O(k)-stable if there exists a positive constant λ > 0 such that We consider now the tube T (r) = M × D(r). The boundary of T (r) is the O(k)invariant cylinder Σ(r) = M × ∂ D(r), with (k − 1) principal curvatures equal to 1/r. Hence its mean curvature is equal to (k − 1)/r and the squared norm of the second fundamental form satisfies |σ| 2 = (k − 1)/r 2 . The inner unit normal to Σ(r) is the normal to ∂ D(r) in k (it is tangent to the factor k ). This implies Ric(ξ, ξ) = 0.
We have the following result This proves the Lemma.
Using results by White [23] and Grosse-Brauckmann [11] we get Proof. Since Σ is strictly O(k)-stable, Grosse-Brauckmann [11, Lemma 5] implies that, for some C > 0, Σ has strictly positive second variation for the functional in the sense that the second variation of F C in the normal direction of a function u satisfies for any smooth O(k)-invariant function u (see the discussion in the proof of Theorem 2 in Morgan and Ros [18]). In White's proof of Theorem 3 in [23] it is observed that a sequence of minimizers of F C in tubular neighborhoods of radius 1/i of Σ are almost minimizing, and hence C 1,α submanifolds that converge Hölder differentiably to Σ, contradicting the positivity of the second variation of Σ. Theorem 1.2 implies that the symmetrization of these minimizers are again minimizers. Thus we get a family of O(k)-minimizers of F C converging Hölder differentiably to Σ, thus contradicting the strict O(k)-stability of Σ.

PROOF OF THEOREM 1.1
First we claim that there exists v 0 > 0 such that, for any isoperimetric region E of volume |E| v 0 , the set sym E is a tube.
To prove this, consider a sequence of isoperimetric regions {E i } i∈ with lim i→∞ |E i | = ∞. We know that {sym E i } i∈ are also isoperimetric regions. Let T = M × D be a strictly O(k)-stable tube, that exists by Lemma 3.1. For large i, we scale down the sets sym E i so that Ω i = ϕ −1 t i (sym E i ) has the same volume as T . As sym E i is isoperimetric and t i > 1, we get from (1.4) and (1.2) that P(Ω i ) P(T ). By Corollary 2.2, the sets {∂ Ω i } i∈ converge to ∂ T in Hausdorff distance. By Theorem 3.2, Ω i = T for large i and so sym E i is a tube. This proves the claim. In particular, H m (E ∩ ({p} × k )) = H m (D) for any p ∈ M .
Hence the isoperimetric profile satisfies I (v) = C v (k−1)/k for the constant C in (1.3)