Isoperimetric inequalities in cylinders with density

Given a compact Riemannian manifold with density $M$ without boundary and the real line $\mathbb{R}$ with constant density, we prove that isoperimetric regions of large volume in $M\times\mathbb{R}$ with the product density are slabs of the form $M\times [a,b]$. We previously prove, as a necessary step, the existence of isoperimetric regions in any manifold of density where a subgroup of the group of transformations preserving weighted perimeter and volume acts cocompactly.


Introduction and preliminaries
In recent years, isoperimetric problems have been considered in manifolds with density. One of the most interesting spaces of this type is the Gauss space, the Euclidean space R n with the Gaussian density Ψ(x) := exp(−π|x| 2 ). Borell [6] and Sudakov and Tirel'son [29] independently proved in 1974 and 1975 that half-spaces minimize perimeter under a volume constraint for this density. A new proof was given in 1983 by Ehrhard [10] using symmetrization. In 1997 Bobkov [4] proved a functional version of this isoperimetric inequality, later extended to the sphere and used to prove isoperimetric estimates for the unit cube by Barthe and Maurey [2]. Following [4], Bobkov and Houdré [5] considered "unimodal densities"with finite total measure on the real line. These authors explicitly computed the isoperimetric profile for such densities and found some of the isoperimetric solutions. Gromov [14,15] studied manifolds with density as "metric measure spaces" and mentioned the natural generalization of mean curvature obtained by the first variation of weighted area. Bakry and Ledoux [1] and Bayle [3] proved generalizations of the Lévy-Gromov isoperimetric inequality and other geometric comparisons depending on a lower bound on the generalized Ricci curvature of the manifold. Isoperimetric comparisons results in manifolds with density were considered by Maurmann and Morgan [17]. Existence of isoperimetric sets in R n with density under various hypotheses on the growth of the density were proven by Morgan and Pratelli [23] and Milman [18], see also De Philippis, Franzina and Pratelli [8]. For regularity of isoperimetric regions with density see Sect. 3.10 in paper of Morgan [19] and see also Pratelli and Saracco [24]. Boundedness of isoperimetric regions was studied by Cinti and Pratelli [7] and Pratelli and Saracco [25]. Symmetrization techniques in manifolds with density developed by Ros [27] and Morgan et al. [22].
For nice surveys on manifolds with density the reader is referred to [20,3,21] and the references therein.
Date: November 29, 2021. The author has been supported by MICINN-FEDER grants MTM2010-21206-C02-01, MTM2013-48371-C2-1-P, MEC-Feder grants MTM2017-84851-C2-1-P and PID2020-118180GB-I00, and Junta de Andalucía grants A-FQM-441-UGR18 and P20-00164. Open Access funding: Universidad de Granada / CBUA. In this paper, (M, g, Ψ) will denote a manifold with density without boundary, where g is a Riemannian metric on M and Ψ : M → R is a smooth function. We define the weighted volume of a set by where dM is the Riemannian volume element on (M, g). The weighted area of a smooth hypersurface Σ is defined by where dΣ is the Riemannian area element on Σ.
If E ⊂ M , we define the weighted perimeter of E in the manifold with density (M, g, Ψ) by where X ∞ 0 (M ) is the set of smooth vector fields on M with compact support and (1.4) div Ψ (X) = div(e Ψ X), and div is the Riemannian divergence in (M, g). If E has smooth boundary Σ, then P Ψ (E) = area Ψ (Σ), see [23]. Given a manifold with density, we shall denote by Isom(M, g, Ψ) the group of isometries of (M, g) preserving the function Ψ (i.e., maps f : M → M such that Ψ • f = Ψ). Such isometries preserve the weighted area and volume.
The isoperimetric profile of (M, g, Ψ) is the function I : [0, +∞) → R + defined by ). This means that E minimizes the weighted perimeter under a weighted volume constraint. Regularity of isoperimetric sets was considered by Morgan and Pratelli [23].
Given a manifold with density (M, g, Ψ), we shall consider the cylinders with density (M × R k , g × g 0 , Ψ × 1), where R k is k-dimensional Euclidean space with its standard Riemannian metric g 0 , and (Ψ×1)(p, x) = Ψ(p) for every (p, x) ∈ M ×R k .
In case M is a compact manifold, the quotient of M × R k by Isom(M × R k , g × g 0 , Ψ × 1) is the compact base M of the product. We focus in this paper in the case k = 1.
The aim of this paper is to prove that isoperimetric sets in (M × R, g × g 0 , Ψ × 1) are slabs of the form M × [a, b], where a, b ∈ R, a < b. This result is proven in Theorem 3.3 in Section 3. As a necessary previous step in our proof we must show existence of isoperimetric regions in manifolds with density such that the action of Isom(M, g, Ψ) is cocompact, that is, the quotient (M, g, Ψ)/ Isom(M, g, Ψ) is compact, like in the case of the cylinders considered in Section 3. The proof of existence is based on Galli and Ritoré's in contact sub-Riemannian manifolds, see [12]. Since this proof has now become standard, we check in Section 2 that the main ingredients are available: a relative isoperimetric inequality for uniform radii, see Theorem 2.3; the doubling property, see Theorem 2.4; and a deformation result for sets of finite perimeter, see Theorem 2.5. In Theorem 3.3 we characterize the isoperimetric regions of large volume in a cylinder with density M × R, where M is a compact Riemannian manifold with density, the real line R is endowed with a constant density, and the product with the product density. In the non-weighted Riemannian case Duzaar and Steffen [9] proved that in M × R isoperimetric sets of large volume are of the form M × [a, b], where a, b ∈ R. For higher dimensional Euclidean factors the problem was considered in [26], where the authors proved that the isoperimetric solutions of large volume in the Riemannian product M × R k are of the form M × B(x, r), where B(x, r) is an Euclidean ball, see also [13].
For more results about variational problems in cylinders the reader is referred to [11] and the references in [28].

Existence of isoperimetric regions in M
In this Section we prove the existence of isoperimetric sets, for any volume, in a manifold with density (M, g, Ψ) such that Isom(M, g, Ψ) acts cocompactly. The scheme of proof devised by Galli and Ritoré in [12] applies to our situation, provided we are able to show that • There exists r 0 > 0 such that a relative isoperimetric inequality holds in all balls B(p, r), with p ∈ M , 0 < r r 0 , with a uniform constant. • The manifold is doubling. This means the existence of r 0 > 0 and a uniform constant C D > 0 such that vol Ψ (B(p, 2r)) C D vol Ψ (B(p, r)) for all p ∈ M and 0 < r r 0 . • A deformation result for finite perimeter sets, see Theorem 2.5, holds in (M, g, Ψ).
Assuming these results hold in (M, g, Ψ), and using the well-known techniques in [12] we have the following result Theorem 2.1. In a manifold with density (M, g, Ψ) such that Isom(M, g, Ψ) acts cocompactly, isoperimetric sets exist for any given volume.
To prove the required ingredients needed for Theorem 2.1 we start with a preliminary result. We recall that the convexity radius conv(K) of a subset K of a Riemannian manifold M is the infimum of positive numbers r such that the geodesic open ball B(p, r) is convex for every p ∈ K. We call d the Riemannian distance in (M, g).
for any p ∈ K and x, y ∈ B(0, r 0 ) ⊂ T p M .
Proof. Given p ∈ K, consider a compact coordinate neighborhood U around p with a global orthonormal basis defined on U . Given q ∈ U , let g q ij be the components of the Riemann tensor in the coordinate neighborhood defined by the exponential map exp q : B(0, r 0 ) ⊂ T q M → B(q, r 0 ) and the global orthornormal basis. The functions g q ij depend smoothly on q. We define It is easy to check that α U (r) is decreasing, β U (r) is increasing, and that Given q ∈ U , we take x, y ∈ B(0, r 0 ). To compute the distance d between the points exp q (x), exp q (y), it is enough to consider curves inside the convex ball B(0, r). Let γ : I → B(0, r 0 ) be a curve joining x and y. Then Observe that The left quantity is larger than or equal to α U (r) |x − y|. Since γ is an arbitrary curve joining x and y, this implies The result follows by covering the compact set K by a finite number of coordinate neighborhoods U , and taking α(r) as the minimum of the α U (r) and β(r) as the maximum of the β U (r). Setting λ(r) = α(r) − 1, Λ(r) = β(r) − 1 the result follows.
Using Lemma 2.2 we are obtain as corollaries the existence of a uniform relative isoperimetric inequality and the existence of a doubling constant. (M, g, Ψ)). Let (M, g, Ψ) be an n-dimensional manifold with density and K ⊂ M a compact subset. Let r 0 > 0 be the radius obtained in Lemma 2.2. Then for all p ∈ K and E ⊂ B(p, r) with 0 < r r 0 , there exists a positive constant C > 0 not depending on p, such that
Proof. By Lemma 2.2, for 0 < r r 0 , the exponential map f = exp p : B(0, r) → B(p, r) is a diffeomorphism. Let F = f −1 (E). Then where H n is the n-dimensional Hausdorff measure in (M, g). Consider positive functions constants a, b > 0 so that a e Ψ b in B(p, r 0 ) for all p ∈ K. So we have As f is Lipschitz, Lemma 2.2 in dimension n implies (2.9) a(1+λ(r)) n H n 0 (F ) a·H n (E) vol Ψ (E) b·H n (E) b(1+Λ(r)) n H n 0 (F ), where H n 0 is the n-dimensional Hausdorff measure with respect to the Euclidean metric.
On the other hand, by § 2 in [23], where ∂ ⋆ E is the reduced boundary of E. And therefore, where P 0 is the Euclidean perimeter. By a similar computation we obtain P Ψ (E, B(p, r)) a(1 + λ(r)) n−1 P 0 (F, B(0, r)).

Theorem 2.5 (Deformation of finite perimeter sets)
. Let E ⊂ M be a set of locally finite weighted perimeter. Assume that P Ψ (E, B(p, r)) > 0 for some p ∈ M and r > 0. Then there exists a deformation {E t } t∈(−δ,δ) of E, with E 0 = E, by sets of locally finite perimeter, and a constant C = C(p, r, δ) such that Proof. Since E is a set of locally finite perimeter and P Ψ (E, B (p, r)) > 0 there exists a vector field X, with X 1, such that X ∈ X ∞ 0 (B (p, r)) and E div Ψ (X) dM > 0. Let {ϕ s } s∈R be the flow associated to X. Since the map is differentiable and its derivative at s = 0 is E div Ψ (X)dM > 0, we can apply the inverse function theorem to find δ > 0 and a function g : (−δ, δ) → R such that g(0) = 0 and vol Ψ (ϕ g(t) (E)) = vol Ψ (E) + t. Let E t = ϕ g(t) (E). This proves (1). If necessary we can reduce δ so that, for |t| δ we have As ϕ t (q) = q for all q ∈ B(p, r) and t ∈ R we trivially have E△E t ⊂ B(p, r) for all t ∈ R. This proves (2).
To prove (3) note that, for all t ∈ (−δ, δ), we have e Ψ e Ψ |Jac(ϕ g(t) )|dH n Hence we have Taking into account (2.11) we have, for t ∈ (−δ, δ)\{0}, and C is a positive constant which depends on p, r and δ. note that (3) is trivially true for t = 0 and any constant C. So we have |P Ψ (E t , B(p, r)) − P Ψ (E, B(p, r))| C |vol Ψ (E t ∩ B(p, r)) − vol Ψ (E ∩ B(p, r))| , and this fact together with (2) implies the inequality of the left side of (3). Note that (3) is trivially true for t = 0 and any constant C.
On the other hand, for any positive measure µ we have This completes the proof of (3).
To conclude this section, we sketch the proof of Theorem 2.1. Since this proof has become standard after [12], we include some basic guidelines for reader's convenience.
Proof of Theorem 2.1. Using the relative isoperimetric inequality (2.6), the doubling property (2.10) and the hypothesis that Isom(M, g, Ψ) acts cocompactly on M , an isoperimetric inequality for small volumes can be obtained in a standard way, see Lemma 3.10 in [12]. Combining the latter with the deformation property of finite perimeter sets proven in Theorem 2.5, and using again that Isom(M, g, Ψ) acts cocompactly, we can prove that the isoperimetric solutions are bounded, see Lemma 4.6 in [12]. The Structure Theorem for minimizing sequences of sets of positive volume v > 0, see Proposition 5.1 in [12], works also in our case without modification. From them the Concentration Lemma 6.2 in [12], and the Existence Theorem 6.1 in [12] work without relevant modifications.

Isoperimetric regions in M × R
In this section we prove existence of isoperimetric regions in a cylinder with density for large volumes. We shall need the following preliminary results in the proof of Theorem 3.3. Proof. It follows easily since the isoperimetric profile of (M, g, Ψ) is strictly positive and asymptotic to the function t → t (n−1)/n for t > 0 small.
is either empty or a vertical segment centered at (p, 0) for all p ∈ M . Notice that a normalized set is invariant by the reflection σ : M × R → M × R defined by σ(p, t) = (p, −t), an isometry of (M × R, g × g 0 ) preserving the weighted volume. Given any set E ⊂ N , we denote by E * the projection of E over M . From now on we denote (E t ) * by E * t to simplify the notation. Notice that for normalized sets one has E * t ⊂ E * s whenever |s| |t|. We denote by P and vol the perimeter and volume in the manifold with density (M × R, g × g 0 , Ψ × 1). Proof. For every t ∈ R we define M t = M × {t} and E t = E ∩ M t . As E is normalized we can choose τ 0 so that vol Ψ (E * t ) vol Ψ (M )/2 for all t τ and vol Ψ (E * t ) > vol Ψ (M )/2 for all t ∈ [0, τ ) if τ > 0. Let us consider first the case τ > 0. We apply the coarea formula and Lemma 3.1 (1) to obtain
In particular, taking limits when t → τ − and using that y(t) is non-decreasing Hence τ n vol Ψ (M ) 1/n c 2 and so (3.4) The last inequality follows 2 vol Ψ (E * 0 ) P (E), which holds since E is normalized and so P (E) is the sum of a lateral area that projects to some set of weighted measure zero on M and the area of the graphs of two C 1 functions u and −u over some set Ω ⊂ M of full measure in E * . So we have Hence (3.1) follows from (3.2) and (3.4). It remains to consider the case τ = 0. In this case, equation (3.2) alone implies the linear isoperimetric inequality (3.1) since vol Ψ (E ∩ (M × [0, +∞))) = 1 2 vol Ψ (E) as E is normalized. where |E p | is the 1-dimensional Lebesgue measure of E p , the volume is preserved when we pass to the Steiner symmetrization of E. To see that (3.6) P (sym(E)) P (E) we consider a function u : Ω ⊂ M → R and the graph G(u) of u and we observe that the weighted area of G(u) is given by area Ψ (G(u)) = Ω e Ψ 1 + |∇u| 2 dM.
So we can reason as in the proof of the Euclidean case to verify (3.6), see again [16, § 14.1]. Equality holds if and only if E = sym(E). If E is an isoperimetric region then also sym(E) is isoperimetric and, moreover, sym(E) * = E * . So from now on we assume that E is normalized replacing E by sym(E) if necessary. If vol Ψ (M \ E * ) > 0 then Lemma 3.2 provides a constant c > 0 independent of vol(E) so that P (E) c vol(E). But this in contradiction to (3. Since P (E) 2 vol Ψ (M ) we should have equality in the above inequality, that implies ∇u = ∇v = 0 and so E is a slab. This completes the proof of the Theorem.
Remark 3.4. Note that Theorem 3.3 does not hold when there is a non-trivial density in the vertical factor. For example, in (S n × R, g × g 0 , 1 × e −t 2 /2 ) the only isoperimetric regions are of the type S n × (−∞, a) or S n × (a, ∞). See Example 4.6 in [28].