On the extremal function of the modulus of a foliation

We investigate the properties of the modulus of a foliation on a Riemannian manifold. We give necessary and sufficient conditions for the existence of the extremal function and state some of its properties. We obtain an integral formula which, in a sense, combines the integral over the manifold with the integral over the leaves. We state a relation between the extremal function and the geometry of the distribution orthogonal to a foliation.

One can show that the q-capacity of a condenser (where q is the coefficient conjugate to p) equals the q-modulus of the family of curves joining plates of the condenser and is the reciprocal to the p-modulus of the hypersurfaces separating the plates [9,13]. Among these surfaces, there is the "smallest" family of surfaces which realizes the modulus. It is given by the level sets of the function realizing the q-capacity. Thus this family forms a foliation on the condenser. Moreover, there is a function, called extremal, which realizes this modulus, and the integral of the function on each leaf of the obtained foliation is equal to one [13].
In this short article, we consider a general case of a foliation on a Riemannian manifold. We study properties of the extremal function for the p-modulus of a foliation. The majority of results in this paper is obtained under the assumption of existence of the extremal function. We give necessary and sufficient conditions for existence of this function (Theorem 3). In particular, we consider foliations given by the level sets of a submersion (Corollary 6). Moreover, we state some properties of the extremal function.
The existence of the extremal function allows one to define a function The main result is the following integral formula (Theorem 9) where f 0 is the extremal function for the p-modulus and ϕ ∈ L p (M ). Moreover, we show that (1) implies that f 0 is extremal. Using (1), we obtain some results concerning the geometry of a foliation. Namely, the tangent gradient of the extremal function is related to the mean curvature of distribution orthogonal to F. In the last section we give some examples. We consider a foliation by circles on a torus in R 3 , a foliation given by the distance function and a foliation by spheres in a ring in R n .
Notice that all results can be generalized to the case of foliations equipped not necessarily with Lebesgue measures but arbitrary Borel measures. Moreover, the "global" Lebesgue measure μ M could be replaced by any Borel measure. This follows from the original definition of p-modulus by Fuglede [8].
In particular, we could consider foliation on a smooth metric measure space as it was pointed out by the anonymous referee. We focus only on the case of Lebesgue measures, since the aim of the author's PhD thesis [11], on which this article is based, was to establish the relation between extrinsic and intrinsic geometry of a foliation and p-modulus.
respectively. Let L p (M ) denote the space of all measurable and p-integrable functions on M (with the norm · p ).

Lemma 1. Let A ⊂ M be a measurable set. Then μ M (A) = 0 if and only if
In particular, if f ∈ L p (M ) is nonnegative, then the integral L f dμ L exists for almost every leaf L ∈ F.
Proof. Let U = {U } be a (countable) cover of M such that for each U ∈ U there is a submersion F U : U → N U with bounded Jacobian such that F defines a foliation F on U . By Fubini's theorem (compare (2)) there are constants c 1 , c 1 (depending on U ) such that Let L be an arbitrary subfamily of F. In the space L p (M ) consider the family adm p (L) of all nonnegative functions f such that L f dμ L ≥ 1 for almost every L ∈ L. Functions belonging to this family are called admissible. The p-modulus of L is defined as follows We say that f 0 ∈ adm p (L) is extremal for the p-modulus of the family L if mod p (L) = f 0 p . We will show later (see the considerations preceding Corollary 11) that an extremal function for the p-modulus of a foliation (i.e L = F) is unique up to sets of measure zero.
For a family L ⊂ F denote the union of the leaves L ∈ L by L. In particular, F = M . Proposition 2 [8]. The modulus has the following properties.

Existence and properties of the extremal function.
Let M be a Riemannian manifold, F a foliation on M .

Theorem 3. There is an extremal function f 0 for the p-modulus of F if and only if for any subfamily
. Indeed, any closed and convex set in a Banach space contains an element of smallest norm and, as can be easily shown, adm p (F) is convex. Take a sequence (f n ) of admissible functions converging to f . Then f ∈ L p (M ) and f ≥ 0. By Proposition 2 there is a subsequence (f ij ) and a subfamily L such that mod p (L) = 0 and Proof. Function f = 1 is admissible for F ∞ and satisfies the condition 3. of Proposition 2 (for F ∞ ). Hence mod p (F ∞ ) = 0. Thus, by Theorem 3, μ M (M ∞ ) = 0.
We will specify conditions for the existence of an extremal function in the case of a foliation given by the level sets of a submersion.
Let Φ : M → N be a submersion between Riemannian manifolds. Then there is the decomposition Denoting its dual as Φ * : T N → H Φ , we define the Jacobian JΦ of Φ as follows We will need the following version of Fubini's theorem [4] Vol. 107 (2016) On the extremal function 93 for any nonnegative and measurable function f . Thus, by Hölder's inequality, we have

Proposition 5. Assume a foliation F is given by a submersion
By (3) and Fubini's theorem (2), Contradiction ends the proof.
Then f is admissible, f ≤ f 0 , and f p ≤ f 0 p . Since f 0 is extremal, we have f p = f 0 p . Therefore f = f 0 , so L f 0 dμ L = 1 for almost every leaf L ∈ F.
(2) Suppose there is a set A of positive and finite measure such that f 0 = 0 on A. Put By Lemma 1 μ M (A 3 ) = 0. Hence we may assume A = A 1 ∪A 2 . Let us consider two cases: (i) μ M (A 2 ) = 0. Then we may assume that A = A 1 . Denote by B the set of all points x ∈ M \A such that the leaf L x through x meets A on the set of positive measure. For a fixed 0 ≤ t ≤ 1 put .
Then f ≥ 0 and L fdμ L = 1 for almost every leaf L ∈ F. Moreover, Hence f p < f 0 p if and only if the function .
Then f ≥ 0 and L fdμ L = ∞ if μ M (A 2 ∩ L) > 0 and L fdμ L = 1 for remaining leaves. Moreover, is negative at some t. This is clear by choosing ε and t small enough (again, as before, f 0 | B p > 0). Remark 1. Proposition 7(1) was first established in [3] but obtained under the assumption of continuity of the extremal function. Proof. It follows immediately by Theorem 3 and Proposition 2(4).
By the above Lemma, for any ϕ ∈ L p (M ) the following function is well-defined.

Theorem 9.
There is the following integral formula for any function ϕ ∈ L p (M ).
Proof. Firstly, notice that in the definition of the p-modulus we may consider admissible functions which are not necessarily non-negative. Indeed, if f ∈ L p (M ) andf ≥ 1, then its non-negative part f + is admissible, sincef Then α ∈ L p (M ) andα = 0. Consider the following one parameter family of functions Since f 0 and α are both p-integrable, it follows that f t is also p-integrable for all t. Moreover, by Proposition 7,f t = 1. Thus f t is admissible. Fix ε ∈ (0, 1) and consider a set Since M ε → M almost everywhere as ε → 0, then, by admissibility of f t , 0 ≤ for ε sufficiently small. Notice that the function θ is measurable, as f t , f 0 , and ϕ are measurable. By the inequality Since f 0 p = g 0 p , we get f 0 = g 0 . As a next corollary we obtain the formula, firstly proven in [10] using a different approach, for p-modulus and the extremal function f 0 of a foliation given by the level sets of a submersion. Therefore formula (4) holds with f 0 = JΦ q−1 JΦ q−1 . Thus, by Corollary 10, f 0 is extremal for the p-modulus of F. Remark 3. It seems that we can derive the formula for the extremal function in a more general case. This follows from the fact that the formula (5) does not depend on a choice of the Riemannian metric on N .
Assume that the holonomy of F is finite. The normal bundle to F can be treated as the quotient T M/T F. Moreover, we have a natural projection Φ * : T M → T M/T F. Since the holonomy is finite, there is a holonomy invariant metric on T M/T F. Thus we may speak about JΦ.
If additionally the leaf space M/F is Hausdorff, then it has a structure of an orbifold and the natural projection Φ : M → M/F is a morphism of orbifolds [14]. Up to sets of measure zero, Φ is smooth, and we may consider the differential Φ * and, again, define JΦ.