A geometric capacitary inequality for sub-static manifolds with harmonic potentials

In this paper, we prove that associated with a sub-static asymptotically flat manifold endowed with a harmonic potential there is a one-parameter family $\{F_{\beta}\}$ of functions which are monotone along the level-set flow of the potential. Such monotonicity holds up to the optimal threshold $\beta=\frac{n-2}{n-1}$ and allows us to prove a geometric capacitary inequality where the capacity of the horizon plays the same role as the ADM mass in the celebrated Riemannian Penrose Inequality.


Introduction
In this paper, the object under investigation is a triple (M, g 0 , u) satisfying the following two conditions: (a) (M, g 0 ) is a smooth, connected, noncompact, complete, asymptotically flat, n-dimensional Riemannian manifold, with n ≥ 3, with one end, and with nonempty smooth compact boundary ∂M , which is a priori allowed to have several connected components.
where Ric g0 , D g0 and ∆ g0 are the Ricci tensor, the Levi-Civita connection, and the Laplace operator of the metric g 0 , respectively.
If the equality holds in the first equation of (1), the triple (M, g 0 , u) is said static. For clarity, we recall the definition to which we refer for asymptotically flat manifolds. Definition 1.1. A smooth, connected, noncompact, n-dimensional Riemannian manifold (with or without compact boundary) (N, h), with n ≥ 3, is said to be asymptotically flat if there exists a compact subset K ⊂ N such that N \ K is a finite disjoint union of ends N k , each of which is diffeomorphic to R n minus a closed ball by a coordinate chart ψ k , through which, if h := (ψ k ) * h = h ij dx i ⊗ dx j , we have for some p > (n − 2)/2. Here, δ is the Kronecker delta, and the coordinate charts ψ k are called charts at infinity.
Throughout the paper, we will refer to a triple (M, g 0 , u) that satisfies conditions (a) and (b) as to a sub-static harmonic triple. A fundamental sub-static harmonic triple is the so called Schwarzschild solution, which is given by It is well-known that both the metric g 0 and the potential u, which a priori are well defined only inM , extend smoothly up to the boundary and (M, g 0 ) is called (spatial) Schwarzschild manifold. The parameter m > 0 is the ADM mass m ADM of the Schwarzschild manifold. We refer the reader to Section 5 for the definition of the m ADM associated with a general asymptotically flat manifold. Here, we limit ourselves to recall that the decay conditions (2)- (5) guarantee that m ADM is a geometric invariant ( [4], [7]).
Associated with a sub-static harmonic triple, specifically with the potential u ranging in [0, 1), let us consider the following family of functions depending on the parameter β ≥ 0: {u=t} |Du| β+1 dσ.
In [2] it was proven that if (M, g 0 , u) is a static triple, then, for every β ≥ 2, the function V β is strictly nonincreasig unless (M, g 0 , u) is the Schwarzschild solution. The main purpose of this paper is to extend this result to the sub-static case and to the optimal threshold β = n−2 n−1 . This is the content of Theorem 3.1, where the monotonicity of the above family -equipped with a corresponding rigidity statement -is expressed in terms of the functions F β (τ ), where τ = 1+t 2 1−t 2 ≥ 1, to be consistent with [3] and in light of the more advanced analysis contained therein. This generalisation suggests that our approach is robust enough and likely to be exported to other contexts. In a similar way, S. Brendle shows in [6] how some structure conditions for the metric are sufficient to prove an Alexandrov-type theorem and how such structure generalises to the sub-static case.
Let us now be slightly more detailed on how our Theorem 3.1 is proved. We adopt the main strategy proposed in [2], which essentially consists in obtaining the monotonicity as a consequence of a fundamental integral identity derived in a suitable conformally-related setting (see Proposition 4.3). A delicate point is justifying such identity in a region where critical points of the potential are present. One of the main differences with [2] is that, whereas in the static case the analyticity of the potential guaranteed the local finiteness of the singular values, which made the argument simpler in many occurrences, in the present sub-static setting the metric and in turn the potential are not a priori analytic. Nevertheless, standard measure properties of the critical set of harmonic functions (summarised in Theorem 2.3) are enough to obtain the fundamental integral identity, which in turn implies the monotonicity of F β and, coupled with Sard's Theorem, also its differentiability. Observe that the difficulty in treating the critical points under the threshold β = 1 can be read off directly from formulae (36) and (52), the first one displaying the derivative of F β and the second one expressing the mean curvature on a equipotential set in terms of the Hessian of the potential itself. In fact, calling Φ β the conformal version of F β and looking at formula (70) containing the equivalent characterisation of Φ ′ β derived from the integral identity (63), one realises that problems arise already when β < 2. Let us stress that the monotonicity is obtained from the nonnegativity of the right-hand side of our fundamental integral identity. It is above the threshold β = n−2 n−1 that this is guaranteed, thanks to the Refined Kato Inequality for harmonic functions. The optimality of such inequality reflects a corresponding optimality of β = n−2 n−1 in our result. Moreover, let us remark that the (nonnegative) right-hand side of (63) is obtained as the divergence a suitable modification of a specific vector filed with nonnegative divergence (see (64)), in the limit of a vanishing neighbourhood of the critical set. The crucial point in the construction is to maintain the divergence of the modified vector field nonnegative. It would be interesting to see whether a similar construction can be performed for other families of metrics, including special solutions as rigid case. A straightforward application of the monotonicity of F β is comparing F β (1) with F β (+∞), in turn yielding a "capacitary version" of the Riemannian-Penrose inequality (Theorem 1.1 below). The capacity comes naturally into play when computing F β (1) and F β (+∞), the latter value via the asymptotic expansions of the metric and of the potential. We recall that the capacity Cap(∂M, g 0 ) of ∂M is defined as Throughout the paper, we will use the short-hand notation C for the capacity. Comparing (6) with either (9) or (10), it is straightforward, in the case of the Schwarzschild solution, that m ADM = C. For a general sub-static harmonic triple, the following inequality holds. Theorem 1.1 (Capacitary Riemannian Penrose Inequality). Let (M, g 0 , u) be a sub-static harmonic triple with associated capacity C and suppose that ∂M is connected. Then Moreover, the equality in (7) holds if and only if (M, g 0 ) is isometric to the Schwarzschild manifold with m ADM = C.
Whereas the above inequality has been obtained as a consequence of the monotonicity of F β , at every fixed β ≥ n−2 n−1 , we remark that one could possible push the above described analysis one step forward, at the same time exploiting the full power of the optimality threshold. Indeed, we believe that considering p-harmonic functions defined at the exterior of a bounded domain Ω lying in M , it may be possible to derive, as done in [1] for the Euclidean case and in the simultaneous limit as β ↓ n−2 n−1 and p ↓ 1, a Minkowski-like inequality for ∂Ω (see [20] for a Minkowski-like inequality in the static, asymptotically flat case). Concerning the treatment of general sub-static metrics and the derivation of related geometric inequalities, besides the already cited [6] we also would like to mention [18], where an integral formula is obtained and applied to prove Hentze-Karcher-type inequalities. For the case of asymptotically hyperbolic sub-static manifolds (specifically, for adS-Reissner-Nordström manifolds), we refer the interested reader to [26] and [12]. We remark that our results are not based on the Positive Mass Theorem. By contrast, we observe that using this celebrated result, more precisely a consequence of it contained in [16,Theorem 1.5], one can prove the following uniqueness statement. We refer the reader to Definition 1.1 for the notation and terminology. Theorem 1.2 (Uniqueness Theorem for sub-static harmonic triples). Let (M, g 0 , u) be a sub-static harmonic triple with associated capacity C. Suppose that there is a chart at infinity such that for some q > n. Then (M, g 0 ) is the Schwarzschild manifold with associated ADM mass given by C.
It remains an open question to see whenever it is possible to remove the assumption on the decay of R g0 and get the same conclusion.
The paper is organised as follows. In Section 2, we recall and discuss some preparatory material, namely the asymptotic expansions of the metric and of the potential, and classical measure properties of the critical set of the potential, with a close look on related integral quantities. In Section 3, we prove the Monotonicity and Outer Rigidity Theorem 3.1, and the consequent Capacitary Riemannian Penrose Inequality contained in Theorem 1.1. To do this, we use from Section 4 some corresponding results obtained in a suitable conformally-related setting. The biggest technical effort is contained in such section. In the Appendix we also provide an alternative proof of the monotonicity of our monotone quantities. Finally, Section 5 is devoted to the proof of Theorem 1.2.

Preliminaries
Let (M, g 0 , u) be a sub-static harmonic triple. We observe, as a first consequence of system (1), that the scalar curvature R g0 is nonnegative. Since u satisfies the last three conditions of system (1), by the Maximum Principle we have Also, by the forth condition in (1), each level set of u is compact. Moreover, from the Hopf Lemma, it follows that |D g0 u| g0 > 0 on ∂M . In particular, zero is a regular value of u. Furthermore, from the first two conditions in (1) restricted to ∂M it is easy to deduce that D 2 g0 u ≡ 0 on ∂M . In turn, the function |D g0 u| g0 attains a positive constant value on each connected component of ∂M , and the boundary ∂M is a totally geodesic hypersurface in M .
We now deal with the asymptotic behaviour of the potential u at ∞. By Theorem 2.2 below, this is given by: being Here, σ g0 is the canonical measure on the boundary ∂M seen as a Riemannian submanifold of (M, g 0 ), and we have used the standard notation o 2 , which means that, in any chart at infinity ψ, denoted by u the function u • ψ −1 , the following conditions hold true.
Let us remark that we can always suppose, without loss of generality, that the considered chart at infinity admits a diffeomorphic extension to the closure of the coordinate domain. We will make this implicit assumption throughout the paper, so that ∂K (see Definition 1.1) is a connected hypersurface of M and the quantities related to the metric can be pushed-forward in R n outside an open ball and be smooth here. We also observe that formula (10) is nothing but an equivalent characterisation of the capacity of ∂M .

Asymptotic expansions
Let (N, h) be a smooth, connected, noncompact, complete, asymptotically flat, n-dimensional Riemannian manifold, with n ≥ 3, with one end and with nonempty smooth compact boundary ∂N . We adopt the following notation.
• B and B R a generic open ball and the open ball of radius R > 0 centred in the origin of (R n , d e ), respectively; • | · | the euclidean norm of R n ; • |S n−1 | the hypersurface area of the unit sphere inside R n with the canonical metric; • D e and ∆ e the Levi-Civita connection and the Laplace operator of (R n , g R n ), respectively; • D h and ∆ h the Levi-Civita connection and the Laplace operator of (N, h), respectively; • σ e the canonical measure on a Riemannian submanifold of (R n , g R n ); • σ h the canonical measure on a Riemannian submanifold of (N, h); • | · | e the norm induced by g R n on the tangent spaces to the manifold R n ; • | · | h the norm induced by h on the tangent spaces to the manifold N .
• If ψ is a chart at infinity of (N, h) according to Definition 1.1, we denote by h the pushforward metric ψ * h of h by ψ, having coordinate expression h ij (x) dx i ⊗ dx j . In this context, D h and ∆ h denote the Levi-Civita connection and the Laplace operator of h, respectively, while σ h is the canonical measure on a Riemannian submanifold of (R n \ B, h) and | · | h is the norm induced by h on the tangent spaces. Moreover, Ric h and R h are the Ricci tensor and the scalar curvature of h, respectively.
hold true for some p > n−2 2 . Moreover, where ν e is the ∞-pointing unit normal with respect to the Euclidean metric and σ e the associated canonical measure on ∂B R , while ν h is the ∞-pointing unit normal with respect to h and σ h the associated canonical measure on ∂B R .
Proof. From h ik h kj = δ i j it is easy to get These formulae coupled with (2), (3) and (4) give (14). Decay (15) is another direct consequence of Definition 1.1, keeping in mind that Decays (16)- (17) are obtained by contractions of the Riemannian tensor. Now, observe that and that Observe also that for some C > 0, for any x ∈ R n \ B. Since trivially |x k x l | ≤ |x| 2 , from (21) and (22), coupled with (14), we get decay (18). Concerning decay (19), recall first that, using a coordinate chart (y 1 , . . . , y n−1 ) on ∂B R , we have that dσ h = det h ∂BR dy 1 . . . dy n−1 with h ∂BR = h ∂BR αβ dy α ⊗ dy β , where h ∂BR αβ = h ∂ ∂y α , ∂ ∂y β . Now, using the specific local parametrization x = x(y 1 , . . . , y n−1 ) of ∂B R , given by the inverse of stereographic projection from its north pole with the diffeomorphism p ∈ S n−1 → Rp ∈ ∂B R , we have that Hence, on ∂B R , where in the last identity we have used the Leibniz formula for the determinant and Taylorexpanded the square root.
The following result is well-known. For completeness, we provide the statement, along with its proof, which is an extension of [19, Lemma A.2.] to every n ≥ 3.
Theorem 2.2. Let (N, h) be a smooth, connected, noncompact, complete, asymptotically flat, n-dimensional Riemannian manifold, with n ≥ 3, with one end, and with nonempty smooth compact boundary ∂N . If We remark that the asymptotic behaviour of the potential u at ∞, given by formula (9), is a simply consequence of the above theorem observing that u = 1 − v when (N, h) = (M, g 0 ). Proof.
Step 1: Construction of a barrier function. Let ψ be a chart at infinity for N . From now on by C we will denote some positive constant, which may change from line to line. By Definition 1.1, there exist p > (n − 2)/2 and R 1 ≥ 1 such that for every x ∈ R n \ B R1 . By (14), the same conditions as in (24) are satisfied by h −1 (x) for all x ∈ R n \ B R1 . Then, for every f ∈ C ∞ (R n \ B R1 ), writing where in R n \ B R1 . For a fixed 0 < ε < p and for a > 0 to be chosen later, consider the function By direct computation one can check that and in turn that Therefore, by (25) and (26), we obtain that and hence there exists Step 2: Asymptotic expansion of v. Note that from (27) one gets in particular that v ≤ C|x| 2−n . We now apply Shauder's Interior estimates ([11, Lemma 6.20]) to ∆ h v = 0 in R n \ B R2 , where the operator ∆ h is defined as in (25) and its coefficients satisfy the estimates in (26). Recalling that the Hölder norms are weighted by the (Euclidean) distance d e ( · , ∂B R2 ) from ∂B R2 and since d e (x, ∂B R2 ) ≃ |x| when |x| >> 1, from such estimates we get in R n \B R2 (up to a bigger R 2 ). Combining (26) and (28), the equation We consider a smooth extension of v on R n , still denoted by v, and the smooth extension of f given by ∆ e v, still denoted by f . By a classical representation formula and due to (29), the function is well-defined and fulfils ∆ e w = f on R n . Now, one can rewrite w in R n \ {O} as and show that each summand can be bounded by C|x| −(n−2+p) , except the first one. Therefore, we have that Since the function v − w is harmonic and bounded on R n , then it is constant and this constant is zero, using the fact that v − w → 0 for |x| → ∞. Hence Therefore, applying Shauder's Interior estimates to in R n \ B R2 (up to a bigger R 2 ). From (30) and (31) we obtain in particular (23).
Step 3: Characterization of C. First of all we remark that 0 < v < 1 onN , v : N → (0, 1] is proper, and, from the Hopf Lemma, |D h v| h > 0 on ∂N . In particular, 1 is a regular value of v. Let K be the compact set on the complement of which the chart ψ is defined. For every R > R 2 , applying the Divergence Theorem to the function v on K ∪ {|ψ| < R} we obtain that where ν h is the outward unit normal vector field with respect to h along ∂N and {|ψ| = R}. Then, it follows that (14), (18) and (19), and also by identity (30) and the second in (31), and keeping in mind that |∂ i v| ≤ C|x| 1−n , we have that For the ease of the reader, we collect in the next theorem some results about the measure of the level sets of the potential u and

Measure of and integration on the level sets of the potential
which are well-known in the Euclidean setting (see, e.g., [14] and [15]).  (ii) Crit(u) is a compact subset of M and its Hausdorff dimension in M is less than or equal to (n−2); (iii) The set of the critical values of u has zero Lebesgue measure, and for every t ∈ [0, 1) regular value of u there exists ǫ t > 0 such that (t − ǫ t , t + ǫ t ) ∩ [0, 1) does not contain any critical value of u.
Proof. Each level set of u is compact, due to the forth condition in (1). Now, consider the nontrivial case where Crit(u) = Ø and let p be a point of a critical level set {u = t}. Take a chart for each (v 1 , . . . , v n ) ∈ R n and for each x ∈ B 1 2 . The same condition is satisfied by the coefficients g ij 0 . In particular, setting u = u • ψ −1 p , we have that . Applying this fact to u − t, one can argue that u − t has finite order of vanishing at O. Then, by using [15,Theorem 1.7], there exists Since u is nonconstant in B 1 2 and by the structure and regularity of ∆ g0 , [14, Theorem 1.1] yields Hence, since the restriction of ψ p to ψ −1 p (B 1/2 ) is bilipschitz due to (32) and since the measures ) coincide on borel sets, statements (i) and (ii) are true locally. In turn, by compactness of Crit(u), they are true globally. To prove (iii), observe that by Sard's Theorem, the set of the critical values of u has zero Lebesgue measure. Now, suppose by contradiction that there exists t ∈ [0, 1) regular value such that, for all m ≥ m with 1 Hence there is a sequence {t m } m≥m of critical values such that t m → t. In particular, there exists a sequence {p m } m≥m of critical points contained in the set {0 ≤ u ≤ t + 1 m } and such that u(p m ) = t m . Then, by compactness and up to a subsequence, p m → p. In turn, 0 = |D g0 u| g0 (p m ) → |D g0 u| g0 (p) and t m = u(p m ) → u(p). Hence |D g0 u| g0 (p) = 0 and u(p) = t, which is absurd. This concludes the proof of (iii).

Remark 2.1.
It is useful to observe that: (i) for every t ∈ (0, 1), the set {u ≥ t} is connected; (ii) for every t ≈ 1, the level set {u = t} is regular and diffeomorphic to S n−1 ; We check (ii) first. We start by observing that due to (9) |D g0 u| g0 = 0 in {u ≥ t 0 }, for some 0 < t 0 < 1. This fact establishes a diffemorphism between {u ≥ t 0 } and {u = t 0 } × [t 0 , 1) and tells us at the same time that the level sets {u = t} are pairwise diffeomorphic, for every t ≥ t 0 . It is thus sufficient to show that {u = t 0 } is connected. Suppose by contradiction that this is not the case. Without loss of generality we can assume that {u = t 0 } can be decomposed into the disjoint union of two connected sets C 1 and C 2 , indeed the same argument works a fortiori if the connected components are more than two. Now, note that by definition of asymptotically flat manifold, there exists a compact set K ⊂ M such that M \K is diffeomorphic to R n \B by a chart at infinity ψ, where B is a suitable ball, and we can suppose, up to a bigger t 0 , that {u ≥ t 0 } ⊆ M \K. Now, in view of the asymptotic expansion of u, there exist two positive In particular, setting is a compact and connected hypersurface of R n , having strictly positive sectional curvature, as Riemannian submanifold of (R n , g R n ) , up to a bigger t 0 and due to (13). Hence, { u = t 0 } is diffeomorphic to S n−1 by the Gauss map (see [9,Section 5.B] for more details). Statement (ii) thus follows, being {u = t 0 } and { u = t 0 } diffeomorphic. To see (i), observe first that if t is a regular value of u, then E t := {u ≥ t} is a n-dimensional submanifold with boundary {u = t}. By Theorem 2.3 and by the Maximum Principle, every connected component C of E t is unbounded. Since u → 1 at ∞, we have that u(C) = [t, 1), and hence C ∩ {u = t 0 } = Ø, for every t 0 ∈ (t, 1). Then, E t is connected by (ii). If t is a critical value of u, we let t > t be a regular value of u such that {u = t} is connected and let {t m } be a nondecreasing sequence of regular value of u such that t m < t and t m → t. Hence, {t m ≤ u ≤ t} m∈N is a nonincreasing family of connected and compact sets in M , which is Hausdorff, and in turn the intersection {t ≤ u ≤ t} is still connected. In particular, we deduce To check (iii), note first that for every t ∈ (0, 1) regular value of u, the equalities are always true.
By Theorem 2.3 and by definition of the Lebesgue integral, we have thatf ∈ L 1 H n−1 Similarly, if f : In the rest of this paper, we will confuse the integrals of cases (33) and (34), denoting both by {u=t} f dσ g0 .

Monotonicity and Outer Rigidity Theorem
In this section, we state and prove our Monotonicity and Outer Rigidity Theorem, which is then used to prove the Capacitary Riemannian Penrose Inequality (7). From now on and unless otherwise stated, (M, g 0 , u) will always be a sub-static harmonic triple, and, when referring to such triple, the subscript g 0 will be dropped. The only exception is |S n−1 |, which always stands for the Euclidean volume of S n−1 .
for every β ≥ 0. Then, the following properties hold true.
(i) Differentiability, Monotonicity and Outer Rigidity: for every β > n−2 n−1 , the function F β is continuously differentiable with nonpositive derivative in (1, ∞). Moreover, if there exists τ 0 ∈ (1, ∞) such that F ′ β (τ 0 ) = 0 for some β > n−2 (ii) Convexity: for every β > n−2 n−1 , the function F β is convex on [1, ∞). We remark that the functions F β are well-defined, in view of Theorem 2.3 and since the integrand function in (35) is bounded on every level set of u. Note that, once Theorem 3.1 is proven, The Dominated Convergence Theorem allows us to extend the monotonicity and convexity of F β also to the case β = n−2 n−1 . Moreover, on the values τ such that u = » τ −1 τ +1 is regular and thus on a.e. τ > 1 due to Theorem 2.3 (iii), the function F β is twice differentiable for each β > n−2 n−1 , with first and second derivative given by In the computation, we have used the first normal variation of the volume and the mean curvature of {u = t}, and the Divergence Theorem. The symbols H and h stand respectively for the mean curvature and the second fundamental form of the smooth (n − 1)-dimensional submanifold u = » τ −1 τ +1 , with respect to the ∞-pointing unit normal vector field ν = Du |Du| . Also, D T denotes the tangential part of the gradient, that is To prove Theorem 3.1, we use the results of Section 4, which are obtained in the conformal setting defined by Denoting by ∇ and ∆ g the Levi-Civita connection and the Laplace-Beltrami operator of g, the triple (M, g, ϕ) satisfies the following system.
Moreover, we have that as we will see in the proof of Lemma 4.1.
for every β ≥ 0. For the convenience of the reader, we anticipate from Section 4 the properties of Φ β that we are going to use.
In the above list we have gathered and summarised the results contained in Lemma 4.5, Proposition 4.6, and Corollary 4.7.
Proof of Theorem 3.1.
Step 2: Outer Rigidity. Let us assume that there exists τ 0 ∈ (1, ∞) such that F ′ β (τ 0 ) = 0 for some β > n−2 n−1 . Then, by equality (41) where ρ is the g-distance function to {ϕ = s 0 } and ϕ = (n − 2) (2C) − 1 n−2 ρ + s 0 , because |∇ρ| g ≡ 1 and in view of the limit in (⋆). Setting t 0 = tanh s0 2 , consider N the submanifold with boundary {ϕ ≥ s 0 } = {u ≥ t 0 }. Writing the Riemannian manifolds in the first row, whose metrics are indicated in the second row, are pairwise isometric through the applications written in the third row. We recall that the application p → (ρ, q) in the third row is the inverse of the diffeomorphism given by the normal exponential map, i.e. the application which associates to every point p of N the couple having as first coordinate the g-distance of p from ∂N and as second coordinate the point q of ∂N that realizes such distance. Then, in view of (37) and with the same notation as above, the following Riemannian manifolds are isometric.
where r 0 = 2C 1−t 2 0 1 n−2 . Doing some computations, we obtain that where the convection followed for the Riemannian curvature tensor is that given in [22], c is a suitable positive constant and q is the point of ∂N that realizes the g-distance of p from ∂N . Denoting by Θ the diffeomorphism from N to [r 0 , +∞) × ∂N introduced in (42), for every q 0 ∈ ∂N we consider the curve γ : r ∈ [r 0 , +∞) → Θ −1 (r, q 0 ) ∈ M and observe from (43) that At the same time, we have that This is because g 0 is asymptotically flat according to Definition 1.1 and by (11), which yields in particular for some p > n−2 2 . Combining (44) and (45), the arbitrariness of the point q 0 in ∂N gives that Hence the sectional curvature of the Riemannian manifold (∂N, g ∂N ) is constant and identically equal to (2C ) − 2 n−2 . Then, being all the level sets {u = t} with t ≈ 1 regular and diffeomorphic to S n−1 as observed in Remark 2.1, for [9, Section 3.F] (∂N, g ∂N ) and (S n−1 , (2C ) 2 n−2 g S n−1 ) are isometric. Then, ({u ≥ t 0 }, g 0 ) is isometric to the submanifold [r 0 , +∞) × S n−1 , dr⊗dr 1−2Cr 2−n + r 2 g S n−1 of the Schwarzschild manifold with associated ADM mass given by C.
Proof of Theorem 1.1. Spep 1: Inequality. By Theorem 3.1, we have that F β (τ 0 ) ≥ lim τ →+∞ F β (τ ), for every τ 0 > 1. In particular, since F β is continuous in [1, +∞) due to the step 1 of Theorem 3.1, we have that for every β > n−2 n−1 . Since D 2 u ≡ 0 on ∂M and since ∂M of M is connected, |Du| is constantly equal to (n−2) C |S n−1 | R n |∂M| , by formula (10). In particular, we have that By (⋆), we know that and the level sets u = » are regular for all τ ≥ τ 0 . Therefore, for every τ ≥ τ 0 we have that where in the second equality we have used the Divergence Theorem couple with the fact that u is harmonic, and in the third equality we have used formula (10). Since ε is arbitrary, we get In a similar way we can obtain the reverse inequality, so that Joining the formulas in (46), (47) and (48), we obtain the desired inequality (7).
Step 2: Rigidity. If (M, g 0 ) is isometric to the Schwarzschild manifold with ADM mass m > 0, then the right-hand side and the left-hand side of (7) are both equal to m, by direct computation. Suppose now that the equality holds in (7). Then, by Step 1 and for every β > n−2 n−1 , the function F β is constant. In turn, Φ β is constant, being Finally, (⋄ ⋄) and the very same argument of the proof of the Outer Rigidity in Theorem 3.1 imply first that (M, g) is isometric to where ρ is the g-distance to ∂M and ϕ is an affine function of ρ, and secondly that (M, g 0 ) is isometric to the Schwarzschild manifold with ADM mass C.

Conformal setting
Let us consider the conformal change g of the metric g 0 introduced in (37) which is well-defined being 0 ≤ u < 1 in M . The metric g is complete, since any g-geodesic γ parametrized by garc length defined on a bounded interval [0, a) can be extended to a continuous path on [0, a]. Indeed, if γ has infinity length with respect to g 0 , there exists a sequence {t m } m∈N such that γ(t m ) → ∞ (being γ not contained in any compact set) and using, in the computation of glength of γ, the passage from g to g 0 , the asymptotic flatness of (M, g 0 ) and the asymptotic expansion of u in (11) we obtain that γ has infinity length with respect to g. Hence γ has finite length with respect to g 0 and, being g 0 complete, it follows that g is complete (see [23, Section 1.1] and [8]). We also recall that the metric g is asymptotically cylindrical (see [2,Section 3.1]). The other main element of the conformal setting is the C ∞ -function ϕ, defined in (37). Now, the reverse changes are Recalling that we denote by the symbols ∇ and ∆ g the Levi-Civita connection and the Laplace-Beltrami operator of g, by the formulas in [5, Theorem 1.159], we obtain Translating system (1) in terms of g and ϕ, we get system (38). Moreover, on {ϕ = s} \ Crit(ϕ) we consider the ∞-pointing normal unit vector fields ν = Du |Du| , ν g = ∇ϕ |∇ϕ| g , the mean curvatures and the second fundamental forms for any X, Y tangent vector fields to the considered submanifold. Reversing formulas (49), (50) and (51), we get These equalities, jointly with the asymptotic flatness of (M, g 0 ) and the asymptotic expansion of u given in Section 2, allow us to obtain an upper bound for the functions |∇ϕ| g and |∇ 2 ϕ| g , and for the g-areas of the level sets of ϕ sufficiently "close" to infinity. This is the content of the following lemma.
We recall that the set of the critical values of |∇ϕ| 2 g has zero Lebesgue measure by Sard's Theorem, whereas we have no information regarding the local H-dimension of Crit(|∇ϕ| 2 g ).
We underline that from now on we will use Remark 3.1 widely.
The following proposition contains the integral identity which is the main tool of our analysis.

Proposition 4.3.
Let (M, g 0 , u) be a sub-static harmonic triple, and let g and ϕ be the metric and the function defined in (37). Then, for every β > n−2 n−1 and for every S > s > 0 regular values of ϕ, it holds where the tensor Q is defined as in (56).
Proof. The case β ≥ 2 is an easy adaptation of the argument used in [2] but it is anyway a consequence of the following argument. We focus on the unknown case n−2 n−1 < β < 2. In M \ Crit(ϕ) we consider the smooth vector field Y β , defined in (59) and satisfying as already explained. Set E S s := {s < ϕ < S}, for every S > s > 0 .
When E S s ∩ Crit(ϕ) = Ø, then the statement is a straightforward application of the Divergence Theorem. Now, suppose that E S s ∩ Crit(ϕ) = Ø. In this case we consider, for every ε > 0 sufficiently small, a smooth nondecreasing cut-off function χ ε : [0, +∞) → [0, 1] satisfying the following conditions where c is a positive real constant independent of ε. We then define the smooth function Ξ ε : and apply the Divergence Theorem to the smooth vector field Ξ ε Y β in E S s . In this way, we get where U µ is defined in (62). Note that {χ ε } can always be chosen to be nondecreasing in ε so that, in turn, {Ξ ε } is nondecreasing. Therefore, applying the Monotone Convergence Theorem, when ε → 0 + , the first term on the right of the second equality tends to For obtaining the desired statement, we show First we observe that where, keeping in mind the properties satisfied by χ ε , in the first inequality we have used the nonnegativity of the integrand function and in the last one the Coarea Formula. Note that there exist ε 0 , c 1 > 0 such that the inequality is true a.e. s ∈ [ 1 2 ε, 3 2 ε] for every 0 < ε < 2 3 ε 0 , by both Sard's Theorem applied to the smooth function |∇ϕ| 2 g and by Lemma 4.2. Then, we get  sinh ϕ dµ g is well-defined. Moreover, it is nonnegative as above and therefore we get the statement.

Remark 4.2.
For every β > n−2 n−1 and for every s > 0 regular value of the function ϕ: For every S big enough, which is a regular value of ϕ, by Lemma 4.1 with (53) we have In particular, Therefore, the desired identity can be obtained by the Monotone Convergence Theorem, by passing to the limit as S → +∞ in (63).

Remark 4.3.
For every β > n−2 n−1 , as consequence of integral identity (63), we have for every K ⊂M compact, by Hölder's Inequality from (66) with (61) we get that We need a final lemma before stating the (last and) most important result of this section. (ii) For every β ≥ 0 and for every s > 0: for every β ≥ 0, is continuous and admits for every s > 0 the integral representation This lemma can be proved as [2,Proposition 4.1]. In the Appendix we provide an alternative proof which is self contained and does not make use of any fine property of the measure of Crit(ϕ): we just need to know very classical properties of it (see Remark 3.1).
Proposition 4.6. Let (M, g 0 , u) be a sub-static harmonic triple, let g and ϕ be the metric and the function defined in (37), and let Φ β : [0, ∞) → R be the function defined by formula (69) for every β ≥ 0. Then for every β > n−2 n−1 , the function Φ β is continuously differentiable. The derivative Φ ′ β is nonpositive and admits for every s > 0 the integral representation where ρ is the g-distance function to {ϕ = s 0 } and ϕ is an affine function of ρ in {ϕ ≥ s 0 }.
The following proof is essentially the same as in [2]. For completeness, we include it here, in a slightly refined version. Proof.
Step 1: Continuous Differentiability and Monotonicity. Let β > n−2 n−1 . Note that the boundary ∂M is a regular level set of ϕ and then, by Theorem 2.3 and the relationship between Crit(u) and Crit(ϕ), there exists ǫ 0 such that the interval [0, ǫ 0 ] doesn't contain critical values of the function ϕ. Therefore, for every 0 < ǫ ≤ ǫ 0 , applying first the Divergence Theorem to the smooth vector field |∇ϕ| β g ∇ϕ in {0 < ϕ < ǫ} and later the Coarea Formula, we get div g ∇|∇ϕ| β g dµ g for every 0 ≤ s 1 < s 2 ≤ ǫ 0 , by the Dominated Convergence Theorem the function is continuous and therefore, by the Fundamental Theorem of Calculus Φ β is continuously differentiable on the closed interval [0, ǫ 0 ].
The first statement follows immediately from Proposition 4.3. As for the second statement, we first observe that for every couple 0 < s < s < +∞. Always by Remark 4.3 and by the Dominated Convergence Theorem, we can deduce the right and the left continuity of Ψ β on the interval (0, +∞). We consider Υ β : s ∈ (0, +∞) → Φ β (s) sinh s ∈ R . For every (s, s) couple of real number such that 0 < s < s < +∞, we have where the first equality follows from Lemma 4.5 (i), the second equality from the Coarea Formula keeping in mind (67). Moreover, the last equality follows from (i) and from Sard's Theorem. Using the continuity of both the functions Υ β and Ψ β , passing to the limit in (⋆) for either s → s or s → s yields that the function Υ β is C 1 , and Since Φ β (s) = sinh(s)Υ β (s) for every s > 0, then Φ β ∈ C 1 (0, +∞) and Φ ′ β (s) = −β sinh(s)Ψ β (s).
This tell us that ϕ is an affine function of ρ on {ϕ ≥ s 0 }.
While the previous proposition contains an outer rigidity result, with the following corollary we provide a global rigidity result.
Corollary 4.7. Let (M, g 0 , u) be a sub-static harmonic triple, let g and ϕ be the metric and the function defined in (37), and let Φ β : [0, ∞) → R be the function defined by formula (69) for every β ≥ 0. If Φ β is constant for some β > n−2 n−1 , then ∂M is connected and (M, g) is isometric to [0, +∞) × ∂M, dρ ⊗ dρ + g ∂M ), where ρ is the g-distance function to ∂M and ϕ is an affine function of ρ.
normal and the canonical measure on ∂B r as Riemannian submanifold of (R n \ B, h), respectively. Also, Ric h and R h are the Ricci tensor and the scalar curvature of h respectively, and X is the Euclidean conformal Killing vector field x i ∂ ∂x i . The ADM mass is well defined as m ADM := lim r→+∞ m(r) , and independent of the chosen chart at infinity. Moreover (see [21]), it can be equivalently expressed as m ADM = lim r→+∞ m I (r) .
From the alternative definition of ADM mass, given by (72), and using the Positive Mass Theorem, more precisely a consequence of it contained in [16,Theorem 1.5], one can prove the following uniqueness statement. For the notation and terminology, we refer the reader to Definition 1.1 and Section 2.

Appendix
In this Appendix we provide a proof of Lemma 4.5 which is alternative and more self contained than the corresponding in [2]. We underline that we will use Remark 3.1 widely.