A Review of the interrelations of terrestrial carbon sequestration and urban forests

1.   Introduction

In this paper, the object under investigation is a triple $(M, g_{0}, u)$ satisfying the following two conditions:

($\mathrm{a}$) $(M, g_{0})$ is a smooth, connected, noncompact, complete, asymptotically flat, $n$-dimensional Riemannian manifold, with $n\geq 3$, with one end, and with nonempty smooth compact boundary $\partial M$, which is a priori allowed to have several connected components.

($\mathrm{b}$) $u\in C^{\infty}(M)$ satisfies the system

where $\mathrm{Ric}_{g_{0}}$, $\mathrm{D}_{g_{0}}$ and $\Delta_{g_{0}}$ 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 (1.1), 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 $n\geq 3$, is said to be asymptotically flat if there exists a compact subset $K\subset N$ such that $N \setminus K$ is a finite disjoint union of ends $N_{k}$ with the following properties. Every $N_{k}$ is diffeomorphic to $\mathbb{R}^{n}$ minus a closed ball by a coordinate chart $\psi_{k}$ and, if $\widetilde{h}: = (\psi_{k})_{*}h = \widetilde{h}_{ij}dx^{i}\otimes dx^{j}$, we have

for some $p > (n-2)/2$. Here, $\delta$ is the Kronecker delta, and the coordinate charts $\psi_{k}$ are called charts at infinity.

Throughout the paper, we will refer to a triple $(M, g_{0}, u)$ that satisfies conditions $(\mathrm{a})$ and $(\mathrm{b})$ 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 in $\mathring{M}$, extend smoothly up to the boundary and $(M, g_{0})$ is called (spatial) Schwarzschild manifold. The parameter $m > 0$ is the ADM mass $m_{\mathrm{ADM}}$ of the Schwarzschild manifold. We refer the reader to Section 5 for the definition of the $m_{\mathrm{ADM}}$ associated with a general asymptotically flat manifold. Here, we limit ourselves to recall that the decay conditions (1.2)–(1.5) guarantee that $m_{\mathrm{ADM}}$ is a geometric invariant ^[4,8].

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 $\beta\geq0$:

In ^[2] it was proven that if $(M, g_0, u)$ is a static triple, then, for every $\beta\geq2$, the function $V_\beta$ 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 $\beta = \frac{n-2}{n-1}$. 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 $F_\beta(\tau)$, where $\tau = \frac{1+t^2}{1-t^2}\geq1$, 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 ^[7] 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 guarantees 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 $F_\beta$ and, coupled with Sard's Theorem, also its differentiability.

Observe that the difficulty in treating the critical points under the threshold $\beta = 1$ can be read off directly from formulæ (3.2) and (4.5), the first one displaying the derivative of $F_\beta$ and the second one expressing the mean curvature on a equipotential set in terms of the Hessian of the potential itself. In fact, calling $\Phi_\beta$ the conformal version of $F_\beta$ and looking at formula (4.26) containing the equivalent characterisation of $\Phi_\beta'$ derived from the integral identity (4.18), one realises that problems arise already when $\beta < 2.$

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 $\beta = \frac{n-2}{n-1}$ that this is guaranteed, thanks to the Refined Kato Inequality for harmonic functions. The optimality of such inequality reflects a corresponding optimality of $\beta = \frac{n-2}{n-1}$ in our result. Moreover, let us remark that the (nonnegative) right-hand side of (4.18) is obtained as the divergence of a suitable modification of a specific vector filed with nonnegative divergence (see (4.19)), 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 $F_\beta$ is comparing $F_\beta(1)$ with $F_\beta(+\infty)$, in turn yielding a "capacitary version" of the Riemannian-Penrose inequality (Theorem 1.1 below). The capacity comes naturally into play when computing $F_\beta(1)$ and $F_\beta(+\infty)$, the latter value via the asymptotic expansions of the metric and of the potential. We recall that the capacity $\mathrm{Cap}(\partial M, g_{0})$ of $\partial M$ is defined as

Throughout the paper, we will use the short–hand notation $\mathcal C$ for the capacity. Comparing (1.6) with either (2.1) or (2.2), it is straightforward, in the case of the Schwarzschild solution, that $m_{\mathrm{ADM}} = \mathcal C$. 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 $\mathcal{C}$ and suppose that $\partial M$ is connected. Then

Moreover, the equality in (1.7) holds if and only if $(M, g_{0})$ is isometric to the Schwarzschild manifold with $m_{\mathrm{ADM}} = \mathcal{C}$.

Whereas the above inequality has been obtained as a consequence of the monotonicity of $F_\beta$, at every fixed $\beta\geq\frac{n-2}{n-1}$, we remark that one could possibly 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 $\Omega$ lying in $M$, it may be possible to derive, as done in ^[1] for the Euclidean case and in the simultaneous limit as $\beta\downarrow\frac{n-2}{n-1}$ and $p\downarrow1$, a Minkowski-like inequality for $\partial\Omega$ (see ^[21] for a Minkowski-like inequality in the static, asymptotically flat case and ^[5] for the nonnegative Ricci case).

Concerning the treatment of general sub-static metrics and the derivation of related geometric inequalities, besides the already cited ^[7] we also would like to mention ^[19], 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 ^[13] and ^[27].

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 [17,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 $\mathcal{C}$. 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 $\mathcal{C}$.

It remains an open question to see whether it is possible to remove the assumption on the decay of $\mathrm{R}_{\widetilde{g}_{0}}$ 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.

2.   Preliminaries

Let $(M, g_{0}, u)$ be a sub-static harmonic triple. We observe, as a first consequence of system (1.1), that the scalar curvature $\mathrm{R}_{g_{0}}$ is nonnegative. Since $u$ satisfies the last three conditions of system (1.1), by the Maximum Principle we have

Also, by the forth condition in (1.1), each level set of $u$ is compact. Moreover, from the Hopf Lemma, it follows that $\vert \mathrm{D}_{g_{0}}u \vert _{g_{0}} > 0$ on $\partial M$. In particular, zero is a regular value of $u$. Furthermore, from the first two conditions in (1.1) restricted to $\partial M$ it is easy to deduce that $\mathrm{D}_{g_{0}}^{2}u\equiv 0$ on $\partial M$. In turn, the function $\vert \mathrm{D}_{g_{0}}u \vert_{g_{0}}$ attains a positive constant value on each connected component of $\partial M$, and the boundary $\partial M$ is a totally geodesic hypersurface in $M$.

We now deal with the asymptotic behaviour of the potential $u$ at $\infty$. By Theorem 2.2 below, this is given by:

being

Here, $\sigma_{g_{0}}$ is the canonical measure on the boundary $\partial M$ 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 $\psi$, denoted by $\widetilde{u}$ the function $u\circ \psi^{-1}$, 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 $\partial K$ (see Definition 1.1) is a connected hypersurface of $M$ and the quantities related to the metric can be pushed–forward in $\mathbb{R}^{n}$ outside an open ball and be smooth here. We also observe that formula (2.2) is nothing but an equivalent characterisation of the capacity of $\partial M$.

2.1.   Asymptotic expansions

Let $(N, h)$ be a smooth, connected, noncompact, complete, asymptotically flat, $n$-dimensional Riemannian manifold, with $n\geq 3$, with one end and with nonempty smooth compact boundary $\partial N$. 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 $(\mathbb{R}^{n}, d_{e})$, respectively;

● $\vert \cdot \vert$ the euclidean norm of $\mathbb{R}^{n}$;

● $\vert \mathbb{S}^{n-1}\vert$ the hypersurface area of the unit sphere inside $\mathbb{R}^{n}$ with the canonical metric;

● $\mathrm{D}_{e}$ and $\Delta_{e}$ the Levi–Civita connection and the Laplace operator of $(\mathbb{R}^{n}, g_{ \mathbb{R}^{n}})$, respectively;

● $\mathrm{D}_{h}$ and $\Delta_{h}$ the Levi–Civita connection and the Laplace operator of $(N, h)$, respectively;

● $\sigma_{e}$ the canonical measure on a Riemannian submanifold of $(\mathbb{R}^{n}, g_{ \mathbb{R}^{n}})$;

● $\sigma_{h}$ the canonical measure on a Riemannian submanifold of $(N, h)$;

● $\vert \cdot \vert_{e}$ the norm induced by $g_{ \mathbb{R}^{n}}$ on the tangent spaces to the manifold $\mathbb{R}^{n}$;

● $\vert \cdot \vert_{h}$ the norm induced by $h$ on the tangent spaces to the manifold $N$.

● If $\psi$ is a chart at infinity of $(N, h)$ according to Definition 1.1, we denote by $\widetilde{h}$ the push–forward metric $\psi_{*}{h}$ of $h$ by $\psi$, having coordinate expression $\widetilde{h}_{ij}(x)\, dx^{i}\otimes dx^{j}$. In this context, $\mathrm{D}_{\widetilde{h}}$ and $\Delta_{\widetilde{h}}$ denote the Levi–Civita connection and the Laplace operator of $\widetilde{h}$, respectively, while $\sigma_{\widetilde{h}}$ is the canonical measure on a Riemannian submanifold of $(\mathbb{R}^{n}\setminus B, \widetilde{h})$ and $\vert \cdot \vert_{\widetilde{h}}$ is the norm induced by $\widetilde{h}$ on the tangent spaces. Moreover, $\mathrm{Ric}_{\widetilde{h}}$ and $\mathrm{R}_{\widetilde{h}}$ are the Ricci tensor and the scalar curvature of $\widetilde{h}$, respectively.

Proposition 2.1. Let $\psi$ be a chart at infinity of $N$ (according to Definition 1.1). The decays

hold true for some $p > \frac{n-2}{2}$. Moreover,

where $\nu_{e}$ is the $\infty$–pointing unit normal with respect to the Euclidean metric and $\sigma_{e}$ the associated canonical measure on $\partial B_{R}\,$, while $\nu_{\, \widetilde{h}}$ is the $\infty$–pointing unit normal with respect to $\widetilde{h}$ and $\sigma_{\widetilde{h}}$ the associated canonical measure on $\partial B_{R}\,$.

Proof. From $\widetilde{h}^{ik} \widetilde{h}_{kj} = \delta^{i}_{j}$ it is easy to get

These formulae coupled with (1.2), (1.3) and (1.4) give (2.6). Decay (2.7) is another direct consequence of Definition 1.1, keeping in mind that

Decays (2.8)–(2.9) are obtained by contractions of the Riemannian tensor. Now, observe that

and that

Observe also that

for some $C > 0$, for any $x\in \mathbb{R}^{n}\setminus B$. Since trivially $\vert x^{k} x^{l}\vert \leq \vert x\vert^{2}$, from (2.13) and (2.14), coupled with (2.6), we get decay (2.10). Concerning decay (2.11), recall first that, using a coordinate chart $(y^{1}, \dots, y^{n-1})$ on $\partial B_{R}$, we have that $d\sigma_{\widetilde{h}} = \sqrt{\det\, \widetilde{h}^{\partial B_{R}}}\, dy^{1}\dots dy^{n-1}$ with $\widetilde{h}^{\partial B_{R}} = \widetilde{h}^{\partial B_{R}}_{\alpha\beta}\, dy^{\alpha}\otimes dy^{\beta}$, where $\widetilde{h}^{\partial B_{R}}_{\alpha\beta} = \widetilde{h}\big(\frac{\partial}{\partial y^{\alpha}}, \frac{\partial}{\partial y^{\beta}}\big)$. Now, using the specific local parametrization $x = x(y^{1}, \dots, y^{n-1})$ of $\partial B_{R}$, given by the inverse of stereographic projection from its north pole with the diffeomorphism $p\in \mathbb{S}^{n-1}\to Rp\in\partial B_{R}$, we have that

because

Hence, on $\partial B_{R}$,

where in the last identity we have used the Leibniz formula for the determinant and Taylor–expanded the square root.

The following result is well–known. For completeness, we provide the statement, along with its proof, which is an extension of [20,Lemma A.2.] to every $n\geq3$.

Theorem 2.2. Let $(N, h)$ be a smooth, connected, noncompact, complete, asymptotically flat, $n$-dimensional Riemannian manifold, with $n\geq 3$, with one end, and with nonempty smooth compact boundary $\partial N$. If $v\in C^{\infty}(N)$ is the solution to

then

We remark that the asymptotic behaviour of the potential $u$ at $\infty$, given by formula (2.1), 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 $\psi$ 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 $R_{1}\geq1$ such that

for every $x\in \mathbb{R}^{n}\setminus B_{R_{1}}$. By (2.6), the same conditions as in (2.17) are satisfied by $\widetilde{h}^{ij}(x)$ for all $x\in \mathbb{R}^{n}\setminus B_{R_{1}}$. Then, for every $f\in C^{\infty}(\mathbb{R}^{n}\setminus B_{R_{1}})$, writing

where

we have that

in $\mathbb{R}^{n}\setminus B_{R_{1}}$. For a fixed $0 < \varepsilon < p$ and for $a > 0$ to be chosen later, consider the function

By direct computation one can check that

and in turn that

Therefore, by (2.18) and (2.19), we obtain that

and hence there exists $R_{2} > R_{1}$ independent of $a$ such that $\Delta_{\widetilde{h}} \phi_{a} < 0$ in $\mathbb{R}^{n}\setminus B_{R_{2}}$, for every $a > 0$. We now choose $a > 0$ so that $\phi_{a} = 1$ on $\partial B_{R_{2}}$, that is $a = \Big[\frac{1}{R_{2}^{n-2}}-\frac{1}{R_{2}^{n-2+\varepsilon}}\Big]^{-1}$. Since $\phi_{a}$ is $\widetilde{h}$–superharmonic in $\mathbb{R}^{n}\setminus B_{R_{2}}$ and since $\widetilde{v}: = v\circ \psi^{-1} < 1$ on $\partial B_{R_{2}}$, by the Maximum Principle

Step 2: Asymptotic expansion of $v$. Note that from (2.20) one gets in particular that $\widetilde{v}\leq C\vert x \vert^{2-n}$. We now apply Shauder's Interior estimates ([12,Lemma 6.20]) to $\Delta_{\widetilde{h}}\widetilde{v} = 0$ in $\mathbb{R}^{n}\setminus \overline{B}_{R_{2}}$, where the operator $\Delta_{\widetilde{h}}$ is defined as in (2.18) and its coefficients satisfy the estimates in (2.19). Recalling that the Hölder norms are weighted by the (Euclidean) distance $d_{e}(\, \cdot\, ,\partial B_{R_{2}})$ from $\partial B_{R_{2}}$ and since $d_{e}(x, \partial B_{R_{2}})\simeq\vert x\vert$ when $\vert x\vert > > 1$, from such estimates we get

in $\mathbb{R}^{n}\setminus B_{R_{2}}$ (up to a bigger $R_{2}$). Combining (2.19) and (2.21), the equation $\Delta_{\widetilde{h}}\widetilde{v} = 0$ can be equivalent written as $\Delta_{e}\widetilde{v} = f\,$ where

We consider a smooth extension of $\widetilde{v}$ on $\mathbb{R}^{n}$, still denoted by $\widetilde{v}$, which is zero in a ball centred in the origin, and the smooth extension of $f$ given by $\Delta_{e}\widetilde{v}$, still denoted by $f$. By a classical representation formula and due to (2.22), the function

is well–defined and fulfils $\Delta_{e} w = f$ on $\mathbb{R}^{n}$. Now, one can rewrite $w$ in $\mathbb{R}^{n}\setminus \{O\}$ as

and show that each summand can be bounded by $C\vert x\vert^{-(n-2+\gamma)}$, where $\gamma = \min\{1, \, p\}$ if $p\neq1$ and $\gamma\in(1/2, 1)$ if $p = 1$, except the first one. Therefore, we have that

in $\mathbb{R}^{n}\setminus \{O\}$. Since the function $\widetilde{v}-w$ is harmonic and bounded on $\mathbb{R}^{n}$, then it is constant and this constant is zero, using the fact that $\widetilde{v}-w \to 0$ for $\vert x\vert \to \infty$. Hence

in $\mathbb{R}^{n}\setminus B_{R_{2}}$. We observe that

and that

Therefore, applying Shauder's Interior estimates to $\Delta_{\widetilde{h}}z = -\mathcal{C}\hat{z}$ in $\mathbb{R}^{n}\setminus \overline{B}_{R_{2}}$, we get

in $\mathbb{R}^{n}\setminus B_{R_{2}}$ (up to a bigger $R_{2}$). From (2.23) and (2.24) we obtain in particular (2.16).

Step 3: Characterization of $\mathcal{C}$. First of all we remark that $0 < v < 1$ on $\mathring{N}$, $v:N\to(0, 1]$ is proper, and, from the Hopf Lemma, $\vert \mathrm{D}_{h}v \vert _{h} > 0$ on $\partial N$. In particular, $1$ is a regular value of $v$. Let $K$ be the compact set on the complement of which the chart $\psi$ is defined. For every $R > R_{2}$, applying the Divergence Theorem to the function $v$ on $K\cup \{\vert \psi\vert < R\}$ we obtain that

where $\nu_{h}$ is the outward unit normal vector field with respect to $h$ along $\partial N$ and $\{\vert \psi\vert = R\}$. Then, it follows that

where $\widetilde{v} = v\circ \psi^{-1}$. Now, thanks to (1.2), (2.6), (2.10) and (2.11), which are true for $\gamma$ too, and also by identity (2.23) and the second in (2.24), and keeping in mind that $\vert\partial_{i}\widetilde{v} \vert\leq C\vert x\vert^{1-n}$, we have that

Hence

2.2.   Measure of and integration on the level sets of the potential

Let $(N, h)$ and $\iota:S \hookrightarrow N$ be respectively a $m$–dimensional Riemannian manifold and a $s$–dimensional Riemannian submanifold of $N$. Let $k$ be a positive real number. We set

$\mathcal{B}(S)$ the smallest $\sigma$–algebra containing all open sets of $S$;

$(S, \Lambda(S), \mu_{\iota^{*}h})$ the canonical space of measure on the Riemannian manifold $(S, \iota^{*}h)$ (see [14,Section 3.4]);

$\mathcal{H}^{k}_{S}$ the $k$-dimensional Hausdorff measure on $(S, d_{S})$, being $d_{S}$ the distance function of $S$;

$\mathcal{H}^{k}_{S; N}$ the $k$-dimensional Hausdorff measure on $(S, d_{S; N})$ where $d_{S; N}$ is the distance function of ${N}$ restricted to $S\times S$;

$\mathcal{H}^{k}_{N} \llcorner S$ the $k$-dimensional Hausdorff measure of $N$ restricted to $S$.

By definition of the Hausdorff measure and by [26,Proposition 12.7], $\mathcal{H}^{k}_{S; N}$, $\mathcal{H}^{k}_{N} \llcorner S$ and $\mathcal{H}^{k}_{S}$ coincide on $\mathcal{B}(S)$, and by [26,Proposition 12.6] and by [25,Proposition 2.17], $\mathcal{H}^{s}_{S}$ and $\mu_{\iota^{*}h}$ coincide on $\Lambda(S)$. The same results still hold when $N$ is a manifold with boundary.

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

which are well–known in the Euclidean setting (see, e.g., ^[15,16]).

Theorem 2.3. Let $(M, g_{0}, u)$ be a sub-static harmonic triple. Then the following statements hold true.

$(i)$ For every $t\in[0, 1)$, the level set $\{u = t\}$ is compact and has finite $(n-1)$–Hausdorff measure in $M$;

$(ii)$ $\mathrm{Crit}(u)$ 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 $t\in[0, 1)$ regular value of $u$ there exists $\epsilon_{t} > 0$ such that $(t-\epsilon_{t}, t+\epsilon_{t})\cap [0, 1)$ does not contain any critical value of $u$.

Proof. Each level set of $u$ is compact, due to the forth condition in (1.1), while the compactness of $\mathrm{Crit}(u)$ follows by (2.4). Now, consider the nontrivial case where $\mathrm{Crit}(u)\neq\emptyset$ and let $p$ be a point of a critical level set $\{u = t\}$. Take a chart $(U_{p}, \psi_{p})$ centred at the point $p$ with $\psi_{p}(U_{p}) = B_{1}$. Setting $\widetilde{g}_{0} = (\psi_{p})_{*}g_{0} = \widetilde{g}_{0;ij}dx^{i}\otimes dx^{j}$, there exists $C > 0$ such that

for each $(v^{1}, \dots, v^{n})\in \mathbb{R}^{n}$ and for each $x\in \overline{B_{\frac{1}{2}}}\,$. The same condition is satisfied by the coefficients $\widetilde{g}_{0}^{\, ij}$. In particular, setting $\widetilde{u} = u\circ\psi_{p}^{-1}$, we have that

We observe that $\Delta_{\widetilde{g}_{0}}v = a_{ij}\partial_{i}\partial_{j}v+b_{i}\partial_{i}v = 0$ is an elliptic partial differential equation with coefficient $C^{\infty}$ in $\overline{ B_{\frac{1}{2}}}$. We recall that if $v$ is a $C^{\infty}$–solution of the above equation and if $v$ vanishes to infinite order at a point $x_{0}\in B_{\frac{1}{2}}$, i.e., for every $k > 0$

then $v$ is identically zero in $B_{\frac{1}{2}}$ (see [11,Theorem 1.2]). Applying this fact to $\widetilde{u}-t$, one can argue that $\widetilde{u}-t$ has finite order of vanishing at $O$. Then, by using [16,Theorem 1.7], there exists $0 < \rho < \frac{1}{2}$ such that

Since $\widetilde{u}$ is nonconstant in $B_{\frac{1}{2}}$ and by the structure and regularity of $\Delta_{\widetilde{g}_{0}}$, [15,Theorem 1.1] yields

Hence, since the restriction of $\psi_{p}$ to $\psi_{p}^{-1}(B_{1/2})$ is bilipschitz due to (2.25) and since the measures $\mathcal{H}^{k}_{\psi_{p}^{-1}(B_{1/2})}$ and $\mathcal{H}^{k}_{M} \llcorner \psi_{p}^{-1}(B_{1/2})$ coincide on borel sets, statements $(i)$ and $(ii)$ are true locally. In turn, by compactness of $\{u = t\}$ and $\mathrm{Crit}(u)$, 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 $\overline{t}\in[0, 1)$ regular value such that, for all $m\geq \overline{m}$ with $\frac{1}{\overline{m}} < 1-\overline{t}$, the interval $(\overline{t}-\frac{1}{m}, \overline{t}+\frac{1}{m})\cap [0, 1)$ contains critical values. Hence there is a sequence $\{t_{m}\}_{m\geq \overline{m}}$ of critical values such that $t_{m}\to\overline{t}$. In particular, there exists a sequence $\{p_{m}\}_{m\geq \overline{m}}$ of critical points contained in the set $\{0\leq u\leq \overline{t}+\frac{1}{m}\}$ and such that $u(p_{m}) = t_{m}$. Then, by compactness and up to a subsequence, $p_{m}\to p$. In turn, $0 = \vert \mathrm{D}_{g_{0}} u\vert_{g_{0}} (p_{m})\to \vert\mathrm{D}_{g_{0}} u\vert_{g_{0}} (p)$ and $t_{m} = u(p_{m})\to u(p)$. Hence $\vert\mathrm{D}_{g_{0}} u\vert_{g_{0}} (p) = 0$ and $u(p) = \overline{t}$, which is absurd. This concludes the proof of $(iii)$.

Remark 2.1. It is useful to observe that:

(i) for every $t\in(0, 1)$, the set $\{u\geq t\}$ is connected;

(ii) for every $t\approx1$, the level set $\{u = t\}$ is regular and diffeomorphic to $\mathbb{S}^{n-1}$;

(iii) For every $t\in(0, 1)$, $\{u\geq t\} = \overline{\{u > t\}}$ and $\{0\leq u\leq t\} = \overline{\{0 < u < t\}}$.

We check $(ii)$ first. We start by observing that due to (2.1) $\vert\mathrm{D}_{g_{0}} u\vert_{g_{0}}\neq0$ in $\{u\geq t_0\}$, for some $0 < t_0 < 1$. This fact establishes a diffemorphism between $\{u\geq t_0\}$ and $\{u = t_0\}\times[t_0, 1)$ and tells us at the same time that the level sets $\{u = t\}$ are pairwise diffeomorphic, for every $t\geq t_0$. 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 $K\subset M$ such that $M\setminus\mathring K$ is diffeomorphic to $\mathbb{R}^n\setminus\mathring B$ by a chart at infinity $\psi$, where $B$ is a suitable ball, and we can suppose, up to a bigger $t_0$, that $\{u\geq t_0\}\subseteq M\setminus\mathring K$. Now, in view of the asymptotic expansion of $u$, there exist two positive constants $A < B$ such that

In particular, setting $R_0 = [B/(1-t_0)]^{1/(n-2)}$, we have that

At the same time, we have that $\{|x| > R_0\}$ is connected and each $C_i\times[t_0, 1)$ is a closed set of $M$, so that indeed $\{|x| > R_0\}\, \subseteq\, C_i\times[t_0, 1),$ for some $i\in\{1, 2\}$.Therefore, we have that

which gives the contradiction that the noncompact set $\{C_j\times[t_0, 1)\big\}$, where $j\in\{1, 2\}\setminus\{i\}$, is contained into the compact one $(M\setminus\mathring K)\cap\{|x|\leq R_0\}$. Therefore, $\{u = t_0\}$ is connected. Now, setting $\widetilde{u}: = u\circ\psi^{-1}$, we have that, up to a bigger $t_{0}$ and due to (2.5), the set $\{\widetilde{u} = t_{0}\}$ is a compact and connected hypersurface of $\mathbb{R}^{n}$ having strictly positive sectional curvature, as a Riemannian submanifold of $(\mathbb{R}^{n}, g_{ \mathbb{R}^{n}})$. Hence, $\{\widetilde{u} = t_{0}\}$ is diffeomorphic to $\mathbb{S}^{n-1}$ by the Gauss map (see [10,Section 5.B] for more details). Statement $(ii)$ thus follows, being $\{u = t_0\}$ and $\{\widetilde{u} = t_{0}\}$ diffeomorphic.

To see $(i)$, observe first that if $t$ is a regular value of $u$, then $E_{t}: = \{u\geq t\}$ 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 $u\to 1$ at $\infty$, we have that $u(C) = [t, 1)$, and hence $C\cap\{u = t_{0}\}\neq\emptyset$, for every $t_{0}\in(t, 1)$. Then, $E_{t}$ is connected by $(ii)$. If $t$ is a critical value of $u$, we let $\overline{t} > t$ be a regular value of $u$ such that $\{u = \overline{t}\}$ is connected and let $\{t_{m}\}$ be a nondecreasing sequence of regular value of $u$ such that $t_{m} < t$ and $t_{m}\to t$. Hence, $\big\{ \, \{t_{m}\leq u\leq \overline{t}\}\, \big\}_{m\in \mathbb{N}}$ is a nonincreasing family of connected and compact sets in $M$, which is Hausdorff, and in turn the intersection $\{t\leq u\leq \overline{t}\}$ is still connected. In particular, we deduce that $E_t = \{t\leq u\leq \overline{t}\}\cup \{u\geq\overline{t}\}$ is connected.

To check $(iii)$, note first that for every $t\in(0, 1)$ regular value of $u$, the equalities are always true. If $t\in(0, 1)$ is a critical value of $u$, by Theorem 2.3 and by the Maximum Principle the interior of $\{0\leq u\leq t\}$ and the interior of $\{u\geq t\}$ are both disjoint from $\{u = t\}$, so that $(iii)$ is still true.

Let $(M, g_{0}, u)$ be a sub-static harmonic triple, and let $t\in[0, 1)$ be a real number. We consider the spaces of measure

Let $f:S\to \mathbb{R}$ be a continuous and consider the zero–extension $\mathring{f}$ of $f$ defined on $\{u = t\}$ as

By Theorem 2.3 and by definition of the Lebesgue integral, we have that $\mathring{f}\in L^{1}\big(\mathcal{H}^{n-1}_{M} \llcorner \{u = t\}\big)$ iff $f \in L^{1}(\sigma_{g_{0}})$ and

Similarly, if $f:\{u = t\}\to \mathbb{R}$ is a continuous function, then $f\in L^{1}\big(\mathcal{H}^{n-1}_{M} \llcorner \{u = t\}\big)$ iff $f|_{S}\in L^{1}(\sigma_{g_{0}})$ and

In the rest of this paper, we will confuse the integrals of cases (2.26) and (2.27), denoting both by $\int\limits_{\{u = t\}}f d\sigma_{g_{0}}$.

3.   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 (1.7). 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 $\vert \mathbb{S}^{n-1}\vert$, which always stands for the Euclidean volume of $\mathbb{S}^{n-1}$.

Theorem 3.1 (Monotonicity and Outer Rigidity Theorem). Let $(M, g_{0}, u)$ be a sub-static harmonic triple, and let $F_{\beta}:\tau \in[1, +\infty)\to [0, +\infty)$ be the function defined by

for every $\beta\geq0$. Then, the following properties hold true.

$(i)$ Differentiability, Monotonicity and Outer Rigidity: for every $\beta > \frac{n-2}{n-1}$, the function $F_{\beta}$ is continuously differentiable with nonpositive derivative in $(1, +\infty)$. Moreover, if there exists $\tau_{0} \in (1, +\infty)$ such that $F'_{\beta}(\tau_{0}) = 0$ for some $\beta > \frac{n-2}{n-1}$, then, setting $t_{0} = \sqrt{\frac{\tau_{0}-1}{\tau_{0}+1}}$, the Riemannian submanifold $\{u\geq t_{0}\}$

is isometric to

$(ii)$ Convexity: for every $\beta > \frac{n-2}{n-1}$, the function $F_{\beta}$ is convex on $[1, \infty)$.

We remark that the functions $F_{\beta}$ are well–defined, in view of Theorem 2.3 and since the integrand function in (3.1) is bounded on every level set of $u$. Note that, from Theorem 3.1 and by a simple argument based on the Dominated Convergence Theorem, the monotonicity and the convexity of $F_{\beta}$ extend to the case $\beta = \frac{n-2}{n-1}$. Moreover, on the values $\tau$ such that $\Big\{u = \sqrt{\frac{\tau-1}{\tau+1}}\Big\}$ is regular and thus on a.e. $\tau > 1$ due to Theorem 2.3 $(iii)$, the function $F_{\beta}$ is twice differentiable for each $\beta > \frac{n-2}{n-1}$, 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 $\mathrm{H}$ and $\mathrm{h}$ stand respectively for the mean curvature and the second fundamental form of the smooth $(n-1)$–dimensional submanifold $\big\{u = \sqrt{\frac{\tau-1}{\tau+1}}\, \big\}$, with respect to the $\infty$–pointing unit normal vector field $\nu = \frac{\mathrm{D}u}{\vert \mathrm{D}u \vert}$. Also, $\mathrm{D}^{\mathrm{T}}$ denotes the tangential part of the gradient, that is

for every $f\in C^{1}(M)$.

To prove Theorem 3.1, we use the results of Section 4, which are obtained in the conformal setting defined by

Denoting by $\nabla$ and $\Delta_{g}$ the Levi–Civita connection and the Laplace–Beltrami operator of $g$, the triple $(M, g, \varphi)$ satisfies the following system.

Moreover, we have that

as we will see in the proof of Lemma 4.1.

Remark 3.1. Since $\mathrm{Crit}(\varphi) = \mathrm{Crit}(u) = \{\vert \nabla \varphi \vert_{g} = 0\}$ by the equality in ($\star$) and since $\{\varphi = s\} = \{u = \tanh\big(\frac{s}{2}\big)\}$ by (3.4), using Theorem 2.3 and Remark 2.1 we deduce that: $\mathrm{Crit}(\varphi)$ has zero $\mu_{g}$–measure and zero $(n-1)$–Hausdorff measure in $(M, g)$; the level sets of $\varphi$ have finite $(n-1)$–Hausdorff measure in $(M, g)$ and in particular the smooth $(n-1)$–dimensional submanifolds $\{\varphi = s\}\setminus\mathrm{Crit}(\varphi)$ have finite $g$–area, i.e., finite $\sigma_{g}$–measure. Moreover, $\{\varphi\geq s\}$ is connected for every $s\geq 0$ and there exists $s_0\geq0$ such that $\{\varphi = s\}$ is regular and diffeomorphic to $\mathbb{S}^{n-1}$, for every $s\geq s_0$. Similar comments as those at the end of Subsection 2.2 can be made, regarding the relation between integration and $\mathrm{Crit}(\varphi)$.

Let $\Phi_{\beta}:[0, \infty)\to \mathbb{R}$ be the function defined by formula

for every $\beta\geq 0$. For the convenience of the reader, we anticipate from Section 4 the properties of $\Phi_{\beta}$ that we are going to use.

($\circ$) For every $\beta\geq0$, the function $\Phi_{\beta}(s)$ is continuous in $[0, +\infty)$.

($\diamond$) For every $\beta > \frac{n-2}{n-1}$, the function $\Phi_{\beta}$ is continuously differentiable in $(0, +\infty)$. The derivative $\Phi'_{\beta}$ is nonpositive, satisfies for every $S > s > 0$

and admits for every $s > 0$ the integral representation

where $Q$ is defined in (4.10).

($\diamond\, \diamond$) If there exists $s_{0} > 0$ such that $\Phi'_{\beta}(s_{0}) = 0$ for some $\beta > \frac{n-2}{n-1}$, then $\{\varphi = s_{0}\}$ is connected and $(\{\varphi\geq s_{0}\}, g)$ is isometric to $\big([0, +\infty)\times \{\varphi = s_{0}\}, d\rho\otimes d\rho+g_{\{\varphi = s_{0}\}})$, where $\rho$ is the $g$–distance function to $\{\varphi = s_{0}\}$ and $\varphi$ is an affine function of $\rho$ in $\{\varphi\geq s_{0}\}$. If $\Phi_{\beta}$ is constant for some $\beta > \frac{n-2}{n-1}$, then $\partial M$ is connected and $(M, g)$ is isometric to $\big([0, +\infty)\times \partial M, d\rho\otimes d\rho+g_{\partial M})$, where $\rho$ is the $g$–distance function to $\partial M$ and $\varphi$ is an affine function of $\rho$.

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 $1$: Differentiability, Monotonicity and Convexity. For every $\beta\geq0$ and for all $\tau \in[1, +\infty)$, we note that

Consequently, by $(\circ)$, we deduce that for every $\beta\geq0$ the function $F_{\beta}$ is continuous in $[1, +\infty)$. By $(\diamond)$ and by (3.7), we obtain immediately that for every $\beta > \frac{n-2}{n-1}$ the function $F_{\beta}$ is continuously differentiable in $(1, +\infty)$, with

In particular, from $(\diamond)$ we get $F'_{\beta}\leq 0$. As for the convexity, noticing that

and that the function $\log\big(\, \frac{\sqrt{\tau+1}+\sqrt{\tau-1}}{\sqrt{\tau+1}-\sqrt{\tau-1}}\, \big)$ is nondecreasing, from (3.6) we obtain that $F'_{\beta}\,$ is nondecreasing in $(1, +\infty)$. Therefore also by the continuity of $F_{\beta}$ at $1$, $F_{\beta}$ is convex in $[1, +\infty)$.

Step $2$: Outer Rigidity. Let us assume that there exists $\tau_{0} \in (1, \infty)$ such that $F'_{\beta}(\tau_{0}) = 0$ for some $\beta > \frac{n-2}{n-1}$. Then, by equality (3.8), $\Phi'_{\beta}(s_{0}) = 0$ for $s_{0} = \log\big(\, \frac{\sqrt{\tau_{0}+1}+\sqrt{\tau_{0}-1}}{\sqrt{\tau_{0}+1}-\sqrt{\tau_{0}-1}}\, \big)$, and hence, by $(\diamond\, \diamond)$, $(\{\varphi\geq s_{0}\}, g)$ is isometric to $\big([0, +\infty)\times \{\varphi = s_{0}\}, d\rho\otimes d\rho+g_{\{\varphi = s_{0}\}})$, where $\rho$ is the $g$–distance function to $\{\varphi = s_{0}\}$ and $\varphi = (n-2)\, (2\mathcal{C})^{-\frac{1}{n-2}}\rho+s_{0}$, because $\vert \nabla \rho\vert_{g}\equiv1$ and in view of the limit in $(\star)$. Setting $t_{0} = \tanh \frac{s_{0}}{2}$, consider $N$ the submanifold with boundary $\{\varphi\geq s_{0}\} = \{u\geq t_{0}\}$. 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 $p\to (\rho, q)$ 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 $\partial N$ and as second coordinate the point $q$ of $\partial N$ that realizes such distance. Then, in view of (3.4) and with the same notation as above, the following Riemannian manifolds are isometric.

where $r_{0} = \big(\frac{2\mathcal{C}}{1-t_{0}^{2}}\big)^{\frac{1}{n-2}}$. Doing some computations, we obtain that

where the convection followed for the Riemannian curvature tensor is that given in ^[23], $c$ is a suitable positive constant and $q$ is the point of $\partial N$ that realizes the $g$–distance of $p$ from $\partial N$. Denoting by $\Theta$ the diffeomorphism from $N$ to $[r_{0}, +\infty)\times \partial N$ introduced in (3.9), for every $q_{0}\in \partial N$ we consider the curve

and observe from (3.10) that

At the same time, we have that

This is because $g_{0}$ is asymptotically flat according to Definition 1.1 and by (2.3), which yields in particular

for some $p > \frac{n-2}{2}$. Combining (3.11) and (3.12), the arbitrariness of the point $q_{0}$ in $\partial N$ gives that

Hence the sectional curvature of the Riemannian manifold $(\partial N, g_{\partial N})$ is constant and identically equal to $(2\mathcal{C}\, )^{-\frac{2}{n-2}}$. Then, being all the level sets $\{u = t\}$ with $t\approx1$ regular and diffeomorphic to $\mathbb{S}^{n-1}$ as observed in Remark 2.1, for [10,Section 3.F] $(\partial N, g_{\partial N})$ and $(\mathbb{S}^{n-1}, (2\mathcal{C}\, )^{\frac{2}{n-2}}g_{ \mathbb{S}^{n-1}})$ are isometric. Then, $(\{u\geq t_{0}\}, g_{0})$ is isometric to the submanifold $\big(\, [r_{0}, +\infty)\times \mathbb{S}^{n-1}\, , \frac{dr \otimes dr}{1-2\mathcal{C}r^{2-n}}+r^{2}g_{ \mathbb{S}^{n-1}} \big)$ of the Schwarzschild manifold with associated ADM mass given by $\mathcal{C}$.

Proof of Theorem 1.1. Spep $1$: Inequality. By Theorem 3.1, we have that $F_{\beta}(\tau_{0})\geq \lim_{\tau\to+\infty}\, F_{\beta}(\tau)$, for every $\tau_{0} > 1$. In particular, since $F_{\beta}$ is continuous in $[1, +\infty)$ due to the step $1$ of Theorem 3.1, we have that

for every $\beta > \frac{n-2}{n-1}$. Since $\mathrm{D}^{2}u \equiv 0$ on $\partial M$ and since $\partial{M}$ of $M$ is connected, $\vert \mathrm{D} u \vert$ is constantly equal to $\frac{(n-2)\, \mathcal{C}\, \vert \mathbb{S}^{n-1}\vert\, }{\vert \partial M\vert}\,$, by formula (2.2). In particular, we have that

By $(\star)$, we know that

Therefore, fixed $\varepsilon > 0$, there exists $1 < \tau_{0} < +\infty$ such that

in $\big\{u\geq\sqrt{\frac{\tau_{0}-1}{\tau_{0}+1}}\, \big\}$ and the level sets $\big\{u = \sqrt{\frac{\tau-1}{\tau+1}}\, \big\}$ are regular for all $\tau\geq \tau_{0}$. Therefore, for every $\tau\geq \tau_{0}$ 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 (2.2). Since $\varepsilon$ is arbitrary, we get

In a similar way we can obtain the reverse inequality, so that

Joining the formulas in (3.13), (3.14) and (3.15), we obtain the desired inequality (1.7).

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 (1.7) are both equal to $m$, by direct computation.

Suppose now that the equality holds in (1.7). Then, by Step 1 and for every $\beta > \frac{n-2}{n-1}$, the function $F_{\beta}$ is constant. In turn, $\Phi_{\beta}$ is constant, being

Finally, $(\diamond\, \diamond)$ 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 $\rho$ is the $g$–distance to $\partial M$ and $\varphi$ is an affine function of $\rho$, and secondly that $(M, g_{0})$ is isometric to the Schwarzschild manifold with ADM mass $\mathcal{C}$.

4.   Conformal setting

Let us consider the conformal change $g$ of the metric $g_0$ introduced in (3.4) which is well–defined being $0\leq u < 1$ in $M$. The metric $g$ is complete, since any $g$–geodesic $\gamma$ parametrized by $g$–arc length defined on a bounded interval $[0, a)$ can be extended to a continuous path on $[0, a]$. Indeed, if $\gamma$ has infinity length with respect to $g_{0}$, there exists a sequence $\{t_{m}\}_{m\in \mathbb{N}}$ such that $\gamma(t_{m})\to \infty$ (being $\gamma$ not contained in any compact set) and using, in the computation of $g$–length of $\gamma$, the passage from $g$ to $g_{0}$, the asymptotic flatness of $(M, g_{0})$ and the asymptotic expansion of $u$ in (2.3) we obtain that $\gamma$ has infinity length with respect to $g$. Hence $\gamma$ has finite length with respect to $g_{0}$ and, being $g_{0}$ complete, it follows that $g$ is complete (see [24,Section 1.1] and ^[9]). 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 $C^{\infty}$–function $\varphi$, defined in (3.4). Now, the reverse changes are

Recalling that we denote by the symbols $\nabla$ and $\Delta_{g}$ the Levi–Civita connection and the Laplace–Beltrami operator of $g$, by the formulas in [6,Theorem 1.159], we obtain

Translating system (1.1) in terms of $g$ and $\varphi$, we get system (3.5). Moreover, on $\{\varphi = s\}\setminus\mathrm{Crit}(\varphi)$ we consider the $\infty$–pointing normal unit vector fields

the mean curvatures

and the second fundamental forms

for any $X, Y$ tangent vector fields to the considered submanifold.

Reversing formulas (4.1), (4.2) and (4.3), 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 $\vert \nabla \varphi \vert_{g}$ and $\vert \nabla^{2} \varphi \vert_{g}$, and for the $g$–areas of the level sets of $\varphi$ sufficiently "close" to infinity. This is the content of the following lemma.

Lemma 4.1. There exists $0\leq s_{0} < +\infty$ such that

Proof. Let $\psi$ be a chart at infinity. Considering $\widetilde{g}_{0} = \psi_{*}g_{0} = \widetilde{g}_{0;ij}dx^{i}\otimes dx^{j}$, by formulas (2.6) and (2.4), the coordinate expression of

is

Hence

Moreover, by limit (4.6) there exist a constant $L > 0$ and a value $s_{0} > 0$ of $\varphi$ such that every $s\geq s_{0}$ is a regular value of $\varphi$ and $(1-u^{2})^{\frac{n-1}{n-2}}\leq L\, \, \vert \mathrm{D}u \vert$ on $\{\varphi\geq s_{0}\}$. Then

where in the last equality we have applied the Divergence Theorem. Consequently, we have that $\sup\limits_{s\geq s_{0}}\int\limits_{\{\varphi = s\}}d\sigma_{g} < +\infty$. Similarly, we have that

due to formulas (2.6), (2.5) and (5.3). Moreover,

All in all,

which gives

In particular, since $\varphi$ is smooth, we have that

Remark 4.1. Note that $\sup\limits_{s\geq 0}\int\limits_{\{\varphi = s\}}d\sigma_{g}\in(0, +\infty]$, since we cannot a priori exclude that there exist a critical value $\overline{s} > 0$ and a sequence $\{s_{m}\}\subset (0, +\infty)$ such that $s_{m}\to\overline{s}$ and

As it will be clear in the proof of the integral identity (4.17), which is at the core of the conformal–monotonicity result (Proposition 4.6), it is useful to introduce a suitable vector field with nonnegative divergence. To do this, let us focus on the set $\mathring{M}\setminus\mathrm{Crit}(\varphi)$ and notice first that the classical Bochner formula, applied to the $g$–harmonic function $\varphi$, becomes

Then, we obtain

where in the third equality we have used (4.7). Now, observe from the nonnegativity of the tensor

(see (3.5)) that

Therefore, by adding and subtracting the term $\beta\, \vert\nabla \varphi \vert^{\beta-2}_{g}\, \coth(\varphi)\, \nabla ^{2}\varphi(\nabla \varphi, \nabla \varphi)$ on the right–hand side of (4.8), we get

Since

and since, setting

there holds

from (4.11) we get

Note that, by the refined Kato inequality for harmonic function

we have that

whenever $\beta\geq\frac{n-2}{n-1}$. Hence, $\mathrm{div}_{g}\, Y_{\beta}\geq 0$ for every $\beta\geq\frac{n-2}{n-1}$. This fact will be heavily used in the proof of the forthcoming results. It will also be useful to have a precise estimate of $\int\limits_{\{\vert \nabla \varphi \vert_{g} = \delta\}}\big\vert\, \nabla\vert \nabla \varphi \vert_{g}\, \, \big\vert_{g}\, d\sigma_{g}$ in terms of a suitable power of $\delta$, close to $\mathrm{Crit}(\varphi)$, that is when $\delta\to 0^{+}$. This is the content of the following lemma.

Lemma 4.2. There exists $\delta_{0} > 0$ such that

We recall that the set of the critical values of $\vert \nabla \varphi \vert^{2}_{g}$ has zero Lebesgue measure by Sard's Theorem, whereas we have no information regarding the local $\mathcal{H}$–dimension of $\mathrm{Crit}(\vert \nabla \varphi \vert^{2}_{g})$.

Proof. Applying Sard's Theorem to the smooth function $\vert \nabla \varphi \vert^{2}_{g}$ there exists $\varepsilon_{0} > 0$ such that $\varepsilon_{0}$ is a regular value of $\vert \nabla \varphi \vert^{2}_{g}$ and

where the limit in the previous expression is the (finite and positive) value computed in (4.6). In particular, $\{\vert \nabla \varphi \vert^{2}_{g}\leq \varepsilon_{0}\}$ is compactly contained in $\mathring{M}$, and for every $0 < \delta < \delta_{0}$ regular value of $\vert \nabla \varphi \vert_{g}$ we have that

Now, consider the smooth vector field

with $\mathrm{div}_{g}Z\geq0$. Set

Then, for every $0 < \varepsilon < \varepsilon_{0}$ regular value of the function $\vert \nabla \varphi \vert^{2}_{g}$, we apply the Divergence Theorem to the smooth vector field $Z$ on $U_{\varepsilon_{0}} \setminus \overline{U_{\varepsilon}}$, and we get

Then, it follows

Therefore, setting

we obtain

Consequently, the desired statement follows keeping in mind that: if $\delta$ is a regular value of $\vert \nabla \varphi \vert_{g}$, then $\delta^{2}$ is a regular value of $\vert \nabla \varphi \vert_{g}^{2}$; in $M\setminus\mathrm{Crit}(\varphi)$ we have $\nabla\vert \nabla \varphi \vert^{2}_{g} = 2\vert \nabla \varphi \vert_{g}\nabla\vert \nabla \varphi \vert_{g}$.

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 $\varphi$ be the metric and the function defined in (3.4). Then, for every $\beta > \frac{n-2}{n-1}$ and for every $S > s > 0$ regular values of $\varphi$, it holds

where the tensor $Q$ is defined as in (4.9).

Proof. The case $\beta\geq 2$ 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 $\frac{n-2}{n-1} < \beta < 2$. In $\mathring{M}\setminus\mathrm{Crit}(\varphi)$ we consider the smooth vector field $Y_{\beta}$, defined in (4.12) and satisfying

as already explained. Set

When $E_{s}^{S}\cap \mathrm{Crit}(\varphi) = \emptyset$, then the statement is a straightforward application of the Divergence Theorem. Now, suppose that $E_{s}^{S}\cap \mathrm{Crit}(\varphi)\neq\emptyset$. In this case we consider, for every $\varepsilon > 0$ sufficiently small, a smooth nondecreasing cut–off function $\chi_{\varepsilon}:[0, +\infty)\to [0, 1]$ satisfying the following conditions

where $c$ is a positive real constant independent of $\varepsilon$. We then define the smooth function $\Xi_{\varepsilon}:M\to [0, 1]$ as

and apply the Divergence Theorem to the smooth vector field $\Xi_{\varepsilon}\, Y_{\beta}$ in $E_{s}^{S}$. In this way, we get

where $U_{\mu}$ is defined in (4.16). Note that $\{\chi_{\varepsilon}\}$ can always be chosen to be nondecreasing in $\varepsilon$ so that, in turn, $\{\Xi_{\varepsilon}\}$ is nondecreasing. Therefore, applying the Monotone Convergence Theorem, when $\varepsilon\to 0^+$, 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 $\chi_{\varepsilon}$, 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 $\varepsilon_{0}, \, c_{1} > 0$ such that the inequality

is true a.e. $s\in[\frac{1}{2}\varepsilon, \frac{3}{2}\varepsilon]$ for every $0 < \varepsilon < \frac{2}{3}\varepsilon_{0}$, by both Sard's Theorem applied to the smooth function $\vert \nabla \varphi \vert^{2}_{g}\,$ and by Lemma 4.2. Then, we get

where $c_{2} > 0$ is sufficiently big constant. This implies the limit in (4.19), because $\beta > \frac{n-2}{n-1}$.

Corollary 4.4. For every $S > s > 0$ regular values of $\varphi$ there exists $r_{s, S}\geq0$ such that

Proof. Let $\{\beta_{m}\}_{m\in \mathbb{N}}$ be a sequence such that $\beta_{m} > \frac{n-2}{n-1}$ and $\beta_{m}\to\frac{n-2}{n-1}$. Due to Proposition 4.3, we have

where the first equality is consequence of the Dominate Converge Theorem keeping in mind that $s$ and $S$ are regular values of $\varphi$ while the inequality follows from Fatou's Lemma. Since $\{\beta_{m}\}_{m\in N}$ is arbitrary, the quantity

is well–defined. Moreover, it is nonnegative as above and therefore we get the statement.

Remark 4.2. For every $\beta > \frac{n-2}{n-1}$ and for every $s > 0$ regular value of the function $\varphi$:

For every $S$ big enough, which is a regular value of $\varphi$, by Lemma 4.5 with (4.6) we have

In particular,

Therefore, the desired identity can be obtained by the Monotone Convergence Theorem, by passing to the limit as $S\to+\infty$ in (4.17).

Remark 4.3. For every $\beta > \frac{n-2}{n-1}$, as consequence of integral identity (4.17), we have

Since

for every $K\subset \mathring{M}$ compact, by Hölder's Inequality from (4.21) with (4.14) we get that

We need a final lemma before stating the (last and) most important result of this section.

Lemma 4.5. Let $(M, g_{0}, u)$ be a sub-static harmonic triple, and let $g$ and $\varphi$ be the metric and the function defined in (3.4). Then, the following statements hold true.

$(i)$ For every $\beta\geq0$ and for every $S > s > 0$:

$(ii)$ For every $\beta\geq0$ and for every $s > 0$:

$(iii)$ The function $\Phi_{\beta}:[0, \infty)\to \mathbb{R}$, defined by formula

for every $\beta\geq0$, 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 $\mathrm{Crit}(\varphi)$: 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 $\varphi$ be the metric and the function defined in (3.4), and let $\Phi_{\beta}:[0, \infty)\to \mathbb{R}$ be the function defined by formula (4.24) for every $\beta\geq0$. Then for every $\beta > \frac{n-2}{n-1}$, the function $\Phi_{\beta}$ is continuously differentiable. The derivative $\Phi'_{\beta}$ is nonpositive and admits for every $s > 0$ the integral representation

Moreover, if there exists $s_{0} > 0$ such that $\Phi'_{\beta}(s_{0}) = 0$ for some $\beta > \frac{n-2}{n-1}$, then $(\{\varphi\geq s_{0}\}, g)$ is isometric to $\big([0, +\infty)\times \{\varphi = s_{0}\}, d\rho\otimes d\rho+g_{\{\varphi = s_{0}\}})$, where $\rho$ is the $g$–distance function to $\{\varphi = s_{0}\}$ and $\varphi$ is an affine function of $\rho$ in $\{\varphi\geq s_{0}\}$.

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 $\beta > \frac{n-2}{n-1}$. Note that the boundary $\partial M$ is a regular level set of $\varphi$ and then, by Theorem 2.3 and the relationship between $\mathrm{Crit}(u)$ and $\mathrm{Crit}(\varphi)$, there exists $\epsilon_{0}$ such that the interval $[0, \epsilon_{0}]$ doesn't contain critical values of the function $\varphi$. Therefore, for every $0 < \epsilon\leq\epsilon_{0}$, applying first the Divergence Theorem to the smooth vector field $\vert \nabla \varphi \vert^{\beta}_{g}\, \nabla \varphi$ in $\{0 < \varphi < \epsilon\}$ and later the Coarea Formula, we get

Being

for every $0\leq s_{1} < s_{2}\leq \epsilon_{0}$, by the Dominated Convergence Theorem the function

is continuous and therefore, by the Fundamental Theorem of Calculus $\Phi_{\beta}$ is continuously differentiable on the closed interval $[0, \epsilon_{0}]$.

Let $s_{0}$ be a regular value of the function $\varphi$. By Remark 4.3, we can define the function $\Psi_{\beta}:(0, +\infty)\to \mathbb{R}$ by

which satisfies the following properties

$(i)$ for every $s > 0$ regular value of the function $\varphi$, we have $\Psi_{\beta}(s) = \int\limits_{\{\varphi = s\}} \frac{\vert \nabla \varphi \vert^{\beta}_{g}\, \, \mathrm{H}_{g}}{\sinh\varphi}\, d\sigma_{g}$;

$(ii)$ the function $\Psi_{\beta}$ is continuous on its definition interval $(0, +\infty)$.

The first statement follows immediately from Proposition 4.3. As for the second statement, we first observe that

for every couple $0 < s < \overline{s} < +\infty$. Always by Remark 4.3 and by the Dominated Convergence Theorem, we can deduce the right and the left continuity of $\Psi_{\beta}$ on the interval $(0, +\infty)$.

We consider $\Upsilon_{\beta}:s\in(0, +\infty)\to\frac{\Phi_{\beta}(s)}{\sinh s}\in \mathbb{R}\,$. For every $(s, \overline{s})$ couple of real number such that $0 < s < \overline{s} < +\infty$, we have

where the first equality follows from Lemma 4.5 $(i)$, the second equality from the Coarea Formula keeping in mind (4.22). Moreover, the last equality follows from $(i)$ and from Sard's Theorem. Using the continuity of both the functions $\Upsilon_{\beta}$ and $\Psi_{\beta}$, passing to the limit in $(\star)$ for either $s\to \overline{s}$ or $\overline{s}\to s$ yields that the function $\Upsilon_{\beta}$ is $C^{1}$, and

Since $\Phi_{\beta}(s) = \sinh(s)\Upsilon_{\beta}(s)$ for every $s > 0$, then $\Phi_{\beta}\in C^{1}(0, +\infty)$ and $\Phi'_{\beta}(s) = -\beta\, \sinh(s)\Psi_{\beta}(s)$. Moreover, by (4.26), we can see

for every $0 < s < S < +\infty$.

Finally the integral representation (4.25) follows in the limit as $S\to +\infty$ of the above identity, by using the Monotone Convergence Theorem, and by the fact that

Step $2$: Outer Rigidity. Let $\beta > \frac{n-2}{n-1}$ and suppose $\Phi'_{\beta}(s_{0}) = 0$ for some $s_{0} > 0$. By (4.25) with (4.14) we deduce that

Consequently $\nabla^{2} \varphi\equiv 0$ in $\{\varphi\geq s_{0}\}$ being $\mu_{g}\big(\mathrm{Crit}(\varphi)\, \big) = 0$, and hence $\vert \nabla \varphi \vert_{g}^{2}\equiv a^{2}$ with $a > 0$ since $\{\varphi\geq s_{0}\}$ is connected, due to Remark 3.1. Then, $\{\varphi\geq s_{0}\}$, with the induced Riemanninan metric, is a noncompact, connected and complete Riemannian manifold (being properly embedded in $M$), with smooth, compact and totally geodesic boundary, and with $\mathrm{Ric}_{g}\geq0$ (from the inequality in (3.5)). Applying [18,Theorem C], we can thus deduce that the level set $\{\varphi \, = \, s_{0}\}$ is connected (this is true in general and not only in the rigid case, if $s_0\gg0$, as observed in Remark 3.1), and that $\{\varphi\geq s_{0}\}$ is isometric to the product $[0, + \infty)\times\{\varphi \, = \, s_{0}\}$. Moreover, the isometry from the product $[0, + \infty)\times\{\varphi \, = \, s_{0}\}$ to $\{\varphi\geq s_{0}\}$ is given by the normal exponential map.

Now we want to prove that $\varphi$ is an affine function of $\rho$ on $\{\varphi\geq s_{0}\}$. First, we remark that every integral curve $\gamma_{p}$ of $\nabla \varphi$ outgoing from a point $p$ of $\{\varphi \, = \, s_{0}\}$ is defined on the interval $[0, +\infty)$, and it is contained in $\{\varphi\geq s_{0}\}$, by the completeness and since $\vert\nabla \varphi\vert_{g} > 0$. Furthermore, $\varphi \circ \gamma_{p}(t) = a^{2}t+s_{0}$ for every $t\in[0, +\infty)$, and all the curves $\gamma_{p}$ realize the distance between the hypersurfaces $\{\varphi = s_{0}\}$ and $\{\varphi = s_{1}\}$ with $s_{1} > s_{0}$. Indeed, for any curve $\sigma:[0, l]\to \{\varphi\geq s_{0}\}$ parametrized by arc–length joining a point of $\{\varphi = s_{0}\}$ to a point of $\{\varphi = s_{1}\}$ we have

where $s_{1}, s_{0}$ and $t$ satisfy $s_{1} = a^{2}t+s_{0}$. Since $\xi = \frac{\, \nabla \varphi }{a}$ is the unit inner normal vector field of the boundary $\{\varphi \, = \, s_{0}\}$ and we just know that the normal exponential map is a diffeomorphism, $\exp^{\bot}(t\xi_{p})$ is a point having distance from $\{\varphi = s_{0}\}$ equal to $t\,$, and therefore

This tell us that $\varphi$ is an affine function of $\rho$ on $\{\varphi\geq s_{0}\}$.

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 $\varphi$ be the metric and the function defined in (3.4), and let $\Phi_{\beta}:[0, \infty)\to \mathbb{R}$ be the function defined by formula (4.24) for every $\beta\geq0$. If $\Phi_{\beta}$ is constant for some $\beta > \frac{n-2}{n-1}$, then $\partial M$ is connected and $(M, g)$ is isometric to $\big([0, +\infty)\times \partial M, d\rho\otimes d\rho+g_{\partial M})$, where $\rho$ is the $g$–distance function to $\partial M$ and $\varphi$ is an affine function of $\rho$.

Proof. We obtain immediately that $\Phi_{\beta}'(s) = 0$ for every $s > 0$. Thus, by formula (4.25) with (4.14) we have that

for every $s > 0$. In turn, by the Monotone Convergence Theorem, we get

Then, we deduce that

due to Kato Inequality for harmonic functions (4.13) and by (4.10). Consequently $\nabla^{2} \varphi\equiv 0$ in $M$. The very same argument of the proof of Outer Rigidity in Proposition 4.6 implies that $\partial M$ is connected and $(M, g)$ is isometric to

where $\rho$ is the $g$–distance to $\partial M$ and $\varphi$ is an affine function of $\rho$.

5.   A Black–Hole uniqueness theorem for sub-static manifolds

This section is devoted to the proof of the Black–Hole uniqueness result for a sub–static harmonic triple, Theorem 1.2. We first recall the classical definition of ADM mass, together with an alternative characterization of it.

Let $(N, h)$ and $\psi$ be an asymptotically flat manifold with one end and a chart at infinity of $N$, respectively. We consider $\widetilde{h}: = \psi_{*}h = \widetilde{h}_{ij}dx^{i}\otimes dx^{j}$ and we set

where $\nu_{e}$ and $\sigma_{e}$ are the $\infty$–pointing unit normal and the canonical measure on $\partial B_{r}$ as Riemannian submanifold of $(\mathbb{R}^{n}\setminus \overline{B}, g_{ \mathbb{R}^{n}})$, respectively, and $\nu_{\widetilde{h}}$ and $\sigma_{\widetilde{h}}$ are the $\infty$–pointing unit normal and the canonical measure on $\partial B_{r}$ as Riemannian submanifold of $(\mathbb{R}^{n}\setminus \overline{B}, \widetilde{h})$, respectively. Also, $\mathrm{Ric}_{\widetilde{h}}$ and $\mathrm{R}_{\widetilde{h}}$ are the Ricci tensor and the scalar curvature of $\widetilde{h}$ respectively, and $X$ is the Euclidean conformal Killing vector field $x^{i}\, \frac{\partial }{\partial x^{i}}$. The ADM mass is well defined as

and independent of the chosen chart at infinity. Moreover (see ^[22]), it can be equivalently expressed as

From the alternative definition of ADM mass, given by (5.1), and using the Positive Mass Theorem, more precisely a consequence of it contained in [17,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.

Proof of Theorem 1.2. By condition (1.8) and by the fact that $\mathrm{D}_{g_{0}}^{2}u\equiv 0$ on $\partial M$, which in turn implies $\mathrm{H}_{\partial M}^{g_{0}}\equiv 0$, we have that the hypothesis of [17,Theorem 1.5] are fulfilled, so that

Now, we want to show that the reverse inequality holds. Let $\psi$ be a chart at infinity of $M$ (according to Definition 1.1) and consider $\widetilde{g}_{0} = \psi_{*}{g_{0}}$. Recalling that $\widetilde{u}$ stands for $u\circ\psi^{-1}$, we rewrite characterization (5.1) as

We note first that since $\nu_{e} = \frac{x^{i}}{\vert x\vert}\, \frac{\partial }{\partial x^{i}} = \frac{1}{\vert x\vert}\, X$ and $\widetilde{u}\, \mathrm{Ric}_{\widetilde{g}_{0}}-\mathrm{D}_{\widetilde{g}_{0}}^{2}\widetilde{u} \geq 0$ from the first equation in (1.1), we have

Secondly, recalling that $(\mathrm{D}_{\widetilde{g}_{0}}^{2}\widetilde{u})_{ij} = \partial_{i}\partial_{j}\widetilde{u}-\Gamma_{ij}^{k}\partial_{k}\widetilde{u}$, where $\Gamma_{ij}^{k}$ are the Christoffel symbols related to $\widetilde{g}_{0}$, and using (1.3), (2.12), and the asymptotic expansions of $\widetilde{u}$, we get

Decay (5.4) coupled with (2.8) (2.10) and (2.11) yields

being $p > \frac{n-2}{2}$. Thirdly, we observe that

Indeed

and one can show, with similar estimates as before, that the first two terms of this sum tend to $0$ for $r\to+\infty$. It is also easy to see, using (2.5) and (2.11), that

Hence, (5.6) is proven. Gathering (5.2), (5.5), and (5.6), we have finally obtained

We remark that the above inequality is true for any $\psi$ chart at infinity of $M$. From now on we assume that $\psi$ satisfies condition (1.8) regarding the decay rate of $\mathrm{R}_{\widetilde{g}_{0}}$ at $\infty$. Since

also using (2.11) we obtain

The fact that $m_{\mathrm{ADM}}\leq \mathcal{C}$ thus follows from (5.7). All in all, the rigidity case $m_{\mathrm{ADM}} = \mathcal{C}$ of [17,Theorem 1.5] holds, which implies that $(M, g_{0})$ is the Schwarzschild manifold.

Acknowledgments

The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA), which is part of the Istituto Nazionale di Alta Matematica (INdAM), and they are partially funded by the GNAMPA project "Aspetti geometrici in teoria del potenziale lineare e nonlineare". F. O. thanks L. Benatti and C. Mantegazza for useful discussions during the preparation of the manuscript.

Conflict of interest

The authors declare no conflict of interest.

A.   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.

Proof of Lemma 4.5 (i). In $\mathring{M}\setminus\mathrm{Crit}(\varphi)$ and for every $\beta\geq 0$, we consider the smooth vector field

which is such that

If $\{s\leq \varphi\leq S\}\cap \mathrm{Crit}(\varphi) = \emptyset$, then the statement is a straightforward application of the Divergence Theorem. Now, suppose that $\{s\leq \varphi\leq S\}\cap \mathrm{Crit}(\varphi) \neq\emptyset$. Since there always exists $\overline{s}\in (s, S)$ regular value of $\varphi$, up to splitting the right–hand side of (4.23) into two subintegrals, we can suppose without loss of generality that one among $s$ and $S$ is a regular value of $\varphi$. To fix the ideas, suppose that $S$ is the regular value. We are going to change the function $\varphi$ in a neighbourhood of the set $\mathrm{Crit}(\varphi)$. To do this, for every $\varepsilon > 0$ sufficiently small, applying Sard's Theorem to the smooth function $\varphi$, we can fix a positive real number $\delta(\varepsilon)$ such that $s+\delta(\varepsilon) < S$ is a regular value of $\varphi$ and $\delta(\varepsilon) < d\, \varepsilon$, where $d > 0$ will be specified later. Then, considering a smooth nonincreasing cut–off function $\zeta_{\varepsilon}:[0, +\infty)\to [0, 1]$ satisfying the conditions

where $c$ is a positive real constant independent of $\varepsilon$, we define

Clearly,

and

Note that $s$ is a regular value for the function $\varphi_{\varepsilon}$. To see this, let $p$ be a point of $\{\varphi_{\varepsilon} = s\}$ and distinguish the two cases

In the first case, $\zeta_{\varepsilon}(\vert \nabla \varphi\vert^{2}_{g})\equiv 1$ so that $s = \varphi_{\varepsilon}(p) = \varphi(p)-\delta(\varepsilon)$ and $\nabla \varphi_{\varepsilon}(p) = \nabla \varphi(p)$. Since $s+\delta(\varepsilon)$ is a regular value for $\varphi$, $\nabla \varphi_{\varepsilon}(p)\neq 0$. In the second case, observing that $s\leq\varphi(p)\leq s+\delta(\varepsilon)$ and therefore $p\in\{s\leq \varphi\leq S\}$, we have from (A.2) that in $p$

where $c$ is the constant appearing in (A.1). Now, observe that $\max\limits_{\{s\leq \varphi\leq S\}}\big\vert\nabla\vert \nabla \varphi \vert_{g}\, \big\vert_{g}\, > 0$, since otherwise, due to the presence of critical points in $\{s\leq \varphi\leq S\}$, there should be a connected component of $\{s\leq \varphi\leq S\}$ where $\nabla \varphi\equiv0$. But this is impossible because $\{s\leq \varphi\leq S\} = \overline{\{s < \varphi < S\}}$ (by Remark 2.1) and by the size of $\mathrm{Crit}(\varphi)$. Hence, choosing

from above we obtain $\vert\nabla \varphi_{\varepsilon}\vert_{g}(p)\geq \frac{\vert\nabla \varphi\vert_{g}}{2}(p)$. In particular, from $(\bigstar)$ we get that $\vert\nabla \varphi_{\varepsilon}\vert_{g}(p) > \frac{\varepsilon}{4}$.

Now, we apply the Divergence Theorem to the smooth vector field $\Xi_{4\varepsilon} X_{\beta}$ on $\{s < \varphi_{\varepsilon} < S\}$, where

Recalling that $U_{\mu}$ is defined as in (4.16), we obtain

Note that $\{\varphi = S\}$ is compactly contained in $\{\vert \nabla \varphi\vert^{2}_{g} > \frac{3}{2}\, \varepsilon\}$ for every $\varepsilon$ sufficiently small, and $\Xi_{4\varepsilon}\equiv0$ in $\{\vert \nabla \varphi\vert^{2}_{g}\leq2\, \varepsilon\}\supset \{\vert \nabla \varphi\vert^{2}_{g}\leq\frac{3}{2}\, \varepsilon\}$. Then, by (A.3) we get

Looking at the left–hand side of (A.4), note that

where in the second inequality we have used Lemma 4.1 and the fact that $U_{\varepsilon}$ is contained in a compact set for every $\varepsilon < < 1$ (which is a consequence of (4.6)). Moreover, by the Dominated Convergence Theorem, we have that

Finally, note that $\{\zeta_{\varepsilon}\}$ can always be chosen to be nonincreasing in $\varepsilon$ so that, in turn, $\{\Xi_{\varepsilon}\}$ in nondecreasing. Therefore, looking at the left–hand side of (A.4), we have that

by the Monotone Convergence Theorem. All in all, passing to the limit as $\varepsilon\to 0^+$ in (A.4), yields the desired identity.

Proof of Lemma 4.5 (ii). Lemma 4.1 implies

Note that

because its absolute value belongs to $L^{1}_{\rm{loc}}\big(\{\varphi\geq s\}, \mu_{g}\big)$ immediately and to $L^{1}\big(\{\varphi\geq S\}, \mu_{g}\big)$ for $S$ sufficiently big, applying the Coarea Formula coupled with (4.6) and Lemma 4.5. Therefore, passing to the limit as $S\to +\infty$ in (4.23) and using the Dominated Convergence Theorem gives the desired identity.

Proof of Lemma 4.5 (iii). Let $\beta\geq0$. We are assuming that the boundary $\partial M$ is a regular level set of $\varphi$ so that there exists $\epsilon > 0$ such that $[0, \epsilon]\cap\mathrm{Crit}(\varphi) = \emptyset$. Therefore, applying the Divergence Theorem to the smooth vector field $\vert \nabla \varphi \vert^{\beta}_{g}\, \nabla \varphi$ in $\{0 < \varphi < \epsilon\}$ yields

In turn, the absolute continuity of the integral implies the continuity of $\Phi_{\beta}$ at $0$. By point $(i)$ and again by the absolute continuity of the integral, we obtain the right and the left continuity of the function

Hence, $\Phi_{\beta}$ is continuous also in $(0, +\infty)$. The integral representation of $\Phi_{\beta}$ follows directly from point $(ii)$.