Mass, Capacitary Functions, and the Mass-to-Capacity Ratio

Abstract

We study connections among the ADM mass, positive harmonic functions, and capacity of the boundary on asymptotically flat 3-manifolds of nonnegative scalar curvature. We start with new formulae detecting the mass via positive harmonic functions. Then we derive a family of monotone quantities and geometric inequalities assuming the manifold has simple topology. As a first application, we observe several additional proofs of the 3-dimensional Riemannian positive mass theorem. One proof leads to new, sufficient conditions implying positivity of the mass via $C^0$-geometry of regions separating the boundary and $\infty$. A special case of such conditions shows if a region enclosing the boundary has relative small volume, then the mass is positive. As further applications, we obtain integral identities for the mass-to-capacity ratio. We also promote the inequalities to become equality on Schwarzschild manifolds outside rotationally symmetric spheres. Among other things, we show the mass-to-capacity ratio is always bounded below by one minus the square root of the normalized Willmore functional of the boundary. Prompted by these findings, we carry out a study of manifolds satisfying a constraint on the mass-to-capacity ratio in the context of the Bartnik quasi-local mass.

1 Introduction and Statement of Results

On an asymptotically flat 3-manifold (M, g), the ADM mass [3] is a flux integral near $\infty$ given by

$$\begin{aligned} \mathfrak {m}= \lim _{ r \rightarrow \infty } \frac{1}{16 \pi } \int _{ |x | = r } \sum _{j, k} ( g_{jk,j} - g_{jj,k} ) \nu ^k. \end{aligned}$$

Here $\{ x_i \}_{1\le i \le 3}$ is a coordinate chart defining the asymptotic flatness of (M, g) and $\nu = |x|^{-1} x$ denotes the coordinate unit normal to $\{ | x | = r \}$. By a result of Bartnik [4], and of Chruściel [11], $\mathfrak {m}$ is independent on the choice of the coordinates $\{ x_i \}$.

On an asymptotically flat 3-manifold (M, g) with boundary $\Sigma$, the capacity (or the $L^2$-capacity) of $\Sigma$ is given by

$$\begin{aligned} \mathfrak {c}_{_\Sigma } = \inf \left\{ \frac{1}{4 \pi } \int _M | \nabla f |^2 \right\} , \end{aligned}$$

where the infimum is taken over all locally Lipschitz functions f that equal 1 at $\Sigma$ and tend to 0 at $\infty$. Equivalently, $\displaystyle \mathfrak {c}_{_\Sigma } = \frac{1}{4 \pi } \int _M | \nabla \phi |^2 = \frac{1}{4\pi } \int _{\Sigma } | \nabla \phi |$, where

$$\begin{aligned} \Delta \phi = 0, \quad \phi |_\Sigma = 1, \quad \text{ and } \,\ \phi \rightarrow 0 \,\ \text{ at } \,\ \infty . \end{aligned}$$

Regarding the mass, a fundamental result is the Riemannian positive mass theorem, first proved by Schoen and Yau [27] and by Witten [31]. The theorem states if (M, g) is a complete, asymptotically flat 3-manifold with nonnegative scalar curvature, without boundary, then

$$\begin{aligned} \mathfrak {m}\ge 0, \end{aligned}$$

and equality holds if and only if (M, g) is isometric to the Euclidean space ${\mathbb {R}}^3$.

Regarding the mass and the capacity, an important result was due to Bray [6]. Bray showed if (M, g) is a complete, asymptotically flat 3-manifold with nonnegative scalar curvature, with minimal surface boundary $\Sigma = \partial M$, then

$$\begin{aligned} \mathfrak {m}\ge \mathfrak {c}_{_\Sigma }, \end{aligned}$$

and equality holds if and only if (M, g) is isometric to a spatial Schwarzschild manifold outside the horizon.

If the mean curvature H of the boundary $\Sigma$ in (M, g) is not assumed to be zero, using the weak inverse mean curvature flow developed by Huisken and Ilmanen [17], Bray and the author [8] showed

$$\begin{aligned} \mathfrak {m}\mathfrak {c}^{-1}_{_\Sigma } \ge 1 - \left( \frac{1}{16 \pi } \int _\Sigma H^2 \right) ^\frac{1}{2} \end{aligned}$$

under the assumptions $\int _\Sigma H^2 \le 16 \pi$ and $H_2 (M, \Sigma ) = 0$, and equality holds if and only if (M, g) is isometric to a spatial Schwarzschild manifold outside a rotationally symmetric sphere with nonnegative (constant) mean curvature.

Recently, level sets of harmonic functions have been found to be an efficient tool to study scalar curvature in 3-dimension. A pioneering work of Stern [30] revealed intriguing analogy between the use of such level sets and the use of stable minimal surfaces instituted by Schoen and Yau [27]. On asymptotically flat 3-manifolds, a new proof of the positive mass theorem was given by Bray, Kazaras, Khuri and Stern [7], which made use of harmonic functions asymptotic to a linear coordinate function.

In terms of monotone quantities along the level sets, Munteanu and Wang in [25] established sharp comparison results on complete, nonparabolic 3-manifolds via the discovery of a monotone quantity along level sets of the minimal positive Green’s function. In [2], Agostiniani, Mazzieri and Oronzio obtained another proof of the Riemannian positive mass theorem through a different monotone quantity along level sets of the Green’s function on asymptotically flat 3-manifolds.

In this paper, we consider harmonic functions u satisfying

$$\begin{aligned} u (x) = 1 - \mathfrak {c}|x|^{-1} + o ( |x|^{-1} ), \quad \text{ as } \,\ x \rightarrow \infty , \end{aligned}$$

for some constant $\mathfrak {c}> 0$, on an asymptotically flat 3-manifold (M, g). In the case (M, g) has boundary $\Sigma$ and u is 0 at $\Sigma$, $\mathfrak {c}= \mathfrak {c}_{_\Sigma }$ and u is referred as the capacitary function on (M, g). We obtain a sequence of new results relating the mass of (M, g), the capacitary function u, and the capacity $\mathfrak {c}_{_\Sigma }$.

We first find formulae that detect the mass of (M, g) via the level sets of such a u, see Theorem 2.1. In particular, Theorem 2.1 (ii) implies

$$\begin{aligned} \lim _{t \rightarrow 1} \frac{1}{ 1 - t } \left[ 4 \pi - \frac{1}{ (1 - t)^2 } \int _{\Sigma _{t}} | \nabla u |^2 \right] = 4 \pi \, \mathfrak {m}\, \mathfrak {c}^{-1}. \end{aligned}$$

(1.1)

Here $\Sigma _t = u^{-1} (t)$.

Besides (1.1), in Theorem 2.1 (i), we find

$$\begin{aligned} \lim _{t \rightarrow 1} \frac{1}{ 1 - t } \left[ 8 \pi - \frac{1}{ 1 - t} \int _{\Sigma _{t}} H | \nabla u | \right] = 12 \pi \, \mathfrak {m}\, \mathfrak {c}^{-1}. \end{aligned}$$

(1.2)

Here H denotes the mean curvature of a regular level set $\Sigma _t$ with respect to $|\nabla u |^{-1} \nabla u$.

An immediate implication of either (1.1) or (1.2) is that the ADM mass $\mathfrak {m}$ is a geometric invariant of (M, g), since the capacitary function and the boundary capacity are independent on the choice of coordinates at $\infty$.

(1.1) and (1.2) indicate that, as $t \rightarrow 1$,

$$\begin{aligned} 8 \pi - \frac{1}{ 1 - t} \int _{\Sigma _{t}} H | \nabla u | = 3 \left[ 4 \pi - \frac{1}{ (1 - t)^2 } \int _{\Sigma _{t}} | \nabla u |^2 \right] + o \left( ( 1 - t ) \right) . \end{aligned}$$

While this asymptotic comparison was made only via information near $\infty$, we show in Theorem 3.1 that, if M has simple topology and g has nonnegative scalar curvature, then, at each regular level set $\Sigma _t$,

$$\begin{aligned} 8 \pi - \frac{1}{ 1 - t} \int _{\Sigma _{t}} H | \nabla u | \le 3 \left[ 4 \pi - \frac{1}{ (1 - t)^2 } \int _{\Sigma _{t}} | \nabla u |^2 \right] , \end{aligned}$$

(1.3)

and “$=$” holds if and only if (M, g) outside $\Sigma _t$ is isometric to ${\mathbb {R}}^3$ minus a round ball.

Inequality (1.3) is derived via a monotone quantity along $\{ \Sigma _t \}$, see Lemma 3.1. Among other things, we apply (1.3) to find that the quantities in the mass formulae (1.1) and (1.2) are actually monotone non-decreasing, that is

$$\begin{aligned} \mathcal {A} (t): = \frac{1}{1-t} \left[ 8 \pi - \frac{1}{ 1 - t } \int _{\Sigma _{t}} H | \nabla u | \right] \nearrow \quad \text{ as } \,\ t \nearrow , \end{aligned}$$

(1.4)

and

$$\begin{aligned} \mathcal {B} (t): = \frac{1}{1-t} \left[ 4 \pi - \frac{1}{ (1 - t )^2 } \int _{\Sigma _{t}} | \nabla u |^2 \right] \nearrow \quad \text{ as } \,\ t \nearrow , \end{aligned}$$

(1.5)

see Theorem 3.2. This, combined with Theorem 2.1, then shows

$$\begin{aligned} 8 \pi - \int _\Sigma H | \nabla u | \le 12 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1} \end{aligned}$$

(1.6)

and

$$\begin{aligned} 4 \pi - \int _\Sigma | \nabla u |^2 \le 4 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1} \end{aligned}$$

(1.7)

for the capacitary function u. Furthermore, “$=$” holds in any of these inequalities if and only if (M, g) is isometric to ${\mathbb {R}}^3$ minus a round ball.

As an immediate application of (1.1)–(1.7), we observe several new arguments implying the 3-dimensional positive mass theorem, see Sect. 4.

Inequalities (1.6) and (1.7) also give rise to sufficient conditions that imply the positivity of the mass via $C^0$-geometry of regions separating the boundary and $\infty$. For instance, as a special case of Theorem 5.1, we show that if M has simply topology and g has nonnegative scalar curvature, then

$$\begin{aligned} H \le \frac{ 8 \pi L^2 }{\text {Vol}(\Omega ) } \ \Longrightarrow \ \mathfrak {m}> 0. \end{aligned}$$

(1.8)

Here $\Omega$ is a region whose boundary has two components $S_0$ and $S_1$, where $S_1$ encloses $S_0$ and $S_0$ encloses $\Sigma$, L is the distance from $S_1$ to $S_0$, and $\text {Vol}(\Omega )$ is the volume of $(\Omega , g)$. Another such sufficient condition in Theorem 5.2 shows

$$\begin{aligned} \int _{S_0} | \nabla v |^2 \le 4 \pi \ \Longrightarrow \ \mathfrak {m}> 0. \end{aligned}$$

(1.9)

Here v is the harmonic function on $\Omega$ with $v = 0$ at $S_0$ and $v = 1$ at $S_1$.

In [2], Agostiniani, Mazzieri and Oronzio showed, along $\{ \Sigma _t \}$,

$$\begin{aligned} {F}(t): = \frac{1}{ 1 - t } \left[ 4 \pi - \frac{1}{ 1 - t} \int _{\Sigma _{t}} H | \nabla u | \ + \frac{1}{ (1 - t)^2 } \int _{\Sigma _{t}} | \nabla u |^2 \right] \nearrow \quad \text{ as } \,\ t \nearrow .\nonumber \\ \end{aligned}$$

(1.10)

We observe that $\mathcal {A}(t)$, $\mathcal {B}(t)$ in our work are related to F(t) related by

$$\begin{aligned} F(t) = \mathcal {A} (t) - \mathcal {B} (t). \end{aligned}$$

(1.11)

In (A.14) and (A.15) of Appendix A, we give integral identities for the differences

$$\begin{aligned} \mathcal {B} (t_2) - \mathcal {B} (t_1), \quad \mathcal {A} (t_2) - \mathcal {A} (t_1), \quad \text{ for } \,\ t_1 < t_2. \end{aligned}$$

The monotonicity of F(t) can also be seen from (1.11), (A.14) and (A.15). Moreover, as a corollary of (1.1), (1.2) and (1.11), one has $\lim _{t \rightarrow 1} F (t) = 8 \pi \, \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1}$. Such a limit was shown in [2] in the case that (M, g) is isometric to a spatial Schwarzschild manifold near infinity.

Applying the limits of $\mathcal {A}(t)$, $\mathcal {B}(t)$ as $t \rightarrow 1$ and the formulae of their differences at $t_1 < t_2$, we derive integral identities for the mass-to-capacity ratio $\displaystyle \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1}$ in Theorem 6.1. Such integral identities can be compared with the mass identity obtained by Bray, Kazaras, Khuri and Stern [7] via harmonic functions having linear asymptotic.

Inspired by Bray’s work [6], in Sect. 7 we promote inequalities (1.3), (1.6) and (1.7) to become equality in spatial Schwarzschild spaces. Among other things, we show in Corollary 7.1 that

$$\begin{aligned} \left( \frac{1}{\pi } \int _\Sigma | \nabla u |^2 \right) ^\frac{1}{2} \le \left( \frac{1}{16\pi } \int _\Sigma H^2 \right) ^\frac{1}{2} + 1. \end{aligned}$$

(1.12)

Moreover, equality holds if and only if (M, g) is isometric to a spatial Schwarzschild manifold outside a rotationally symmetric sphere with nonnegative mean curvature. In Theorem 7.3, we show, given the same triple (M, g, u),

$$\begin{aligned} \begin{aligned} \frac{1}{2} \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \ge 1 - \left( \frac{1}{4 \pi } \int _{\Sigma } | \nabla u |^2 \, \textrm{d}\sigma \right) ^\frac{1}{2}, \end{aligned} \end{aligned}$$

(1.13)

and equality holds if and only if (M, g) is isometric to a spatial Schwarzschild manifold outside a rotationally symmetric sphere. As a result of (1.12) and (1.13), we obtain in Theorem 7.4

$$\begin{aligned} \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \ge 1 - \left( \frac{1}{16\pi } \int _\Sigma H^2 \right) ^\frac{1}{2}, \end{aligned}$$

(1.14)

regardless of the mean curvature H of $\Sigma$. (1.14) improves the earlier mentioned result of Bray and the author in [8]. Moreover, if applied to the exterior of small geodesic balls, (1.14) yields another proof of the positive mass theorem, see Remark 7.1.

Prompted by (1.14), in Sect. 8 we carry out a study of manifolds with boundary satisfying a mass-capacity relation

$$\begin{aligned} \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \le 1. \end{aligned}$$

(1.15)

Under this assumption, in Theorem 8.1 we promote (1.6) to

$$\begin{aligned} (2 - \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} ) ( 1 - \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} ) \le \frac{1}{4\pi } \int _{\Sigma } H | \nabla u |, \end{aligned}$$

(1.16)

which picks up an intriguing quadratic term $( \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} )^2$. Equality in (1.16) holds if and only if (M, g) is isometric to a spatial Schwarzschild manifold outside a rotationally symmetric sphere with nonnegative mean curvature.

In Corollary 8.1, we give a capacity-comparison result for manifolds satisfying (1.15) under a condition

$$\begin{aligned} \mathfrak {m}H_{\textrm{max}} \le \frac{ 2}{3 \sqrt{3} }. \end{aligned}$$

Here $H_{\textrm{max}}$ is the maximum of the mean curvature of the boundary and the number $\textstyle \frac{2}{3 \sqrt{ 3} }$ is the maximum value of $\mathfrak {m}H$ evaluated along rotationally symmetric spheres in a spatial Schwarzschild manifold with positive mass.

In Corollary 8.2, we show manifolds satisfying (1.15) have the mass bounded by

$$\begin{aligned} \mathfrak {m}\le \frac{ r_{_\Sigma }}{2} \left[ 1 + \left( \frac{1}{16 \pi } \int _\Sigma H^2 \right) ^\frac{1}{2} \right] , \end{aligned}$$

(1.17)

where $r_{_\Sigma }$ is the area–radius of $\Sigma$. Moreover, the capacitary functions u on these manifolds satisfy

$$\begin{aligned} \int _{\Sigma } | \nabla u |^2 \ge \pi . \end{aligned}$$

(1.18)

Heuristically, this suggests such manifolds may not have long cylindrical regions shielding the boundary, see Remark 8.6.

Toward the end of Sect. 8, we place condition (1.15) in the context of the Bartnik quasi-local mass [5]. We point out manifolds satisfying (1.15) do not contain closed minimal surfaces enclosing the boundary and static metric extensions with a positive static potential necessarily satisfy (1.15), see Proposition 8.1.

We finish this paper with an appendix, including regularization arguments that can be used to verify various monotonicity in Sect. 3.

2 Detecting the Mass at $\infty$

Let (M, g) denote an asymptotically flat 3-manifold (with one end) with boundary. By this, we mean there is a compact set $K \subset M$ such that $M {\setminus } K$ is diffeomorphic to ${\mathbb {R}}^3$ minus a ball and, with respect to the standard coordinates on ${\mathbb {R}}^3$, g satisfies

$$\begin{aligned} g_{ij} = \delta _{ij} + O ( |x |^{-\tau } ), \quad \partial g_{ij} = O ( |x|^{-\tau -1}), \quad \partial \partial g_{ij} = O (|x|^{-\tau -2} ) \end{aligned}$$

(2.1)

for some constant $\tau > \frac{1}{2}$. The scalar curvature R of g is also assumed to be integrable so that the mass $\mathfrak {m}$ of (M, g) exists (see [4, 11] for instance).

Let $\Sigma$ denote the boundary of M. Let u be the function on (M, g) given by

$$\begin{aligned} \Delta u = 0 \,\ \text{ on } \,\ M, \quad u = 0 \,\ \text{ at } \,\ \Sigma , \quad \text{ and } \,\ u \rightarrow 1 \,\ \text{ at } \,\ \infty . \end{aligned}$$

(2.2)

Given any $t \in [0,1]$, let $\Sigma _t = \{ x \in M \ | \ u (x) = t \}$ denote the level set of u. Below, we collect some basic facts about u and $\Sigma _t$.

By the maximum principle, $\max _{K} u < 1$, hence |x| is defined on $\Sigma _t$ for t close to 1; moreover, $\min _{\Sigma _t} | x | \rightarrow \infty$ as $t \rightarrow 1$. Now suppose $\tau \in (\frac{1}{2}, 1)$. As $x \rightarrow \infty$, it is known u has an asymptotic expansion (see Lemma A.2 in [22] for instance)

$$\begin{aligned} u = 1 - \mathfrak {c}_{_\Sigma } | x |^{-1} + O_2 ( | x |^{-1 -\tau } ). \end{aligned}$$

(2.3)

Here $\mathfrak {c}_{_\Sigma } > 0$ is a positive constant equal to the capacity of $\Sigma$ in (M, g). Let $\nabla u$ and $\nabla ^2 u$ denote the gradient and the Hessian of u on (M, g), respectively. By (2.3),

$$\begin{aligned}&| \nabla u |^2 = \mathfrak {c}_{_\Sigma }^2 |x|^{-4} + O ( |x|^{- 4 - \tau } ), \end{aligned}$$

(2.4)

$$\begin{aligned}&(\nabla ^2 u )_{ij}= {} \ \mathfrak {c}_{_\Sigma } |x|^{-3} \bigl ( - 3 |x|^{-2} x_i x_j + \delta _{ij} \bigr ) + O ( |x|^{-3 - \tau } ). \end{aligned}$$

(2.5)

Thus, t is a regular value if t is close to 1 and the mean curvature H of $\Sigma _t$ satisfies

$$\begin{aligned} H = \textrm{ div }\big ( | \nabla u |^{-1} \nabla u \big ) = 2 |x|^{-1} + O (|x|^{-1 - \tau } ). \end{aligned}$$

(2.6)

As a result, for t close to 1, $\Sigma _t$ has positive mean curvature, and consequently $\Sigma _t$ is area outer-minimizing as its exterior is foliated by mean-convex surfaces $\{ \Sigma _s \}_{s > t }$. Here we say a surface S is area outer-minimizing if every surface ${\tilde{S}}$ which encloses S has greater area, see [6] for instance.

Lemma 2.1

Let $| \Sigma _t |$ be the area of $\Sigma _t$ in (M, g) if t is a regular value of u. Then, as $t \rightarrow 1$,

$$\begin{aligned} | \Sigma _t | = 4 \pi \mathfrak {c}^2_{_\Sigma } ( 1 - t )^{-2} + O ( (1 - t)^{ \tau - 2 } ). \end{aligned}$$

(2.7)

Proof

By (2.3), as $t \rightarrow 1$,

$$\begin{aligned} |x| = \mathfrak {c}_{_\Sigma } (1 -t)^{-1} + O ( (1 - t )^{ \tau - 1 } ). \end{aligned}$$

(2.8)

Let $r_- (t) = \min _{\Sigma _t} |x|$ and $r_+(t) = \max _{\Sigma _t} |x|$. Since $\Sigma _t$ and the coordinate sphere $S_r: = \{ |x|= r \}$ are both area outer-minimizing in (M, g), for t close to 1 and large r, respectively, we have

$$\begin{aligned} | S_{r_- (t) } | \le |\Sigma _t| \le | S_{r_+ (t) } |. \end{aligned}$$

(2.9)

For large r, (2.1) implies

$$\begin{aligned} | S_{r} | = 4 \pi r^2 + O ( r^{2 - \tau } ). \end{aligned}$$

(2.10)

Thus, (2.7) follows from (2.8)–(2.10). $\square$

Lemma 2.2

As $t \rightarrow 1$,

$$\begin{aligned} \frac{1}{(1-t)} \int _{\Sigma _t} H | \nabla u | = 8 \pi + O ( ( 1 - t)^\tau ) \end{aligned}$$

and

$$\begin{aligned} \frac{1}{(1-t)^2} \int _{\Sigma _t } | \nabla u |^2 = 4 \pi + O ( ( 1 - t)^\tau ). \end{aligned}$$

Proof

By (2.6) and (2.8),

$$\begin{aligned} H = 2 \mathfrak {c}_{_\Sigma }^{-1} ( 1 - t) + O ( (1-t)^{1+\tau }). \end{aligned}$$

(2.11)

Therefore, using the fact $\int _{\Sigma _t} | \nabla u | = 4 \pi \mathfrak {c}_{_\Sigma }$, one has

$$\begin{aligned} \begin{aligned} \left( \frac{1}{1-t} \int _{\Sigma _t} H | \nabla u | \right) - 8\pi&= \ \int _{\Sigma _t} \left( \frac{H}{1-t} - \frac{2}{\mathfrak {c}_{_\Sigma } } \right) | \nabla u | \\&= \ O ( (1-t)^\tau ). \end{aligned} \end{aligned}$$

Similarly, by (2.4) and (2.8),

$$\begin{aligned} | \nabla u | = \mathfrak {c}_{_\Sigma }^{-1} (1-t)^2 + O ( ( 1 - t )^{2 + \tau } ). \end{aligned}$$

(2.12)

Therefore,

$$\begin{aligned} \begin{aligned} \left( \frac{1}{(1-t)^2} \int _{\Sigma _t} | \nabla u |^2 \right) - 4\pi&= \ \left( \int _{\Sigma _t} \frac{| \nabla u |}{(1-t)^2} - \frac{1}{\mathfrak {c}_{_\Sigma } } \right) | \nabla u | \\&= \ O ( ( 1 - t )^{\tau } ). \end{aligned} \end{aligned}$$

$\square$

Lemma 2.3

As $t \rightarrow 1$, the gradient of $| \nabla u |$ on $\Sigma _t$ satisfies

$$\begin{aligned} \big | \nabla _{_{\Sigma _t}} | \nabla u | \big | = O ( | x |^{- 3 - \tau } ). \end{aligned}$$

(2.13)

Proof

Write $\nabla u = (\nabla u)^j \partial _j$. By (2.3),

$$\begin{aligned} (\nabla u)^j = \mathfrak {c}_{_\Sigma } | x |^{-2 } |x|^{-1} x_j + O( | x |^{-2 -\tau } ). \end{aligned}$$

Let $V = V^i \partial _i$ denote any unit vector tangent to $\Sigma _t$. Then $V^i = O (1)$ and the fact $\langle V, \nabla u \rangle = 0$ shows

$$\begin{aligned} \sum _i V^i ( \nabla u )^i = O ( | x |^{-2 - \tau } ), \quad \text{ and } \text{ hence } \ \sum _i V^i x_i = O ( | x |^{1- \tau } ). \end{aligned}$$

(2.14)

Therefore, by (2.5) and (2.14),

$$\begin{aligned} V ( | \nabla u |^2 )&= {} 2 ( \nabla ^2 u )( V, \nabla u ) \nonumber \\ {}&= {} 2 \mathfrak {c}_{_\Sigma } |x|^{-3} \bigl [ - 3 |x|^{-2} x_i V^i x_j (\nabla u )^j + \delta _{ij} V^i (\nabla u )^j \bigr ] \big ( 1 + O ( |x|^{ - \tau } ) \big ) \nonumber \\ {}&= {} O ( |x|^{- 5 - \tau } ). \end{aligned}$$

(2.15)

Thus, (2.13) follows from (2.15) and (2.4). $\square$

Lemma 2.4

As $t \rightarrow 1$, the traceless part of the second fundamental form $\displaystyle {\mathbb{I}\mathbb{I}}$ of $\Sigma _t$, denoted by $\mathring{\displaystyle {\mathbb{I}\mathbb{I}}}$, satisfies

$$\begin{aligned} | \mathring{\displaystyle {\mathbb{I}\mathbb{I}}} | = O ( | x |^{ - 1 - \tau } ), \end{aligned}$$

(2.16)

and the Gauss curvature K of $\Sigma _t$ satisfies $K = |x|^{-2} + O ( |x|^{-2 - \tau } ).$

Proof

Let $V =V^i \partial _i$ and $W = W^j \partial _j$ be any two unit vectors tangent to $\Sigma _t$ at a given point. Then $\delta _{ij} V^i W^j = g( V, W ) + O ( |x|^{-\tau } )$. As V, W are tangential to $\Sigma _t$, one has

$$\begin{aligned} \nabla ^2 u (V, W) = \langle \nabla _{V} \nabla u, W \rangle = | \nabla u | \displaystyle {\mathbb{I}\mathbb{I}}( V, W ). \end{aligned}$$

Hence,

$$\begin{aligned} | \nabla u | \, \displaystyle {\mathbb {I}\mathbb {I}}( V, W)&= {} ( \nabla ^2 u)_{ij} V^i W^j \nonumber \\ {}&= {} \mathfrak {c}_{_\Sigma } |x|^{-3} \bigl ( - 3 |x|^{-2} x_i V^i x_j W^j + \delta _{ij} V^i W^j \bigr ) \bigl ( 1 + O ( |x|^{ - \tau } ) \bigr ) \nonumber \\ {}&= {} \mathfrak {c}_{_\Sigma } |x|^{-3} g( V, W) + O ( |x|^{-3 - \tau }), \end{aligned}$$

(2.17)

where one used (2.5) and (2.14). Therefore, by (2.4),

$$\begin{aligned} \begin{aligned} \displaystyle {\mathbb{I}\mathbb{I}}(V, W) = \ |x|^{-1} g( V, W ) + O ( | x|^{-1 - \tau } ). \end{aligned} \end{aligned}$$

(2.18)

This combined with (2.6) shows

$$\begin{aligned} \mathring{\displaystyle {\mathbb{I}\mathbb{I}}} (V, W) = \displaystyle {\mathbb{I}\mathbb{I}}(V, W) - \frac{1}{2} H g( V, W ) = O ( | x|^{-1 - \tau } ), \end{aligned}$$

which proves (2.16). The conclusion on the Gauss curvature follows from (2.18), (2.6) and the Gauss equation. $\square$

Lemma 2.5

If (M, g) satisfies $\partial \partial \partial g_{ij} = O (|x|^{-3 - \tau } )$ in (2.1), then

$$\begin{aligned} | D \, \mathring{\displaystyle }{\mathbb{I}\mathbb{I}}| = O ( | x|^{-2 - \tau } ). \end{aligned}$$

(2.19)

Here D denotes covariant differentiation on $\Sigma _t$.

Proof

If g satisfies the higher order derivatives decay assumption, then u satisfies

$$\begin{aligned} u = 1 - \mathfrak {c}_{_\Sigma } |x|^{-1} + O_3 (|x|^{-1-\tau } ) \end{aligned}$$

(see the proof of Lemma A.2 in [22] for instance). The terms $O(|x|^{-3-\tau })$ in (2.5) and $O (|x|^{-1-\tau })$ in (2.6) are then replaced by $O_1 ( |x|^{-3-\tau } )$ and $O_1 (|x|^{-1-\tau })$, respectively.

To prove (2.19), let $\{ V_\alpha \}_{\alpha = 1, 2}$ be a local orthonormal frame around a given point p on $\Sigma _t$. By definition,

$$\begin{aligned} ( D_{V_\mu } \mathring{\displaystyle }{\mathbb{I}\mathbb{I}}) (V_\alpha , V_\beta ) = \left( D_{V_\mu } \displaystyle {\mathbb{I}\mathbb{I}}\right) (V_\alpha , V_\beta ) - \frac{1}{2} V_\mu ( H ) \delta _{\alpha \beta }. \end{aligned}$$

By (2.6) and (2.14),

$$\begin{aligned} \begin{aligned} V_{\mu } ( H) =&\ 2 (-1) |x|^{-2} V_\mu ( |x| ) + O ( |x|^{-2 - \tau } ) = O ( |x|^{-2 - \tau } ). \end{aligned} \end{aligned}$$

To estimate $D \displaystyle {\mathbb{I}\mathbb{I}}$, one may assume $\{ V_\alpha \}$ is normal at p, i.e., $D_{V_\alpha } V_\beta = 0$ at p, then

$$\begin{aligned} \begin{aligned} ( D_{V_\mu } {\displaystyle {\mathbb {I}\mathbb {I}}}) (V_\alpha , V_\beta )&= \ V_\mu ( \displaystyle {\mathbb {I}\mathbb {I}}( V_\alpha , V_\beta ) ) \\ {}&= \ V_\mu ( |\nabla u |^{-1} ) \, ( \nabla ^2 u )_{\alpha \beta } + | \nabla u |^{-1} \, V_\mu \bigl ( ( \nabla ^2 u )_{\alpha \beta } \bigr ) . \end{aligned} \end{aligned}$$

By (2.13) and (2.17),

$$\begin{aligned} V_\mu ( |\nabla u |^{-1} ) \, ( \nabla ^2 u )_{\alpha \beta } = O ( | x|^{-2 - \tau } ). \end{aligned}$$

By (2.17) and (2.14),

$$\begin{aligned} | \nabla u |^{-1} \, V_\mu \bigl ( ( \nabla ^2 u )_{\alpha \beta } \bigr ) = | \nabla u |^{-1} \, O ( |x|^{-4-\tau } ) = O ( |x|^{-2 - \tau } ). \end{aligned}$$

Thus, $( D_{V_\mu } {\displaystyle {\mathbb{I}\mathbb{I}}}) (V_\alpha , V_\beta ) = O ( | x|^{-2-\tau } )$. This proves (2.19). $\square$

Let $\mathfrak {m}_{_H} ( \Sigma _t)$ denote the Hawking mass [14] of $\Sigma _t$ if t is a regular value of u. That is

$$\begin{aligned} \mathfrak {m}_{_H} ( \Sigma _t) = \frac{r_t}{2} \left( 1 - \frac{1}{16 \pi } \int _{\Sigma _t} H^2 \right) . \end{aligned}$$

(2.20)

Here $r_t = \sqrt{ \frac{ | \Sigma _t |}{4\pi } }$ is the area–radius of $\Sigma _t$. By Lemma 2.1,

$$\begin{aligned} r_t = \mathfrak {c}_{_\Sigma } ( 1 - t )^{-1} + O ( (1 - t)^{ \tau - 1 } ). \end{aligned}$$

(2.21)

Proposition 2.1

If $\lim _{ t \rightarrow 1} \mathfrak {m}_{_H} ( \Sigma _t) = \mathfrak {m}$, where $\mathfrak {m}$ is the mass of (M, g), then

$$\begin{aligned} \lim _{t \rightarrow 1} \frac{1}{ 1 - t } \left[ 8 \pi - \frac{1}{ 1 - t} \int _{\Sigma _{t}} H | \nabla u | \right] = 12 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1} \end{aligned}$$

(2.22)

and

$$\begin{aligned} \lim _{t \rightarrow 1} \frac{1}{ 1 - t } \left[ 4 \pi - \frac{1}{ (1 - t)^2 } \int _{\Sigma _{t}} | \nabla u |^2 \right] = 4 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1}. \end{aligned}$$

(2.23)

Proof

For regular values t, define

$$\begin{aligned} A(t) = 8 \pi - \frac{1}{ 1 - t} \int _{\Sigma _{t}} H | \nabla u |. \end{aligned}$$

(2.24)

Then

$$\begin{aligned} - A'(t) = \frac{1}{ 1 - t } \left[ - A(t) + 8 \pi + \left( \int _{\Sigma _{t}} H | \nabla u | \right) ' \, \right] . \end{aligned}$$

Applying formulae for the first and second variation of area, we have

$$\begin{aligned} \left( \int _{\Sigma _t} H | \nabla u | \right) '=&{} \int _{\Sigma _t} H' | \nabla u | + H | \nabla u |' + H | \nabla u | H | \nabla u |^{-1} \nonumber \\ =&{} \int _{\Sigma _t} - | \nabla u |^{-2} \big | \nabla _{_{\Sigma _t}} | \nabla u | \big |^2 + K - \frac{3}{4} H^2 - \frac{1}{2} | \mathring{\displaystyle {\mathbb {I}\mathbb {I}}} |^2 - \frac{1}{2} R,\nonumber \\ \end{aligned}$$

(2.25)

where $| \nabla u |' = - H$ and $H' = - \Delta _{\Sigma _t} | \nabla u |^{-1} - ( \text{ Ric }(\nu , \nu ) + | \displaystyle {\mathbb{I}\mathbb{I}}|^2 ) | \nabla u |^{-1}$.

By (2.20) and the Gauss–Bonnet theorem,

$$\begin{aligned} 8 \pi + \left( \int _{\Sigma _{t}} H | \nabla u | \right) ' = \frac{24 \pi \mathfrak {m}_{_H} (\Sigma _t) }{r_t} - E (t), \end{aligned}$$

(2.26)

where

$$\begin{aligned} E (t) = \int _{\Sigma _t} | \nabla u |^{-2} | \nabla _{_\Sigma } | \nabla u | |^2 + \frac{1}{2} | \mathring{\displaystyle {\mathbb{I}\mathbb{I}}} |^2 + \frac{1}{2} R. \end{aligned}$$

Therefore,

$$\begin{aligned} - A' (t) = - \frac{ A(t) }{ 1 - t } + \frac{ 24 \pi \mathfrak {m}_{_H} ( \Sigma _t) }{ (1-t) r_t} - \frac{1}{1 - t} E(t). \end{aligned}$$

(2.27)

By Lemma 2.2, $\lim _{t \rightarrow 1} A (t) = 0$. Hence,

$$\begin{aligned} A(t) = \frac{1}{1-t} \int _t^1 \left[ \frac{24\pi \mathfrak {m}_{_H} ( \Sigma _s ) }{ r_s} - E(s) \right]. \end{aligned}$$

(2.28)

As $t \rightarrow 1$, $\mathfrak {m}_{_H} ( \Sigma _t) = \mathfrak {m}+ o (1)$ by the assumption. Thus, by (2.21),

$$\begin{aligned} \frac{\mathfrak {m}_{_H} ( \Sigma _t ) }{ r_t} = \mathfrak {m}\, \mathfrak {c}^{-1}_{_\Sigma } (1 - t) + (1 - t ) o (1). \end{aligned}$$

(2.29)

Consequently,

$$\begin{aligned} \int _t^1 \frac{\mathfrak {m}_{_H} ( \Sigma _s ) }{ r_s} = \frac{1}{2} \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1} ( 1 - t)^2 + o ( (1-t)^2). \end{aligned}$$

(2.30)

To estimate $\int _t^1 E(s)$, we note Lemmas 2.3 and 2.4, combined with (2.8), show

$$\begin{aligned} | \nabla u |^{-2} \big | \nabla _{_\Sigma } | \nabla u | \big |^2 + \frac{1}{2} | \mathring{\displaystyle {\mathbb {I}\mathbb {I}}} |^2 = O ( | x |^{-2 - 2 \tau } ) = O ( ( 1 - t )^{2 + 2 \tau } ). \end{aligned}$$

Thus, by Lemma 2.1,

$$\begin{aligned} \int _{\Sigma _t} | \nabla u |^{-2} \big | \nabla _{_\Sigma } | \nabla u | \big |^2 + \frac{1}{2} | \mathring{\displaystyle {\mathbb {I}\mathbb {I}}} |^2 = O ( ( 1 - t )^{ 2 \tau } ). \end{aligned}$$

(2.31)

Therefore,

$$\begin{aligned} \int _{t}^1 \int _{\Sigma _s} | \nabla u |^{-2} \big | \nabla _{_\Sigma } | \nabla u | \big |^2 + \frac{1}{2} | \mathring{\displaystyle {\mathbb {I}\mathbb {I}}} |^2 = O ( ( 1 - t )^{1 + 2 \tau } ). \end{aligned}$$

(2.32)

To handle the scalar curvature term, we use the assumption R is integrable. As $t \rightarrow 1$,

$$\begin{aligned} o (1) = \int _{ u \ge t} | R | = \int _t^1 \int _{\Sigma _s} | R | | \nabla u |^{-1}, \end{aligned}$$

where we also used the coarea formula. By (2.12), $| \nabla u |^{-1} \ge \frac{1}{2} \mathfrak {c}_{_\Sigma } ( 1 - t )^{-2}$ for t close to 1. Hence,

$$\begin{aligned} \begin{aligned} \int _t^1 \int _{\Sigma _s} | R | | \nabla u |^{-1} \ge&\ \frac{1}{2} \mathfrak {c}_{_\Sigma } ( 1 - t )^{-2} \int _t^1 \int _{\Sigma _s} | R |. \end{aligned} \end{aligned}$$

These imply

$$\begin{aligned} \int _t^1 \int _{\Sigma _s} | R | = o ( ( 1 - t )^2 ). \end{aligned}$$

(2.33)

It follows from (2.28), (2.30), (2.32) and (2.33) that

$$\begin{aligned} \frac{1}{1-t} A(t) = 12 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1} + o (1) + O ( (1-t)^{2 \tau - 1 }) . \end{aligned}$$

(2.34)

Since $\tau > \frac{1}{2}$, this proves (2.22).

Similarly, define

$$\begin{aligned} B (t) = 4 \pi - \frac{1}{ (1 - t )^2 } \int _{\Sigma _{t}} | \nabla u |^2. \end{aligned}$$

(2.35)

At any regular value t,

$$\begin{aligned} \begin{aligned} - B'(t)&= \ \frac{1}{ 1 - t } \left[ 2 ( - B(t) + 4 \pi ) + \frac{1}{1-t} \left( - \int _{\Sigma _t} H | \nabla u | \right) \, \right] \\&= \ \frac{1}{ 1 - t } \left[ - 2 B(t) + A (t) \, \right] . \end{aligned} \end{aligned}$$

(2.36)

By Lemma 2.2, $\lim _{t \rightarrow 1} B(t) =0$. Thus,

$$\begin{aligned} B (t) = \frac{1}{ (1 - t)^2} \int _t^1 ( 1 - s ) A(s). \end{aligned}$$

Therefore, as $t \rightarrow 1$, by (2.34),

$$\begin{aligned} \begin{aligned} \frac{1}{1-t} B(t)&= \ 4 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1} + o ( 1 ) + O ( (1 - t)^{2 \tau - 1 } ) . \end{aligned} \end{aligned}$$

This proves (2.23). $\square$

Theorem 2.1

Let (M, g) be an asymptotically flat 3-manifold with boundary $\Sigma$, with $\partial \partial \partial g_{ij} = O ( |x|^{-3-\tau } )$ at $\infty$. Let u be the harmonic function that tends to 1 at $\infty$ and vanishes at $\Sigma$. Then

• (i) $\displaystyle \lim _{t \rightarrow 1} \frac{1}{ 1 - t } \left[ 8 \pi - \frac{1}{ 1 - t} \int _{\Sigma _{t}} H | \nabla u | \right] = 12 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1};$

• (ii) $\displaystyle \lim _{t \rightarrow 1} \frac{1}{ 1 - t } \left[ 4 \pi - \frac{1}{ (1 - t)^2 } \int _{\Sigma _{t}} | \nabla u |^2 \right] = 4 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1}.$

Here $\mathfrak {m}$ is the mass of (M, g) and $\mathfrak {c}_{_\Sigma }$ is the capacity of $\Sigma$ in (M, g).

Proof

It suffices to show $\lim _{t \rightarrow 1} \mathfrak {m}_{_H} (\Sigma _t) = \mathfrak {m}$. For t close to 1, let $r_- (t)= \min _{\Sigma _t} | x|$ and $r_+ (t) = \max _{\Sigma _t} |x|$. By (2.8), $r_+ (t) \le C r_- (t)$. Here and below, $C > 0$ denotes some constant independent on t. By Lemma 2.1, $| \Sigma _t | \le C r_-^2$. By Lemma 2.4, $K \ge C r_-^{-2}$, hence $\text {diam} (\Sigma _t) \le C r_-$. By Lemmas 2.4 and Lemma 2.5, $| \mathring{\displaystyle {\mathbb{I}\mathbb{I}}} | \le C r_-^{-1-\tau }$ and $| D \mathring{\displaystyle {\mathbb{I}\mathbb{I}}} | \le C r_-^{-2-\tau }$. Hence, $\{ \Sigma _t \}$ is a family of nearly round surfaces near $\infty$ in (M, g) according to Definition 1.3 in [29]. By Theorem 2 in [29], $\lim _{t \rightarrow 1} \mathfrak {m}_{_H} (\Sigma _t) = \mathfrak {m}$.

Theorem 2.1 now follows from Proposition 2.1. $\square$

We can indeed interpret the mass-to-capacity ratio as the derivatives at $\infty$ of the two functions

$$\begin{aligned} A(t) = 8 \pi - \frac{1}{ 1 - t} \int _{\Sigma _{t}} H | \nabla u | \quad \text{ and } \quad B (t) = 4 \pi - \frac{1}{ (1 - t )^2 } \int _{\Sigma _{t}} | \nabla u |^2.\qquad \end{aligned}$$

(2.37)

Corollary 2.1

Let (M, g) be an asymptotically flat 3-manifold with boundary $\Sigma$, with $\partial \partial \partial g_{ij} = O ( |x|^{-3-\tau } )$ at $\infty$. Let u be the harmonic function that tends to 1 at $\infty$ and vanishes at $\Sigma$. Then the functions A(t) and B(t) have $C^1$ extensions to $t = 1$ with

$$\begin{aligned} A (1) = 0, \quad A' (1) = - 12 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1}, \quad B (1) = 0, \quad B'(1) = - 4 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1}. \end{aligned}$$

Proof

By Lemma 2.2, A(t) and B(t) extend continuously to $t = 1$ with $A(1) =0$ and $B(1)= 0$. By Theorem 2.1 (i), (2.27), (2.29) and (2.31),

$$\begin{aligned} \begin{aligned} \lim _{t \rightarrow 1} A'(t)&= \ \lim _{t \rightarrow 1} \left[ \frac{1}{1-t} A(t) - \frac{ 24 \pi \mathfrak {m}_{_H} (\Sigma _t ) }{ (1-t) r_t} + \frac{1}{1-t} E(t) \right] \\&= \ 12 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1} - 24 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1} \\&= \ \lim _{t \rightarrow 1} \frac{1}{t-1} A(t). \end{aligned} \end{aligned}$$

Similarly, by Theorem 2.1 (i), (ii) and (2.36),

$$\begin{aligned} \lim _{t \rightarrow 1} B'(t) = \lim _{t \rightarrow 1} \frac{1}{1-t} \left[ 2 B(t) - A(t) \right] = - 4\pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1} = \lim _{ t \rightarrow 1} \frac{1}{t-1} B(t). \end{aligned}$$

This shows $A'(t)$ and $B'(t)$ are continuous at $t = 1$ with $A'(1) = - 12 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1}$ and $B'(1) = - 4\pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1}$. $\square$

Remark 2.1

Proofs in this section essentially only use the structure near $\infty$ of (M, g). The arguments indeed establish (1.1) and (1.2) for any harmonic function u, satisfying

$$\begin{aligned} u (x) = 1 - \mathfrak {c}|x|^{-1} + o ( |x|^{-1} ), \quad \text{ as } \,\ x \rightarrow \infty , \end{aligned}$$

for some constant $\mathfrak {c}> 0$. Another way to see this is to derive (1.1) and (1.2) as a corollary of Theorem 2.1. For instance, we may assume $0< u < 1$ on $\{ |x | \ge r \}$ for some large r. Let T be a regular value of u so that $\max _{ |x| = r } u< T < 1$. On $M_T = \{ u \ge T \}$, consider $u_{_T} = \frac{ 1}{1 - T} ( u - T )$, which is the capacitary function on $(M_T, g)$. By Theorem 2.1 (ii),

$$\begin{aligned} \lim _{s \rightarrow 1} \frac{1}{ 1 - s } \left[ 4 \pi - \frac{1}{ (1 - s )^2 } \int _{ \{ u_{_T} = s \} } | \nabla u_{_T} |^2 \right] = 4 \pi \, \mathfrak {m}\, {\mathfrak {c}}_{_T}^{-1}, \end{aligned}$$

(2.38)

where ${\mathfrak {c}}_{_T} = \frac{1}{1-T} \mathfrak {c}$ is the capacity of $\{ u = T \}$. Re-writing (2.38) in terms of u, we then obtain (1.1). Similarly, (1.2) follows from Theorem 2.1 (i).

3 Inequalities Along the Level Sets

In this section, we establish a family of geometric inequalities along $\{ \Sigma _t \}$ under assumptions that g has nonnegative scalar curvature and M has simple topology.

We first compare

$$\begin{aligned} A(t) = 8 \pi - \frac{1}{ 1 - t} \int _{\Sigma _{t}} H | \nabla u | \quad \text {and} \quad B(t) = 4 \pi - \frac{1}{ (1 - t)^2 } \int _{\Sigma _{t}} | \nabla u |^2. \end{aligned}$$

Theorem 3.1

Let (M, g) be a complete, orientable, asymptotically flat 3-manifold with boundary $\Sigma$. Suppose $\Sigma$ is connected and $H_2 (M, \Sigma ) = 0$. Let u be the harmonic function that tends to 1 at $\infty$ and vanishes at $\Sigma$. If g has nonnegative scalar curvature, then

$$\begin{aligned} 4 \pi + \frac{1}{1-t} \int _{\Sigma _t} H | \nabla u | \ge \frac{3}{ (1-t)^2 } \int _{\Sigma _t} | \nabla u |^2 \end{aligned}$$

(3.1)

for all regular values t, and equality holds at some t if and only if (M, g), outside $\Sigma _t$, is isometric to ${\mathbb {R}}^3$ minus a round ball.

In particular, at $\Sigma$,

$$\begin{aligned} 4 \pi + \int _\Sigma H | \nabla u | \ge 3 \int _\Sigma | \nabla u |^2, \end{aligned}$$

(3.2)

and equality holds if and only if (M, g) is isometric to ${\mathbb {R}}^3$ minus a round ball.

Remark 3.1

Inequality (3.1) is equivalent to

$$\begin{aligned} A(t) \le 3 B(t). \end{aligned}$$

(3.3)

We will use Theorem 3.1 in this form later to derive other inequalities along $\{ \Sigma _t \}$.

Remark 3.2

To the author’s knowledge, inequality (3.2) represents a new result even in the 3-dimensional Euclidean space.

To prove Theorem 3.1, we begin with a lemma which may be derived directly from the work of Stern in [30].

Lemma 3.1

Let $(\Omega , g)$ be a compact, orientable, Riemannian 3-manifold with nonnegative scalar curvature, with boundary $\partial \Omega$. Suppose $\partial \Omega$ has two connected components $S_1$ and $S_2$. Let u be a harmonic function on $(\Omega , g)$ such that $u = c_i$ on $S_i$, $i = 1, 2$, where $c_1$, $c_2$ are constants with $c_1< c_2 < 1$. If the level set $\Sigma _s: = u^{-1} (s)$ is connected for $s \in [c_1, c_2]$, then

$$\begin{aligned} \Psi (t): = 4 \pi ( 1 - t ) + \int _{\Sigma _t} H | \nabla u | - \frac{3}{1-t} \int _{\Sigma _t} | \nabla u |^2 \searrow \quad \text{ as } \,\ t \nearrow , \end{aligned}$$

i.e., $\Psi (t)$ is monotone nonincreasing. Here $t \in [ c_1, c_2]$ denotes a regular value of u and H is the mean curvature of $\Sigma _t$ with respect to the unit normal $\nu = | \nabla u |^{-1} \nabla u$.

Proof

Let $t_1 < t_2$ be two regular values of u. On $\Omega _{[t_1, t_2] }: = \{ x \in \Omega \ | \ t_1 \le u(x) \le t_2 \}$, one has

$$\begin{aligned} \int _{\Sigma _{t_1} } H | \nabla u | - \int _{\Sigma _{t_2} } H | \nabla u | \ge \ \int _{t_1}^{t_2} \int _{\Sigma _t} \frac{1}{2} \left( \frac{ | \nabla ^2 u |^2 }{ | \nabla u |^2 } + R \right) - 2 \pi \int _{t_1}^{t_2} \chi (\Sigma _t).\nonumber \\ \end{aligned}$$

(3.4)

Here $\nabla ^2 u$, $\nabla u$ denote the Hessian, the gradient of u on (M, g), respectively, R is the scalar curvature of g, and $\chi (\Sigma _t)$ is the Euler characteristic of $\Sigma _t$. Relation (3.4) is a direct consequence of Stern’s computations in Section 2 of [30], and can also be found explicitly from (4.7) in [7] and (2.18) in [16].

Let $\displaystyle {\mathbb{I}\mathbb{I}}$ denote the second fundamental form of $\Sigma _t$ w.r.t. $\nu$. Let X, Y denote vectors tangent to $\Sigma _t$. Along $\Sigma _t$, one has

$$\begin{aligned} \nabla ^2 u (X, Y) = | \nabla u | \displaystyle {\mathbb {I}\mathbb {I}}(X, Y), \quad \nabla ^2 u (X, \nu ) = X ( | \nabla u | ), \quad \nabla ^2 u (\nu , \nu ) = - H | \nabla u |.\nonumber \\ \end{aligned}$$

(3.5)

Here the first two equations follow from definitions of $\nabla ^2 u$ and $\displaystyle {\mathbb{I}\mathbb{I}}$, and the last equation follows from $0 = \Delta u = \Delta _{_{\Sigma _t} } u + H \partial _\nu u + \nabla ^2 u (\nu , \nu )$. As a result,

$$\begin{aligned} | \nabla u |^{-2} \, | \nabla ^2 u |^2 = | \displaystyle {\mathbb {I}\mathbb {I}}|^2 + 2 | \nabla u |^{-2} \big | \nabla _{_{\Sigma _t}} | \nabla u | \big |^2 + H^2. \end{aligned}$$

(3.6)

Under the assumption $\Sigma _t$ is connected, it follows from (3.4) and (3.6) that

$$\begin{aligned} \begin{aligned}&\ 4 \pi ( t_2 - t_1) + \int _{\Sigma _{t_1} } H | \nabla u | - \int _{\Sigma _{t_2} } H | \nabla u | \\ {}&\quad \ge \ \int _{t_1}^{t_2} \int _{\Sigma _t} \frac{1}{2} | \mathring{\displaystyle {\mathbb {I}\mathbb {I}}} |^2 + | \nabla u |^{-2} \big | \nabla _{_{\Sigma _t} } | \nabla u | \big |^2 + \frac{3}{4} H^2 + \frac{1}{2} R, \end{aligned} \end{aligned}$$

(3.7)

where $\mathring{\displaystyle {\mathbb{I}\mathbb{I}}}$ denotes the traceless part of $\displaystyle {\mathbb{I}\mathbb{I}}$.

To handle the term of $H^2$ in (3.7), we follow the idea in [2, 25] to replace it with $\left( H - 2 | \nabla u | ( 1 - u)^{-1} \right) ^2$. A motivation to this may be seen in the model case in which $\Omega = \{ R_1 \le |x| \le R_2 \} \subset {\mathbb {R}}^3$ and $u = 1 - |x|^{-1}$. In this special setting, H and $|\nabla u |$ satisfy $H = 2 | \nabla u | ( 1 - u)^{-1}$ along any level set sphere.

Thus, one can rewrite (3.7) as

$$\begin{aligned}&{} \ 4 \pi ( t_2 - t_1) + \int _{\Sigma _{t_1} } H | \nabla u | - \int _{\Sigma _{t_2} } H | \nabla u | \qquad \quad \nonumber \\{}&{} \qquad + 3 \int _{t_1}^{t_2} \left[ - \frac{1}{1 - t} \int _{\Sigma _t} H | \nabla u | + \frac{1}{ ( 1 - t)^2 } \int _{\Sigma _t} | \nabla u |^2 \right] \qquad \quad \nonumber \\{}&{} \quad \ge \ \int _{t_1}^{t_2} \int _{\Sigma _t} \frac{1}{2} | \mathring{\displaystyle {\mathbb {I}\mathbb {I}}} |^2 + | \nabla u |^{-2} \big | \nabla _{_{\Sigma _t} } | \nabla u | \big |^2 + \frac{3}{4} \left( H - \frac{ 2 | \nabla u | }{ 1 - u } \right) ^2 + \frac{1}{2} R.\qquad \quad \end{aligned}$$

(3.8)

At each regular value t, one has $\big ( \int _{\Sigma _t} | \nabla u |^2 \big ) ' = - \int _{\Sigma _t} H | \nabla u |$, and therefore,

$$\begin{aligned} \begin{aligned} - \frac{1}{1 - t} \int _{\Sigma _t} H | \nabla u | + \frac{1}{ ( 1 - t)^2 } \int _{\Sigma _t} | \nabla u |^2 = \frac{\textrm{d}}{\textrm{d}t} \left( \frac{1}{1- t} \int _{\Sigma _t} | \nabla u |^2 \right) . \end{aligned} \end{aligned}$$

Thus, if $[t_1, t_2]$ has no critical values, the above directly shows

$$\begin{aligned} \begin{aligned}&\ \int _{t_1}^{t_2} \left( - \frac{1}{1 - t} \int _{\Sigma _t} H | \nabla u | + \frac{1}{ ( 1 - t)^2 } \int _{\Sigma _t} | \nabla u |^2 \right) \\&\quad = \ \frac{1}{1- t_2} \int _{\Sigma _{t_2}} | \nabla u |^2 - \frac{1}{1- t_1} \int _{\Sigma _{t_1}} | \nabla u |^2. \end{aligned} \end{aligned}$$

(3.9)

In general, if $[t_1, t_2]$ has critical values, one may use a regularization argument to still obtain (3.9). For instance, applying Lemma A.1 of Appendix A to u on $\Omega _{[t_1, t_2] }$, one has

$$\begin{aligned}{} & {} \frac{1}{1 - t_2} \int _{\Sigma _{t_2} } | \nabla u |^2 - \frac{1}{1 - t_1} \int _{\Sigma _{t_1}} | \nabla u |^2 \qquad \quad \nonumber \\{} & {} \quad = \ \int _{\Omega _{[t_1, t_2]} } \frac{ | \nabla u |^3 }{ ( 1 - u )^2 } + \int _{ \{ \nabla u \ne 0 \} \subset \Omega _{[t_1, t_2]} } \frac{1}{ 1 - u} | \nabla u |^{-1} \nabla ^2 u ( \nabla u, \nabla u ).\qquad \quad \end{aligned}$$

(3.10)

This, together with the coarea formula and (3.5), gives (3.9).

By (3.8) and (3.9),

$$\begin{aligned} \Psi (t_1) - \Psi (t_2) \ge \int _{t_1}^{t_2} \int _{\Sigma _t} \frac{1}{2} | \mathring{\displaystyle {\mathbb {I}\mathbb {I}}} |^2 + | \nabla u |^{-2} \big | \nabla _{_{\Sigma _t} } | \nabla u | \big |^2 + \frac{3}{4} \left( H - \frac{ 2 | \nabla u | }{ 1 - u } \right) ^2 + \frac{1}{2} R.\nonumber \\ \end{aligned}$$

(3.11)

For the later purpose in Appendix A, we note that (3.11) holds without assumptions on the scalar curvature R.

If the scalar curvature R is nonnegative, then (3.11) implies $\Psi (t_1) \ge \Psi (t_2)$, which proves the lemma. $\square$

In the context of Theorem 3.1, the assumption $\Sigma$ is connected and $H_2 (M, \Sigma ) = 0$ is a sufficient condition to ensure $\chi (\Sigma _t) \le 2$ for a regular $\Sigma _t$. Under this condition, u being harmonic and the maximum principle guarantee $\Sigma _t$ is connected. (The same assumption was used by Bray and the author [8] in estimating the capacity of $\Sigma$ in (M, g) via the solution to the weak inverse mean curvature (1/H) flow [17]. In that setting, a different reasoning shows the level set of the 1/H flow is connected.)

Proof of Theorem 3.1

Let $\Psi (t)$ be given from Lemma 3.1. On an asymptotically flat (M, g), a corollary of Lemma 2.2 shows

$$\begin{aligned} \lim _{t \rightarrow 1} \Psi (t) = 0. \end{aligned}$$

Thus, letting $t_2 \rightarrow 1$ in (3.11) gives

$$\begin{aligned} \begin{aligned} \Psi (t) \ge \ \int _{t}^{1} \int _{\Sigma _s} \frac{1}{2} | \mathring{\displaystyle {\mathbb {I}\mathbb {I}}} |^2 + | \nabla u |^{-2} \big | \nabla _{_{\Sigma _t} } | \nabla u | \big |^2 + \frac{3}{4} \left( H - \frac{ 2 | \nabla u | }{ 1 - u } \right) ^2 + \frac{1}{2} R \end{aligned}\nonumber \\ \end{aligned}$$

(3.12)

for every regular value t. In particular, if $R \ge 0$, then $\Psi (t) \ge 0$.

Inequality (3.1) follows from (3.12) by noting that

$$\begin{aligned} \frac{1}{1-t} \Psi (t) = 3 B(t) - A (t). \end{aligned}$$

(3.13)

To show the rigidity case of (3.1), it suffices to establish it for the case $t = 0$. Suppose the equality in (3.2) holds, then, by (3.12) and its proof, for every regular value $t \in [0,1]$, $\Sigma _t$ is connected (orientable) with $\chi (\Sigma _t) = 2$, hence $\Sigma _t$ is a 2-sphere; moreover, $R =0$, $| \nabla u |$ only depends on t, $\Sigma _t$ is totally umbilic, and $H = \frac{ 2 }{ 1 - t } | \nabla u |$.

To show (M, g) is isometric to ${\mathbb {R}}^3$ minus a round ball, we start from a neighborhood of the boundary $\Sigma$. For convenience, we normalize (M, g) so that $| \Sigma | = 4 \pi$. It follows from the equality

$$\begin{aligned} 4 \pi + \int _\Sigma H | \nabla u | = 3 \int _\Sigma | \nabla u |^2 \end{aligned}$$

that $| \nabla u | = 1$ and $H = 2$ at $\Sigma = \Sigma _0$. Locally, g takes the form of $g = \eta (t) ^{-2} \textrm{d} t^2 + \gamma _t$ near $\Sigma _0$, where $t = u$, $\eta (t) = | \nabla u |$ and $\gamma _t$ denotes the induced metric on $\Sigma _t$, which satisfies $\partial _t \gamma _t = 2 \eta (t)^{-1} \displaystyle {\mathbb{I}\mathbb{I}}_t = \eta (t)^{-1} H \gamma _t = 2(1-t)^{-1} \gamma _t.$ Thus, $(1 - t)^2 \gamma _t =$ a fixed metric. Similarly, since $| \nabla u |' = - H$, $\eta (t)$ satisfies $\eta '(t) = \frac{- 2}{1-t} \eta (t)$. Hence, $( 1 - t )^{-2} \eta =$ a constant. As $| \nabla u | = 1$ at $\Sigma$, we thus have $\eta = (1-t)^2$ and $g = (1-t)^{-4} \textrm{d} t^2 + (1-t)^{-2} \sigma _o$ for some fixed metric $\sigma _o$ on the 2-sphere $\Sigma$. Invoking the fact $R = 0$ near $\Sigma$, we see $\sigma _o$ is a round metric with Gauss curvature 1 on $\Sigma$.

Now, if u has a critical value, let $t_0 \in (0,1)$ be the smallest critical value of u. The above argument then shows $( u^{-1} ([0, t_0) ), g )$ is isometric to

$$\begin{aligned} \left( \Sigma \times [0, t_0), (1-t)^{-4} \textrm{d} t^2 + (1-t)^{-2} \sigma _o \right) . \end{aligned}$$

In particular, this implies $| \nabla u | = (1- t_0)^{2} \ne 0$ on the set $\partial \{ u < t_0 \} = \partial \{ u \ge t_0 \}$. As a result, $\partial \{ u \ge t_0 \}$ is an embedded surface in M. Therefore, $\partial \{ u \ge t_0 \} = \{ u = t_0 \}$ by the strong maximum principle. In summary, this shows $\nabla u \ne 0$ on the set $\{ u = t_0 \}$, which contradicts to the assumption $t_0$ is a critical value. Hence, u has no critical values. We conclude (M, g) is isometric to

$$\begin{aligned} \left( \Sigma \times [0, 1), (1-t)^{-4} \textrm{d} t^2 + (1-t)^{-2} \sigma _o \right) , \end{aligned}$$

which, upon a change of variable $1 - t = r^{-1}$, is isometric to ${\mathbb {R}}^3$ minus a unit ball. $\square$

Remark 3.3

In Theorem 3.1, if $u = c$ at $\Sigma$ for some constant $c < 1$, (3.1) and its equality case still hold. This can be seen either from the above proof, or from considering the function $\frac{1}{1-c} (u-c)$.

Theorem 3.1 implies an upper bound of $\frac{1}{(1-t)^2} \int _{\Sigma _t} | \nabla u |^2$ via $\int _{\Sigma _t} H^2$.

Corollary 3.1

Let (M, g) be a complete, orientable, asymptotically flat 3-manifold with boundary $\Sigma$. Suppose $\Sigma$ is connected and $H_2 (M, \Sigma ) = 0$. Let u be the harmonic function such that $u = 0$ at $\Sigma$ and $u \rightarrow 1$ at $\infty$. If g has nonnegative scalar curvature, then

$$\begin{aligned} \frac{1}{4\pi } \int _\Sigma | \nabla u |^2 \le \frac{1}{9} \bigl [ 2 W + 2 \sqrt{ W^2 + 3 W } + 3 \bigr ] , \end{aligned}$$

(3.14)

where $W = \frac{1}{16 \pi } \int _{\Sigma } H^2$, and equality holds if and only if (M, g) is isometric to ${\mathbb {R}}^3$ minus a round ball.

Proof

Let $z = \bigl ( \int _\Sigma | \nabla u |^2\bigr ) ^\frac{1}{2}$. By Theorem 3.1 and Hölder’s inequality,

$$\begin{aligned} 4 \pi + \sqrt{ 16 \pi W} z \ge 3 z^2. \end{aligned}$$

This implies the bound of z in (3.14) by elementary reason. The equality case follows from the equality case in Theorem 3.1. $\square$

We next apply Theorem 3.1 to show that the quantities in Theorem 2.1, which approach to constant multiples of $\mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1}$ at $\infty$, are actually monotone.

Theorem 3.2

Let (M, g) be a complete, orientable, asymptotically flat 3-manifold with boundary $\Sigma$. Suppose $\Sigma$ is connected and $H_2 (M, \Sigma ) = 0$. Let u be the harmonic function such that $u = 0$ at $\Sigma$ and $u \rightarrow 1$ at $\infty$. If g has nonnegative scalar curvature, then

• (i) $\textstyle \mathcal {A}(t): = \frac{1}{1-t} \big [ 8 \pi - \frac{1}{ 1 - t } \int _{\Sigma _{t}} H | \nabla u | \big ]\nearrow \ \text{ as } \ t \nearrow ,$ i.e., $\mathcal {A}(t)$ is monotone non-decreasing in t. If in addition $\partial \partial \partial g_{ij} = O ( |x|^{-3-\tau } )$ at $\infty$, then $\mathcal {A} (t) \le 12 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1}.$ In particular, at $\Sigma$,

  $$\begin{aligned} 8 \pi - \int _\Sigma H | \nabla u | \le 12 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1}, \end{aligned}$$

  (3.15)

  and equality holds if and only if (M, g) is isometric to ${\mathbb {R}}^3$ minus a round ball.

• (ii) $\textstyle \mathcal {B}(t): = \frac{1}{1-t} \big [ 4 \pi - \frac{1}{ (1 - t )^2 } \int _{\Sigma _{t}} | \nabla u |^2 \big ] \ \nearrow \ \text{ as } \ t \nearrow ,$ i.e., $\mathcal {B}(t)$ is monotone non-decreasing in t. If in addition $\partial \partial \partial g_{ij} = O ( |x|^{-3-\tau } )$ at $\infty$, then $\mathcal {B} (t) \le 4 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1}.$ In particular, at $\Sigma$,

  $$\begin{aligned} 4 \pi - \int _\Sigma | \nabla u |^2 \le 4 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1}, \end{aligned}$$

  (3.16)

  and equality holds if and only if (M, g) is isometric to ${\mathbb {R}}^3$ minus a round ball.

Proof

We first show (ii) as it is more straightforward. By (2.36) and (3.3), at every regular value t, we have

$$\begin{aligned} \begin{aligned} - B'(t) = \frac{1}{ 1 - t } \left[ - 2 B(t) + A (t) \, \right] \le \frac{1}{1-t} B(t). \end{aligned} \end{aligned}$$

Therefore, $\textstyle \bigl [ \frac{1}{1-t} B(t)\bigr ] ' \ge 0,$ which implies the monotonicity of $\mathcal {B} (t) = \frac{1}{1-t} B(t)$ in the case that u has no critical values. If u has critical values, we may again apply a regularization argument to show that $\mathcal {B}(t_2) - \mathcal {B}(t_1) \ge 0$ for $t_2 > t_1$, see Proposition A.1 in Appendix A for details.

By Theorem 2.1 (ii),

$$\begin{aligned} \lim _{t \rightarrow 1} \mathcal {B} (t) = 4 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1}. \end{aligned}$$

Therefore, the monotonicity of $\mathcal {B} (t)$ shows

$$\begin{aligned} \mathcal {B} (t) \le 4 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1}. \end{aligned}$$

At $t =0$, this gives

$$\begin{aligned} \mathcal {B} (0) = 4 \pi - \int _{\Sigma } | \nabla u |^2 \le 4 \pi \, \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1}. \end{aligned}$$

The rigidity part follows from the rigidity part of Theorem 3.1.

To show (i), we calculate, at a regular value t,

$$\begin{aligned} \begin{aligned} \mathcal {A}'(t) = \ \frac{1}{(1- t)^2} \left[ A (t) - \frac{1}{1-t} \int _{\Sigma _t} H | \nabla u | - \left( \int _{\Sigma _t} H | \nabla u | \right) ' \right] . \end{aligned} \end{aligned}$$

By (2.25) and the Gauss–Bonnet theorem,

By (3.3),

$$\begin{aligned} I(t) = 3 B(t) - A (t) \ge 0. \end{aligned}$$

Therefore, $\mathcal {A}'(t) \ge 0$, which implies the monotonicity of $\mathcal {A}(t)$ in the absence of critical values. The general case can be again handled by a regularization argument that shows $\mathcal {A}(t_2) - \mathcal {A}(t_1) \ge 0$ for $t_2 > t_1$, see Proposition A.1 in Appendix A.

The remaining conclusions in (i) follow from Theorem 2.1 (i) and Theorem 3.1. $\square$

Remark 3.4

If $u=c$ at $\Sigma$ for some constant $c < 1$, Theorem 3.2 still holds with $\mathfrak {c}_{_\Sigma }$ replaced by $\mathfrak {c}$, the constant in the asymptotic expansion $u = 1 - \mathfrak {c}|x|^{-1} + o ( |x|^{-1} )$ as $x \rightarrow \infty$.

Remark 3.5

In the above proof of (3.15) and (3.16), we assumed $\partial ^3 g_{ij} = O (|x|^{-\tau -3} )$ to obtain $\lim _{ t \rightarrow 1} \mathcal {A}(t) = 12 \pi \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1}$ and $\lim _{t \rightarrow 1} \mathcal {B} (t) = 4 \pi \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1}$ by applying Theorem 2.1. We mention that this assumption on $\partial ^3 g_{ij}$ can indeed be dropped, due to some recent developments since the appearance of this work. In [15], it was shown that, under the general asymptotic condition (2.1), one has

$$\begin{aligned} \limsup _{ t \rightarrow 1} \mathcal {A}(t) \le 12 \pi \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \quad \text {and} \quad \limsup _{t \rightarrow 1} \mathcal {B} (t) \le 4 \pi \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1}. \end{aligned}$$

(See [15, Proposition 4.1], which made use of [1, Lemma 2.5].) Combined with the monotonicity, such inequalities are sufficient to obtain (3.15) and (3.16) in Theorem 3.2. For this reason, theorems in later sections of this paper, which made use of (3.15) and (3.16), do not need the assumption on $\partial ^3 g_{ij}$ neither.

Remark 3.6

Comparing (3.2), (3.15) and (3.16), we have (3.2) + (3.16) $\Rightarrow$ (3.15).

4 Proofs of the Positive Mass Theorem

The 3-dimensional Riemannian positive mass theorem (PMT), first proved by Schoen–Yau [27] and later by Witten [31], asserts that if (M, g) is a complete, asymptotically flat 3-manifold without boundary, with nonnegative scalar curvature, then $\mathfrak {m}\ge 0$, and $\mathfrak {m}= 0$ if and only if $(M^3, g)$ is isometric to ${\mathbb {R}}^3$.

Since the work of Schoen–Yau and Witten, other proofs of this theorem have been given by Huisken–Ilmanen [17], by Li [21], by Bray–Kazaras–Khuri–Stern [7], and by Agostiniani–Mazzieri–Oronzio [2]. (Agostiniani–Mantegazza–Mazzieri–Oronzio [1] also gave a new proof of the Riemannian Penrose inequality, first proved by Bray [6] and Huisken–Ilmanen [17].)

To show the 3-dimensional PMT, it is known it suffices to assume M is topologically ${\mathbb {R}}^3$, see [7, Section 2] for instance; it also suffices to assume g has higher order derivatives decay near $\infty$, see [26, 28]. For this reason, one can make these assumptions in proving $\mathfrak {m}\ge 0$. Once the inequality is shown, the rigidity case of $m = 0$ follows from a variational argument, see [27].

As applications of Theorems 2.1 and 3.2, we observe a few additional arguments that demonstrate $\mathfrak {m}\ge 0$. We first outline the tools and features of the proofs to be given:

• Proof I uses Theorem 2.1 (ii) and a result of Munteanu–Wang [25].

• Proof II is self-contained. It makes use of Theorems 2.1 and 3.2.

• Proof III is self-contained. It uses the inequalities in Theorem 3.2. Proof III leads to new sufficient conditions that guarantee the positivity of the mass, see Sect. 5.

In what follows, let (M, g) be a complete, asymptotically flat 3-manifold with nonnegative scalar curvature. Suppose M is topologically ${\mathbb {R}}^3$.

Proof

I   Take $p \in M$. Let G(x) be the minimal positive Green’s function with a pole at p, with $G (x) \rightarrow 0$ as $x \rightarrow \infty$. Let $u = 1 - G$. By Theorem 1.1 of Muntenau–Wang [25],

$$\begin{aligned} \begin{aligned} 4\pi ( 1 - t ) - \frac{1}{1 - t} \int _{\Sigma _t} | \nabla u |^2 \searrow \quad \text{ as } \,\ t \nearrow , \end{aligned} \end{aligned}$$

i.e., it is monotone non-increasing in t.

As $x \rightarrow \infty$, $u = 1 - \frac{1}{ 4 \pi } |x|^{-1} + O (|x|^{-1-\tau })$. By Lemma 2.2, $\frac{1}{1 - t} \int _{\Sigma _t} | \nabla u |^2 \rightarrow 0$ as $t \rightarrow 1$. Hence,

$$\begin{aligned} 4\pi ( 1 - t ) - \frac{1}{1 - t} \int _{\Sigma _t} | \nabla u |^2 \ge 0. \end{aligned}$$

Consequently,

$$\begin{aligned} \frac{1}{ (1-t) } \left[ 4\pi - \frac{1}{ (1 - t)^2 } \int _{\Sigma _t} | \nabla u |^2 \right] \ge 0. \end{aligned}$$

By Theorem 2.1 (ii),

$$\begin{aligned} \lim _{t \rightarrow 1} \frac{1}{ (1-t) } \left[ 4\pi - \frac{1}{ (1 - t)^2 } \int _{\Sigma _t} | \nabla u |^2 \right] = ( 4 \pi )^2 \mathfrak {m}. \end{aligned}$$

Therefore, $\mathfrak {m}\ge 0$. $\square$

Proof

II   Take $p \in M$. Let G(x) be the minimal positive Green’s function with a pole at p. Let d(x) denote the distance from x to p in (M, g). As $x \rightarrow p$, it is known

$$\begin{aligned} G (x) = \frac{1}{4\pi } d(x)^{-1} + o ( d(x)^{-1} ), \quad | \nabla G (x) | = \frac{1}{4 \pi } d(x)^{-2} + o ( d (x)^{-2} ). \end{aligned}$$

(4.1)

(See [25, Equation (3.3)] or [23, Theorem 2.4] for instance.)

Consider $u = 1 - G$. By Theorem 3.2 (ii),

$$\begin{aligned} \mathcal {B}(t) = \frac{1}{1-t} \left[ 4 \pi - \frac{1}{ (1 - t )^2 } \int _{\Sigma _{t}} | \nabla u |^2 \right] \nearrow \quad \text{ as } \,\ t \nearrow , \end{aligned}$$

(4.2)

i.e., it is monotone non-decreasing in t. Note this is different from the monotonicity of Munteanu–Wang [25]. The latter asserts $(1-t)^2 \mathcal {B} (t)$ is monotone non-increasing.

As $t \rightarrow - \infty$, by (4.1), $\frac{1}{ (1 - t )^2 } \int _{\Sigma _{t}} | \nabla u |^2$ is bounded, hence $\frac{1}{ (1 - t )^3 } \int _{\Sigma _{t}} | \nabla u |^2 \rightarrow 0$. Thus,

$$\begin{aligned} \lim _{t \rightarrow - \infty } \mathcal {B} (t) = 0. \end{aligned}$$

(4.3)

Therefore, by (4.2) and (4.3), $\mathcal {B} (t) \ge 0$. By Theorem 2.1 (ii),

$$\begin{aligned} (4 \pi )^2 \, \mathfrak {m}= \lim _{t \rightarrow 1} \mathcal {B} (t) \ge 0. \end{aligned}$$

$\square$

Remark 4.1

Proof II is similar to that of Agostiniani–Mazzieri–Oronzio [2]. The difference is the use of different monotone quantities, i.e., $\mathcal {B}(t)$ compared to F(t). A feature of $\mathcal {B} (t)$ used here is that it does not involve derivatives of the metric.

Remark 4.2

One can also work with $\mathcal {A}(t)$, and apply Theorems 3.2 (i) and 2.1 (i). In this case, one checks $\lim _{t \rightarrow - \infty } \frac{1}{(1-t)^2} \int _{\Sigma _t} H | \nabla u | = 0$, which follows from known estimates on $\nabla ^2 G$ near the pole (see [2, 23, 25] for instance).

Proof

III   Take $p \in M$. Given a small $r > 0$, let $B_r$ denote the geodesic ball of radius r centered at p. Let $\Sigma _r = \partial B_r$ and $u = u_r$ be the harmonic function with $u=0$ at $\Sigma _r$ and $u \rightarrow 1$ at $\infty$. Let $\mathfrak {c}_{r}$ be the capacity of $\Sigma _r$ in (M, g).

Applying (3.15) of Theorem 3.2 (i) to $(M \setminus B_r, g)$, we have

$$\begin{aligned} c_r \left( 8 \pi - \int _{\Sigma _r} H | \nabla u | \right) \le 12 \pi \, \mathfrak {m}. \end{aligned}$$

(4.4)

It remains to check, as $r \rightarrow 0$,

$$\begin{aligned} c_r = O (r) \quad \text{ and } \quad \int _{\Sigma _r} H | \nabla u | = O (1). \end{aligned}$$

(4.5)

A conclusion $\mathfrak {m}\ge 0$ will follow from (4.4) and (4.5).

To estimate $c_r$, we may use the variational characterization of the capacity, i.e.,

$$\begin{aligned} c_r = \inf _{f} \left\{ \frac{1}{4\pi } \int _{ M \setminus B_r} | \nabla f |^2 \right\} , \end{aligned}$$

(4.6)

where f is a Lipschitz function with $f= 0$ at $\Sigma _r$ and $f \rightarrow 1$ at $\infty$. Consider a test function $f (x) = r^{-1} \left( d (x) - r \right)$ in $B_{2 r} {\setminus } B_r$ and extend f to be 1 outside $B_{2 r}$. Here d(x) is the distance from x to p. Then

$$\begin{aligned} \begin{aligned} c_r \le&\ \frac{1}{4\pi } \int _{B_{2r} \setminus B_r} | \nabla f |^2 = \frac{1}{4\pi r^2 } \textrm{Vol}(B_{2 r}{\setminus } B_r ) = O (r). \end{aligned} \end{aligned}$$

(4.7)

For $\int _{\Sigma _r} H | \nabla u |$, we have

$$\begin{aligned} \left| \int _{\Sigma _r} H | \nabla u | \right| \le \max _{\Sigma _r} |H | \, \int _{\Sigma _r} | \nabla u | = \max _{\Sigma _r} |H | \, \mathfrak {c}_r = O (1) \end{aligned}$$

by (4.7) and the fact $H = 2 r^{-1} + O (r)$ (see (3.34) in [13] for instance).

This verifies (4.5) and completes the proof. $\square$

Remark 4.3

In the above proof, we estimated $\mathfrak {c}_r$ by the so-called relative capacity of $\Sigma _r$ in $B_{2r}$. By a result of Jauregui [18], one can indeed check

$$\begin{aligned} \limsup _{r \rightarrow 0} \left( 8 \pi - \int _{\Sigma _r} H | \nabla u | \right) \ge 0. \end{aligned}$$

Remark 4.4

Alternatively one may use (3.16) of Theorem 3.2 (ii) to have

$$\begin{aligned} c_r \left( 4 \pi - \int _{\Sigma _r} | \nabla u |^2 \right) \le 4 \pi \, \mathfrak {m}\end{aligned}$$

and check $\int _{\Sigma _r} | \nabla u |^2 = O (1)$. For instance, by the maximum principle, $| \nabla u | \le | \nabla v |$ at $\partial B_r$, where v is the harmonic function with $v = 0$ at $\partial B_r$ and $v = 1$ at $\partial B_{2r}$. By scaling and elliptic boundary estimates, $\int _{\partial B_r} | \nabla v |^2 = O (1)$ which shows $\int _{\Sigma _r} | \nabla u |^2 = O (1)$.

In addition to the above proofs, we want to mention there is a more geometric proof of PMT, which makes use of the full strength of our mass-capacity inequality Theorem 7.4 in Sect. 7. We give that proof in Remark 7.1.

5 Positive Mass Theorems with Boundary

Inspired by Proof III in the preceding section, we give some sufficient conditions that imply positive mass on manifolds with boundary.

Theorem 5.1

Let (M, g) be a complete, orientable, asymptotically flat 3-manifold with nonnegative scalar curvature, with boundary $\Sigma$. Suppose $\Sigma$ is connected and $H_2 (M, \Sigma ) = 0$. Let $\Omega \subset M$ be a bounded region separating $\Sigma$ and $\infty$. More precisely, this means $\partial \Omega$ has two connected components $S_0$ and $S_1$, where $S_0$ encloses $\Sigma$ (and is allowed to coincide with $\Sigma$) and $S_1$ encloses $S_0$. Let $u_{_\Omega }$ be the function on $\Omega$ with

$$\begin{aligned} \Delta u_{_\Omega } = 0, \quad u_{_\Omega } |_{S_0} = 0, \quad \text{ and } \ u_{_\Omega } |_{S_1 } = 1. \end{aligned}$$

Let $\mathfrak {c}(\Omega ) = \frac{1}{4\pi } \int _{\Omega } | \nabla u_{_\Omega } |^2 = \frac{1}{4\pi } \int _{S_0} | \nabla u_{_\Omega } |$. Then

$$\begin{aligned} H \le \frac{2}{ \mathfrak {c}(\Omega ) } \ \Longrightarrow \ \mathfrak {m}> 0. \end{aligned}$$

(5.1)

In particular, this implies

$$\begin{aligned} H \le \frac{ 8 \pi L^2 }{\textrm{Vol}(\Omega )} \ \Longrightarrow \ \mathfrak {m}> 0. \end{aligned}$$

(5.2)

Here H is the mean curvature of $\Sigma$ in (M, g), $\textrm{Vol}(\Omega )$ is the volume of $(\Omega , g)$, and L is the distance between $S_0$ and $S_1$.

Proof

Let u be the harmonic function on M with $u=0$ at $\Sigma$ and $u \rightarrow 1$ at $\infty$. By (3.15) of Theorem 3.2 (i),

$$\begin{aligned} \begin{aligned} 12 \pi \, \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1}&\ge \ 8 \pi - \int _{\Sigma } H | \nabla u | \\ {}&\ge \ 4 \pi \Big ( 2 - \mathfrak {c}_{\Sigma } \max _{\Sigma } H \Big ), \end{aligned} \end{aligned}$$

(5.3)

where $\mathfrak {c}_{_\Sigma }$ is the capacity of $\Sigma$ in (M, g). This shows

$$\begin{aligned} \max _{\Sigma } H \le 2 \mathfrak {c}_{_\Sigma }^{-1} \Longrightarrow \mathfrak {m}\ge 0, \quad \max _{\Sigma } H < 2 \mathfrak {c}_{_\Sigma }^{-1} \Longrightarrow \mathfrak {m}> 0, \end{aligned}$$

(5.4)

respectively.

Let D denote the region enclosed by $S_1$ with $\Sigma$. Extending $u_{_\Omega }$ to be 1 on $M \setminus D$ and to be 0 on $D \setminus \Omega$. By the variational characterization of the capacity,

$$\begin{aligned} \mathfrak {c}_{_\Sigma } < \frac{1}{4 \pi } \int _M | \nabla u_{_\Omega } |^2 = \mathfrak {c}(\Omega ). \end{aligned}$$

(5.5)

Therefore, (5.1) follows from (5.4) and (5.5).

To see (5.2), it suffices to estimate $\mathfrak {c}(\Omega )$. On $\Omega$, consider a test function f(x) which equals $L^{-1} d(x)$ if $d (x) \le L$ and is identically 1 if $d (x) \ge L$. Here d(x) denotes the distance from x to $S_0$. Then

$$\begin{aligned} \mathfrak {c}(\Omega ) \le \ \frac{1}{4\pi } \int _\Omega | \nabla f |^2 \le \frac{1}{4\pi L^2} \text {Vol}(\Omega ). \end{aligned}$$

(5.6)

Hence, (5.2) follows from (5.1) and (5.6). $\square$

The next result does not involve the mean curvature of the boundary. It makes use of (3.16) in Theorem 3.2 (ii).

Theorem 5.2

Let (M, g), $\Omega$, $S_0$, $S_1$ and $u_{_\Omega }$ be given as in Theorem 5.1. Then

$$\begin{aligned} \int _{S_0} | \nabla u_{_\Omega } |^2 \le 4 \pi \Longrightarrow \ \mathfrak {m}> 0. \end{aligned}$$

(5.7)

Proof

Let ${\tilde{M}}$ denote the region outside $S_0$. Let ${\tilde{u}}$ be the harmonic function on ${\tilde{M}}$ with ${\tilde{u}} =0$ at $S_0$ and ${\tilde{u}} \rightarrow 1$ at $\infty$. Applying (3.16) of Theorem 3.2 (ii) to $({\tilde{M}}, g, {\tilde{u}})$, we have

$$\begin{aligned} \int _{S_0} | \nabla {\tilde{u}} |^2 \le 4 \pi \Longrightarrow \mathfrak {m}\ge 0, \quad \int _{S_0} | \nabla {\tilde{u}} |^2 < 4 \pi \Longrightarrow \mathfrak {m}> 0, \end{aligned}$$

(5.8)

respectively. On $(\Omega , g)$, the maximum principle shows

$$\begin{aligned} | \nabla {\tilde{u}} | < | \nabla u_{_\Omega } | \quad \text{ at } \ S_0. \end{aligned}$$

(5.9)

Therefore, (5.7) follows from (5.8) and (5.9). $\square$

Remark 5.1

It may be worthy of noting that the condition in (5.7) and the upper bound of H in (5.1) only involve the $C^0$-geometry of $(\Omega , g)$.

It is conceivable that Theorems 5.1 and 5.2 may be used to study the mass of incomplete asymptotically flat 3-manifolds. Recently Cecchini–Zeidler [10] and Lee–Lesourd–Unger [19] have given sufficient conditions, involving a positive lower bound of the scalar curvature on suitable regions in a manifold $(M^n, g)$ that is spin or of dimension $3 \le n \le 7$, which guarantee the positivity of the mass. If such conditions are interpreted as shielding the incomplete part by regions with sufficiently positive scalar curvature, conditions in (5.1), (5.2) and (5.7) may be thought as shielding conditions in terms of the $C^0$-geometry of a separating region.

We end this section with the following proposition which was known and proved previously via the weak inverse mean curvature (1/H) flow developed by Huisken–Ilmanen [17]. We include it here to show that the result can also be proved using harmonic functions.

Proposition 5.1

Let (M, g) be a complete, orientable, asymptotically flat 3-manifold with boundary $\Sigma$. Suppose $\Sigma$ is connected and $H_2 (M, \Sigma ) = 0$. If g has nonnegative scalar curvature, then

$$\begin{aligned} \int _{\Sigma } H^2 \le 16 \pi \ \Longrightarrow \ \mathfrak {m}\ge 0, \end{aligned}$$

and $\mathfrak {m}= 0$ if and only if (M, g) is isometric to ${\mathbb {R}}^3$ minus a round ball.

Proof

By Corollary 3.1,

$$\begin{aligned} \int _{\Sigma } H^2 \le 16 \pi \ \Longrightarrow \ \int _\Sigma | \nabla u |^2 \le 4 \pi . \end{aligned}$$

Hence, $\mathfrak {m}\ge 0$ by (3.16). The rigidity case follows from that of Corollary 3.1. $\square$

6 Integral Identities for the Mass-to-Capacity Ratio

In [7], Bray–Kazaras–Khuri–Stern found an integral identity for the mass of an asymptotically flat manifold. More precisely, if (E, g) denotes the exterior region of a complete, asymptotically flat Riemannian 3-manifold (M, g) with mass $\mathfrak {m}$, then

$$\begin{aligned} 16 \pi \mathfrak {m}\ge \int _{E} \left( \frac{ | \nabla ^2 u |}{ | \nabla u | } + R | \nabla u | \right) , \end{aligned}$$

(6.1)

where u is a harmonic function on (E, g) satisfying Neumann boundary conditions at $\partial E$, and which is asymptotic to one of the asymptotically flat coordinate functions at $\infty$. In particular, if the scalar curvature is nonnegative, then $\mathfrak {m}\ge 0$.

In this section, we derive mass identities analogous to (6.1) with u being a harmonic function that equals 0 at the boundary and is asymptotic to 1 at $\infty$.

Theorem 6.1

Let (M, g) be a complete, orientable, asymptotically flat 3-manifold with boundary $\Sigma$. Suppose $\Sigma$ is connected and $H_2 (M, \Sigma ) = 0$. Let u be the harmonic function such that $u = 0$ at $\Sigma$ and $u \rightarrow 1$ at $\infty$. Let $\Phi _u$ be a symmetric (0, 2) tensor given by

$$\begin{aligned} \begin{aligned} \Phi _u = \frac{ | \nabla u |^2 }{ 1 - u } g - \frac{ 3 \textrm{d} u \otimes \textrm{d} u }{1-u}. \end{aligned} \end{aligned}$$

Let $\mathfrak {m}$ be the mass of (M, g) and $\mathfrak {c}_{_\Sigma }$ be the capacity of $\Sigma$ in (M, g). Then

$$\begin{aligned} \begin{aligned}&\ \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} - \left( 1 - \frac{1}{ 4 \pi } \int _{\Sigma } | \nabla u |^2 \right) \\&\quad \ge \ \frac{1}{ 16 \pi } \int _M \left[ \frac{1}{ (1- u )^2} - 1 \right] \left( \frac{ | \nabla ^2 u - \Phi _u |^2 }{ | \nabla u | } + R | \nabla u | \right) \end{aligned} \end{aligned}$$

(6.2)

and

$$\begin{aligned} \begin{aligned}&\ \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} - \frac{2}{3} \left( 1 - \frac{1}{8\pi } \int _{\Sigma } H | \nabla u | \right) \\&\quad \ge \ \frac{1}{ 16 \pi } \int _M \left[ \frac{1}{ (1- u )^2} - \frac{1}{3} \right] \left( \frac{ | \nabla ^2 u - \Phi _u |^2 }{ | \nabla u | } + R | \nabla u | \right) . \end{aligned} \end{aligned}$$

(6.3)

Proof

By (3.5), along a regular level set $\Sigma _t$, $(\nabla ^2 u - \Phi _u )$ satisfies

$$\begin{aligned}{} & {} \left( \nabla ^2 u - \Phi _u \right) (\nu , \nu ) = - H | \nabla u | + \frac{ 2 | \nabla u |^2}{1-u}, \ \\{} & {} \left( \nabla ^2 u - \Phi _u \right) (\nu , \cdot ) |_{_{\Sigma _t} } = \langle \nabla _{_{\Sigma _t}} | \nabla u |, \cdot \rangle , \\{} & {} \left( \nabla ^2 u - \Phi _u \right) ( \cdot , \cdot ) |_{_{\Sigma _t} } = | \nabla u | \left( \displaystyle {\mathbb{I}\mathbb{I}}- \frac{ | \nabla u | }{ 1 - u } \gamma \right) , \end{aligned}$$

where $\gamma$ denotes the induced metric on $\Sigma _t$. Therefore,

$$\begin{aligned} \begin{aligned}&\ | \nabla u |^{-2} \left| \nabla ^2 u - \Phi _u \right| ^2 \\ {}&\quad = \ \frac{3}{2} \left( H - \frac{ 2 | \nabla u |}{ 1 - u } \right) ^2 + 2 | \nabla u |^{-2} \big | \nabla _{_{\Sigma _t} } | \nabla u | \big |^2 + | \mathring{\displaystyle {\mathbb {I}\mathbb {I}}} |^2. \end{aligned} \end{aligned}$$

(6.4)

Given two regular values $t_1 < t_2$, by (A.14) in Proposition A.1 of Appendix A, we have

$$\begin{aligned} \mathcal {B} (t_2) - \mathcal {B} (t_1) = \int _{t_1}^{t_2} \frac{1}{(1-t)^2} \left[ 3 B(t) - A (t) \right] . \end{aligned}$$

(6.5)

By (3.13) and (3.12),

$$\begin{aligned} 3B(t) - A (t) = \frac{1}{1-t} \Psi (t) \quad \text {and} \quad \Psi (t) \ge \int _{t}^{1} \psi (s), \end{aligned}$$

(6.6)

where

$$\begin{aligned} \begin{aligned} \psi (t)&= \ \int _{\Sigma _t} \left[ \frac{3}{4} \left( H - \frac{ 2 | \nabla u | }{ 1 - u } \right) ^2 + | \nabla u |^{-2} \big | \nabla _{_{\Sigma _t} } | \nabla u | \big |^2 + \frac{1}{2} | \mathring{ \displaystyle {\mathbb {I}\mathbb {I}}} |^2 + \frac{1}{2} R \right] \\ {}&= \ \frac{1}{2} \int _{\Sigma _t} \left( \frac{ | \nabla ^2 u - \Phi _u |^2 }{ | \nabla u |^{2} } + R \right) . \end{aligned} \end{aligned}$$

(6.7)

Taking $t_1 = 0$ and letting $t_2 \rightarrow 1$, applying Theorem 2.1, we hence have

$$\begin{aligned} \begin{aligned} 4 \pi \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} - \mathcal {B} ( 0 )&= \int _{0}^{1} \frac{1}{(1-t)^3} \Psi (t) \\&\ge \int _{0}^{1} \frac{1}{(1-t)^3} \left( \int _t^1 \psi (s) \right) . \end{aligned} \end{aligned}$$

(6.8)

Integration by parts gives

$$\begin{aligned} \begin{aligned}&\ \int _{0}^{1} \frac{1}{(1-t)^3} \left( \int _t^1 \psi (s) \right) \\&\quad = \ \frac{1}{2} \left[ \lim _{t \rightarrow 1} \frac{1}{ (1-t)^2 } \int _t^1 \psi (s) - \int _0^1 \psi (s) + \int _{0}^{1} \frac{ \psi (t) }{ (1-t)^2} \right] . \end{aligned} \end{aligned}$$

(6.9)

We claim

$$\begin{aligned} \lim _{t \rightarrow 1} \frac{1}{ (1-t)^2 } \int _t^1 \psi (s) = 0. \end{aligned}$$

(6.10)

This is because, by (2.32),

$$\begin{aligned} \int _{t}^1 \int _{\Sigma _s} | \nabla u |^{-2} \big | \nabla _{_\Sigma } | \nabla u | \big |^2 + \frac{1}{2} | \mathring{\displaystyle {\mathbb {I}\mathbb {I}}} |^2 = O ( ( 1 - t )^{1 + 2 \tau } ), \end{aligned}$$

and, by (2.33),

$$\begin{aligned} \int _t^1 \int _{\Sigma _s} |R| = o ( ( 1 - t )^2 ). \end{aligned}$$

Also, by (2.11), (2.12) and Lemma 2.1,

$$\begin{aligned} \int _t^1 \int _{\Sigma _s} \left( H - \frac{2 | \nabla u |}{ 1 - u } \right) ^2 = O ( ( 1 - t)^{1 + 2 \tau } ). \end{aligned}$$

(6.11)

Therefore, (6.10) holds.

Now it follows from (6.8)–(6.10) that

$$\begin{aligned} \begin{aligned}&\ 4 \pi \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} - \mathcal {B} (0) \\&\quad \ge \ \frac{1}{2} \int _0^1 \left[ \frac{1}{ (1-t)^2} - 1 \right] \psi (t) \\&\quad = \ \frac{1}{4} \int _0^1 \left[ \frac{1}{ (1-t)^2} - 1 \right] \int _{\Sigma _t} \left( \frac{ | \nabla ^2 u - \Phi _u |^2 }{ | \nabla u |^{2} } + R \right) \\&\quad = \ \frac{1}{4} \int _M \left[ \frac{1}{ (1- u )^2} - 1 \right] \left( \frac{ | \nabla ^2 u - \Phi _u |^2 }{| \nabla u | } + R | \nabla u | \right) . \end{aligned} \end{aligned}$$

(6.12)

This proves (6.2).

Similarly, by (A.17) following Proposition A.1 of Appendix A,

$$\begin{aligned} \left[ \mathcal {A}(t_2) - \mathcal {B}(t_2) \right] - \left[ \mathcal {A}(t_1) - \mathcal {B}(t_1) \right] \ge \int _{t_1}^{t_2} \frac{1}{(1-t)^2} \psi (t). \end{aligned}$$

(6.13)

Taking $t_1 = 0$, letting $t_2 \rightarrow 1$ and applying Theorem 2.1, we have

$$\begin{aligned} \begin{aligned}&\ 8 \pi \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} - (\mathcal {A}(0) - \mathcal {B}(0) ) \\&\quad \ge \ \frac{1}{2} \int _{0}^{1} \frac{1}{(1-t)^2} \int _{\Sigma _t} \left( \frac{ | \nabla ^2 u - \Phi _u |^2 }{ | \nabla u |^{2} } + R \right) \\&\quad = \ \frac{1}{2} \int _M \frac{1}{(1-u)^2} \left( \frac{ | \nabla ^2 u - \Phi _u |^2 }{ | \nabla u | } + R | \nabla u | \right) . \end{aligned} \end{aligned}$$

(6.14)

This together with (6.12) proves (6.3). $\square$

Remark 6.1

If the scalar curvature R is nonnegative, then (6.2) implies (3.16), (6.3) implies (3.15), and (6.14) implies

$$\begin{aligned} 4 \pi - \int _{\Sigma } H | \nabla u | + \int _{\Sigma } | \nabla u |^2 \le 8 \pi \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1}. \end{aligned}$$

(6.15)

For manifolds that are spatial Schwarzschild manifolds near infinity, (6.15) also follows from the work of Agostiniani–Mazzieri–Oronzio [2]. On the other hand, one sees (6.15) is an algebraic consequence of (3.2) and (3.16).

Remark 6.2

If the manifold M in Theorem 6.1 has no boundary, let G be a minimal positive Green’s function with a pole at some $p \in M$. Taking $u = 1 - 4 \pi G$ in (6.13), and letting $t_2 \rightarrow 1$, $t_1 \rightarrow - \infty$, one finds

$$\begin{aligned} \begin{aligned} \mathfrak {m}\ge&\ \frac{1}{ ( 8 \pi )^2 } \int _M \frac{1}{ G^2} \left( \frac{ | \nabla ^2 G + \Phi _G |^2 }{ | \nabla G | } + R | \nabla G | \right) . \end{aligned} \end{aligned}$$

(6.16)

Here $\Phi _G$ is the (0, 2) tensor given by

$$\begin{aligned} \Phi _G = \frac{ | \nabla G |^2 }{ G } g - \frac{ 3 \textrm{d} G \otimes \textrm{d} G }{ G }. \end{aligned}$$

Inequality (6.16) gives the integral version of the proof of the 3-dimensional PMT in [2].

7 Promoting Inequalities via Schwarzschild Models

Inequalities in Sect. 3 are derived via monotone quantities that become constant in Euclidean spaces outside round balls. As a result, they are strict inequalities when evaluated in spatial Schwarzschild manifolds with nonzero mass outside rotationally symmetric spheres.

Inspired by Bray’s proof of the Riemannian Penrose inequality [6], in this section we apply results from the previous sections to derive inequalities that become equality in Schwarzschild spaces.

We first outline the idea. Given a tuple (M, g, u) satisfying assumptions in Theorem 3.1 (or equivalently in Theorem 3.2), let v be any other harmonic function on (M, g) with $v \rightarrow 1$ at $\infty$ and $v > 0$ at $\Sigma$. The following facts hold:

• 1. the metric ${\bar{g}}: = v^4 g$ is asymptotically flat, with nonnegative scalar curvature;

• 2. the function ${\bar{u}}: = v^{-1} u$ is a harmonic function with respect to the metric ${\bar{g}}$, and satisfies ${\bar{u}} = 0$ at $\Sigma$ and ${\bar{u}} \rightarrow 1$ at $\infty$.

Thus, results from the previous sections are applicable to M with the conformally deformed metric ${\bar{g}}$ and the ${\bar{g}}$-harmonic function ${\bar{u}}$.

To proceed, we compute the quantities involved. Let $\bar{\nabla }$ denote the gradient on $(M, {\bar{g}})$, let ${\bar{H}}$ be the mean curvature of $\Sigma$ in $(M, {\bar{g}})$ with respect to the $\infty$-pointing normal. Let $\textrm{d} \sigma$, $\textrm{d} \bar{\sigma }$ denote the surface measure on $\Sigma$ in (M, g), $(M, {\bar{g}})$, respectively. As $\Sigma$ has dimension two, it can be checked

$$\begin{aligned} \int _{\Sigma } | \bar{\nabla }{\bar{u}} |_{{\bar{g}}}^2 \, \textrm{d} \bar{\sigma }= \int _{\Sigma } | \nabla {\bar{u}} |^2 \, \textrm{d} \sigma . \end{aligned}$$

(7.1)

(We omitted writing the area and volume measures in previous integrals as there was only one metric g involved therein.) The mean curvature ${\bar{H}}$ is related to the mean curvature H of $\Sigma$ in (M, g) via ${\bar{H}} = v^{-2} ( 4 v^{-1} \partial _\nu v + H )$. Thus,

$$\begin{aligned} \int _\Sigma {\bar{H}} | \bar{\nabla }{\bar{u}} |_{{\bar{g}}} \,\text {d} \bar{\sigma }= \int _\Sigma \big ( 4 v^{-1} \partial _\nu v + H \big ) | \nabla {\bar{u}} | \,\text {d} \sigma . \end{aligned}$$

(7.2)

Let ${\bar{\mathfrak {m}}}$ denote the mass of $(M, {\bar{g}})$. ${\bar{\mathfrak {m}}}$ and $\mathfrak {m}$ are related by

$$\begin{aligned} {\bar{\mathfrak {m}}} = \mathfrak {m}- 2 \mathfrak {c}_v, \end{aligned}$$

(7.3)

where $\mathfrak {c}_v$ is the constant in the expansion

$$\begin{aligned} v = 1 - \frac{ \mathfrak {c}_v}{ |x| } + o (|x|^{-1}), \end{aligned}$$

as $x \rightarrow \infty$. Since ${\bar{u}} = v^{-1} u$, ${\bar{u}}$ satisfies

$$\begin{aligned} {\bar{u}} = 1 - \frac{ \mathfrak {c}_{_\Sigma } - \mathfrak {c}_v }{ | x|} + o ( |x|^{-1} ), \end{aligned}$$

where $\mathfrak {c}_{\Sigma } > \mathfrak {c}_v$ by the fact $v > u$ and the maximum principle. The capacity of $\Sigma$ in $(M, {\bar{g}})$, which we denote by ${\bar{\mathfrak {c}}}_{_\Sigma }$, is then given by

$$\begin{aligned} {\bar{\mathfrak {c}}}_{_\Sigma } = \mathfrak {c}_{_\Sigma } - \mathfrak {c}_v. \end{aligned}$$

(7.4)

Finally, we note, as $u = 0$ at $\Sigma$,

$$\begin{aligned} | \nabla {\bar{u}} | = v^{-1} | \nabla u | \quad \text{ at } \ \Sigma . \end{aligned}$$

(7.5)

We want to seek implications of the inequalities (3.2), (3.16), (3.15) and (6.15), i.e.,

$$\begin{aligned}{} & {} 4 \pi + \int _{\Sigma } H | \nabla u | \ge 3 \int _{\Sigma } | \nabla u |^2, \end{aligned}$$

(7.6)

$$\begin{aligned}{} & {} 4 \pi - \int _{\Sigma } | \nabla u |^2 \le 4 \pi \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1}, \end{aligned}$$

(7.7)

$$\begin{aligned}{} & {} 8 \pi - \int _{\Sigma } H | \nabla u | \le 12 \pi \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1}, \end{aligned}$$

(7.8)

$$\begin{aligned}{} & {} 4 \pi - \int _{\Sigma } H | \nabla u | + \int _{\Sigma } | \nabla u |^2 \le 8 \pi \mathfrak {m}\, \mathfrak {c}_{_\Sigma }^{-1}, \end{aligned}$$

(7.9)

when they are applied to the conformally deformed triple $(M, {\bar{g}}, {\bar{u}} )$. As mentioned in Remark 3.6 and Remark 6.15, one knows

$$\begin{aligned} (7.6) \ + \ (7.7) \ \Longrightarrow \ (7.8) \ \ \text {and} \ \ (7.9). \end{aligned}$$

For this reason, we focus on the use of (7.6) and (7.7) below.

Theorem 7.1

Let (M, g) be a complete, orientable, asymptotically flat 3-manifold with boundary $\Sigma$. Suppose $\Sigma$ is connected and $H_2 (M, \Sigma ) = 0$. Let u be the harmonic function such that $u = 0$ at $\Sigma$ and $u \rightarrow 1$ at $\infty$. If g has nonnegative scalar curvature, then, for any constant $k > 0$,

$$\begin{aligned} 4 \pi + k \int _\Sigma H | \nabla u | \ge k ( 4 - k ) \int _\Sigma | \nabla u |^2. \end{aligned}$$

(7.10)

Moreover, equality in (7.10) holds for some k if and only if (M, g) is isometric to a spatial Schwarzschild manifold outside a rotationally symmetric sphere, that is, up to isometry,

$$\begin{aligned} (M, g) = \left( {\mathbb {R}}^3 \setminus \{ | x | < r \}, \left( 1 + \frac{\mathfrak {m}}{ 2 | x |} \right) ^4 g_{_E} \right) , \end{aligned}$$

where $r > 0$ is a constant, $g_{_E} = \delta _{ij} \textrm{d} x^i \textrm{d} x^j$ is the Euclidean metric, and $\mathfrak {m}$, k, r are related by $\mathfrak {m}= 2 r ( k - 1 )$.

Proof

Given any positive harmonic function v on (M, g), let ${\bar{g}} = v^4\,g$ and ${\bar{u}} = v^{-1} u$. Applying (3.2) in Theorem 3.1 to the triple $(M, {\bar{g}}, {\bar{u}} )$, we have

$$\begin{aligned} 4 \pi + \int _\Sigma {\bar{H}} | \bar{\nabla }{\bar{u}} |_{{\bar{g}} } \, \textrm{d} \bar{\sigma }\ge 3 \int _\Sigma | \bar{\nabla }{\bar{u}} |_{{\bar{g}}}^2 \, \textrm{d} \bar{\sigma }. \end{aligned}$$

(7.11)

By (7.1)–(7.5), (7.11) shows

$$\begin{aligned} 4 \pi + \int _\Sigma \bigl ( 4 v^{-1} \partial _\nu v + H \bigr ) v^{-1} | \nabla u | \, \text {d} \sigma \ge 3 \int _\Sigma v^{-2} | \nabla u |^2 \, \text {d} \sigma . \end{aligned}$$

(7.12)

Given any constant $k > 0$, choose

$$\begin{aligned} v = u + \frac{1}{k} ( 1 - u ). \end{aligned}$$

(7.13)

It follows from (7.12) and the fact $\partial _\nu u = | \nabla u |$ at $\Sigma$ that

$$\begin{aligned} 4 \pi + k \int _\Sigma H | \nabla u | \, \textrm{d} \sigma \ge k ( 4 - k ) \int _\Sigma | \nabla u |^2 \, \textrm{d} \sigma , \end{aligned}$$

which proves (7.10).

The above also shows equality in (7.10) holds for some k if and only if equality in (7.11) holds for the corresponding $(M, {\bar{g}}, {\bar{u}} )$. By Theorem 3.1, this occurs if and only if $(M, {\bar{g}})$ is isometric to $\left( {\mathbb {R}}^3 {\setminus } B_r, g_{_E} \right)$, where $B_r = \{ x \in {\mathbb {R}}^3 \ | \ |x| < r \}$ for some constant $r > 0$. In this case,

$$\begin{aligned} {\bar{u}} = 1 - \frac{r}{ | x| }. \end{aligned}$$

(7.14)

This combined with (7.13) and the fact ${\bar{u}} = v^{-1} u$ shows

$$\begin{aligned} v^{-1} = 1 + \frac{ r (k-1)}{ |x| }. \end{aligned}$$

(7.15)

As a result,

$$\begin{aligned} g = v^{-4} g_{_E} = \left( 1 + \frac{ r (k-1)}{ |x| } \right) ^4 \delta _{ij} \, \textrm{d} x^i \textrm{d} x^j, \end{aligned}$$

(7.16)

which is a spatial Schwarzschild metric with mass $\mathfrak {m}= 2 r ( k -1 )$. $\square$

Theorem 7.1 implies a sharp bound of $\int _\Sigma | \nabla u |^2$ by the Willmore functional of $\Sigma$, with the bound achieved by Schwarzschild spaces outside mean-convex round spheres.

Corollary 7.1

Let (M, g) be a complete, orientable, asymptotically flat 3-manifold with boundary $\Sigma$. Suppose $\Sigma$ is connected and $H_2 (M, \Sigma ) = 0$. Let u be the harmonic function such that $u = 0$ at $\Sigma$ and $u \rightarrow 1$ at $\infty$. If g has nonnegative scalar curvature, then

$$\begin{aligned} \left( \frac{1}{\pi } \int _\Sigma | \nabla u |^2 \right) ^\frac{1}{2} \le \left( \frac{1}{16\pi } \int _\Sigma H^2 \right) ^\frac{1}{2} + 1. \end{aligned}$$

(7.17)

Moreover, equality holds if and only if (M, g) is isometric to a spatial Schwarzschild manifold outside a rotationally symmetric sphere with nonnegative constant mean curvature.

Proof

Consider the following quadratic form of k,

$$\begin{aligned} Q (k): = \alpha (u) \, k^2 + \beta (u) \, k + 4 \pi , \end{aligned}$$

(7.18)

where

$$\begin{aligned} \alpha (u) = \int _\Sigma | \nabla u |^2, \quad \beta (u) = \int _\Sigma H | \nabla u | - 4 \int _\Sigma | \nabla u |^2. \end{aligned}$$

We have $Q(0) = 4 \pi$, and Theorem 7.1 shows

$$\begin{aligned} Q( k ) \ge 0, \quad \forall k > 0. \end{aligned}$$

Thus, by elementary reasons, either

$$\begin{aligned} \beta (u)^2 - 16 \pi \alpha (u) \le 0 \end{aligned}$$

(7.19)

or

$$\begin{aligned} \beta (u)^2 - 16 \pi \alpha (u) > 0 \quad \text{ and }\quad -\beta (u) + \sqrt{ \beta (u)^2 - 16 \pi \alpha (u) } < 0. \end{aligned}$$

(7.20)

The latter case is equivalent to

$$\begin{aligned} \beta (u) > \sqrt{ 16 \pi \alpha (u) }, \end{aligned}$$

that is

$$\begin{aligned} \frac{ \int _\Sigma H | \nabla u | }{ \left( \int _\Sigma | \nabla u |^2 \right) ^\frac{1}{2} } - 4 \left( \int _\Sigma | \nabla u |^2 \right) ^\frac{1}{2} > \sqrt{ 1 6 \pi }. \end{aligned}$$

(7.21)

If (7.21) holds, then, by Hölder’s inequality,

$$\begin{aligned} \left( \frac{1}{16\pi } \int _\Sigma H^2 \right) ^\frac{1}{2} > 1 + \left( \frac{1}{\pi } \int _\Sigma | \nabla u |^2 \right) ^\frac{1}{2}. \end{aligned}$$

(7.22)

If (7.19) holds, then

$$\begin{aligned} \left| \int _\Sigma H | \nabla u | - 4 \int _\Sigma | \nabla u |^2 \right| \le 4 \left( \pi \int _\Sigma | \nabla u |^2 \right) ^\frac{1}{2}, \end{aligned}$$

which in particular implies

$$\begin{aligned} 4 \int _\Sigma | \nabla u |^2 \le 4 \left( \pi \int _\Sigma | \nabla u |^2 \right) ^\frac{1}{2} + \int _\Sigma H | \nabla u |. \end{aligned}$$

(7.23)

Combined with Hölder’s inequality, this shows

$$\begin{aligned} \left( \frac{1}{\pi } \int _\Sigma | \nabla u |^2 \right) ^\frac{1}{2} \le 1 + \left( \frac{1}{16\pi } \int _\Sigma H^2 \right) ^\frac{1}{2}. \end{aligned}$$

(7.24)

Therefore, in either case, we conclude (7.17) holds.

If equality in (7.17) holds, then (7.20) does not hold; (7.23) holds with equality; and $H = c | \nabla u |$ for some constant $c \ge 0$. In particular, this gives

$$\begin{aligned} - \beta (u) = 4 \int _\Sigma | \nabla u |^2 - \int _\Sigma H | \nabla u | = \sqrt{ 16 \pi \alpha (u) } > 0. \end{aligned}$$

As a result, $Q (k_0) = 0$ at

$$\begin{aligned} k_0 = - \frac{\beta (u)}{2 \alpha (u) } = 2 \left( \frac{1}{\pi } \int _\Sigma | \nabla u |^2 \right) ^{-\frac{1}{2}} = \frac{2}{ 1 + \left( \frac{1}{16 \pi } \int _{\Sigma } H^2 \right) ^\frac{1}{2} } > 0. \end{aligned}$$

By Theorem 7.1, (M, g) is isometric to a spatial Schwarzschild manifold

$$\begin{aligned} (M, g) = \left( {\mathbb {R}}^3 \setminus \{ | x | < r \}, \left( 1 + \frac{\mathfrak {m}}{ 2 | x |} \right) ^4 g_{_E} \right) , \end{aligned}$$

where $r > 0$ and $\mathfrak {m}= 2 r ( k_0 - 1 )$. As $k_0 \le 2$, the boundary $\{ |x| = r \}$ has nonnegative mean curvature in (M, g).

On such an (M, g), direct calculation shows

$$\begin{aligned} \left( \frac{1}{16 \pi } \int _\Sigma H^2 \right) ^\frac{1}{2} = \left| \frac{2}{k} - 1 \right| \quad \text {and} \quad \left( \frac{1}{ \pi } \int _{\Sigma } | \nabla u |^2 \right) ^\frac{1}{2} = \frac{2}{k}. \end{aligned}$$

As $k \le 2$, equality holds in (7.17). This completes the proof. $\square$

An immediate application of Corollary 7.1 yields a result of Bray and the author [8] on the estimate of the capacity-to-area–radius ratio.

Theorem 7.2

([8]) Let (M, g) be a complete, orientable, asymptotically flat 3-manifold with boundary $\Sigma$. Suppose $\Sigma$ is connected and $H_2 (M, \Sigma ) = 0$. If g has nonnegative scalar curvature, then

$$\begin{aligned} \frac{ 2 \mathfrak {c}_{_\Sigma } }{ r_{_\Sigma } } \le \left( \frac{1}{16\pi } \int _\Sigma H^2 \right) ^\frac{1}{2} + 1. \end{aligned}$$

(7.25)

Here $\mathfrak {c}_{_\Sigma }$ is the capacity of $\Sigma$ in (M, g) and $r_{_\Sigma } = \bigl (\frac{ | \Sigma | }{4 \pi }\bigr ) ^\frac{1}{2}$ is the area–radius of $\Sigma$. Moreover, equality holds if and only if (M, g) is isometric to a spatial Schwarzschild manifold outside a rotationally symmetric sphere with nonnegative constant mean curvature.

Proof

This follows directly from

$$\begin{aligned} \left( \int _\Sigma | \nabla u |^2 \right) ^\frac{1}{2} \ge \frac{ \int _{\Sigma } | \nabla u | }{ | \Sigma |^\frac{1}{2} } = \sqrt{\pi } \frac{ 2 \mathfrak {c}_{_\Sigma } }{ r_{_\Sigma } } \end{aligned}$$

and Corollary 7.1. $\square$

Next, we proceed to find implications of (3.16) in Theorem 3.2.

Theorem 7.3

Let (M, g) be a complete, orientable, asymptotically flat 3-manifold with boundary $\Sigma$. Suppose $\Sigma$ is connected and $H_2 (M, \Sigma ) = 0$. Let u be the harmonic function such that $u = 0$ at $\Sigma$ and $u \rightarrow 1$ at $\infty$. If g has nonnegative scalar curvature, then

$$\begin{aligned} \begin{aligned} \frac{ \mathfrak {m}}{ 2 \mathfrak {c}_{_\Sigma } } \ge 1 - \left( \frac{1}{4 \pi } \int _{\Sigma } | \nabla u |^2 \, \textrm{d} \sigma \right) ^\frac{1}{2}. \end{aligned} \end{aligned}$$

(7.26)

Moreover, equality holds if and only if (M, g) is isometric to a spatial Schwarzschild manifold outside a rotationally symmetric sphere.

Proof

Given any positive harmonic function v on (M, g), let ${\bar{g}} = v^4\,g$ and ${\bar{u}} = v^{-1} u$. Applying (3.16) in Theorem 3.2 to $(M, {\bar{g}}, {\bar{u}} )$, we have

$$\begin{aligned} 4 \pi - \int _{\Sigma } | \bar{\nabla }{\bar{u}} |^2_{{\bar{g}}} \, \textrm{d} \bar{\sigma }\le 4 \pi {\bar{\mathfrak {m}}} {{\bar{\mathfrak {c}}}_{_\Sigma } }^{-1}. \end{aligned}$$

(7.27)

By (7.1)–(7.5), (7.27) becomes

$$\begin{aligned} 4 \pi - \int _{\Sigma } v^{-2} | \nabla u |^2 \, \textrm{d} \sigma \le 4 \pi \frac{ \mathfrak {m}- 2 \mathfrak {c}_v}{ \mathfrak {c}_{_\Sigma } - \mathfrak {c}_v }. \end{aligned}$$

(7.28)

Given any constant $k > 0$, choose

$$\begin{aligned} v = u + \frac{1}{k} ( 1 - u ). \end{aligned}$$

(7.29)

Then $v = k^{-1}$ at $\Sigma$, $\mathfrak {c}_{v} = (1 - k^{-1} ) \mathfrak {c}_{_\Sigma }$, and (7.28) shows

$$\begin{aligned} \begin{aligned} \frac{ \mathfrak {m}}{ \mathfrak {c}_{_\Sigma } } \ge 2 - \frac{1}{k} - \frac{k}{4 \pi } \int _{\Sigma } | \nabla u |^2 \, \textrm{d} \sigma . \end{aligned} \end{aligned}$$

(7.30)

Maximizing the right side of (7.30) over all $k > 0$, we have

$$\begin{aligned} \begin{aligned} \frac{ \mathfrak {m}}{ 2 \mathfrak {c}_{_\Sigma } } \ge 1 - \left( \frac{1}{4 \pi } \int _{\Sigma } | \nabla u |^2 \, \textrm{d} \sigma \right) ^\frac{1}{2}, \end{aligned} \end{aligned}$$

(7.31)

which proves (7.26).

If equality in (7.26) holds, then equality in (7.27) holds for $v = u + k^{-1} ( 1 - u )$ with the constant k given by

$$\begin{aligned} k = \left( \frac{1}{4 \pi } \int _{\Sigma } | \nabla u |^2 \, \textrm{d} \sigma \right) ^{ - \frac{1}{2} }. \end{aligned}$$

(7.32)

By Theorem 3.2, $(M, {\bar{g}})$ is isometric to $\left( {\mathbb {R}}^3 \setminus B_r, g_{_E} \right)$, where $B_r = \{ x \in {\mathbb {R}}^3 \ | \ |x| < r \}$ for some $r > 0$, and

$$\begin{aligned} {\bar{u}} = 1 - \frac{r}{ | x| }. \end{aligned}$$

(7.33)

This combined with ${\bar{u}} = v^{-1} u$ and (7.29) shows

$$\begin{aligned} v^{-1} = 1 + \frac{ r (k-1)}{ |x| }. \end{aligned}$$

(7.34)

As a result,

$$\begin{aligned} g = v^{-4} g_{_E} = \left( 1 + \frac{ r (k-1)}{ |x| } \right) ^4 \delta _{ij} \, \textrm{d} x^i \textrm{d} x^j, \end{aligned}$$

(7.35)

which is a spatial Schwarzschild metric with the mass $\mathfrak {m}= 2 r ( k -1 )$.

On any such an (M, g), direct calculation shows

$$\begin{aligned} \frac{\mathfrak {m}}{2 \mathfrak {c}_{_\Sigma } } = 1 - \frac{1}{k} \quad \text {and} \quad \left( \frac{1}{ 4 \pi } \int _{\Sigma } | \nabla u |^2 \right) ^\frac{1}{2} = \frac{1}{k}, \end{aligned}$$

which verifies equality in (7.26). This completes the proof. $\square$

We now have a succinct lower bound of the mass-to-capacity ratio by the Willmore functional.

Theorem 7.4

Let (M, g) be a complete, orientable, asymptotically flat 3-manifold with boundary $\Sigma$. Suppose $\Sigma$ is connected and $H_2 (M, \Sigma ) = 0$. If g has nonnegative scalar curvature, then

$$\begin{aligned} \begin{aligned} \frac{ \mathfrak {m}}{ \mathfrak {c}_{_\Sigma } } \ge 1 - \left( \frac{1}{16\pi } \int _\Sigma H^2 \right) ^\frac{1}{2}. \end{aligned} \end{aligned}$$

(7.36)

Moreover, equality holds if and only if (M, g) is isometric to a spatial Schwarzschild manifold outside a rotationally symmetric sphere with nonnegative constant mean curvature.

Proof

This is a direct consequence of Corollary 7.1 and Theorem 7.3. $\square$

We give a few remarks.

Remark 7.1

Theorem 7.4 gives another way to see the 3-dimensional positive mass theorem. In the context of Proof III in Sect. 4, Theorem 7.4 gives

$$\begin{aligned} \frac{ \mathfrak {m}}{ \mathfrak {c}_{r} } \ge 1 - \left( \frac{1}{16\pi } \int _{\Sigma _r} H^2 \right) ^\frac{1}{2} = o (1), \quad \text{ as } \,\ r \rightarrow 0, \end{aligned}$$

where $c_{r}$ is the capacity of a small geodesic ball of radius r. Hence, $\mathfrak {m}\ge 0$.

Remark 7.2

Theorem 7.4 improves the result of Bray and the author in [8]. Under an additional assumption of $\int _{\Sigma } H^2 \le 16 \pi$, in [8] the capacity estimate (7.25) was converted into a Hawking mass estimate

$$\begin{aligned} \mathfrak {m}_{_H} (\Sigma ) \ge \left[ 1 - \left( \frac{1}{16\pi } \int _\Sigma H^2 \right) ^\frac{1}{2} \right] \mathfrak {c}_{_\Sigma } \end{aligned}$$

and the relation $\mathfrak {m}\ge \mathfrak {m}_{_H} (\Sigma )$ was applied (if $\Sigma$ is outer-minimizing) to obtain (7.36).

In the current derivation of (7.36), we bound the ratio $\mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1}$ via $\int _{\Sigma } | \nabla u |^2$ and bound $\int _{\Sigma } | \nabla u |^2$ via $\int _\Sigma H^2$, hence bypassing the use of $\mathfrak {m}_{_H} (\Sigma )$ in relating $\mathfrak {m}$ and $\mathfrak {c}_{_\Sigma }$.

Remark 7.3

One may re-write (7.36) as

$$\begin{aligned} \begin{aligned} \mathfrak {M} (g): = \frac{ \mathfrak {m}}{ \mathfrak {c}_{_\Sigma } } + \left( \frac{1}{16\pi } \int _\Sigma H^2 \right) ^\frac{1}{2} - 1 \ge 0. \end{aligned} \end{aligned}$$

This gives a nonnegative quantity $\mathfrak {M} (g)$ on asymptotically flat 3-manifolds (M, g) with boundary (under the curvature and topological assumptions). $\mathcal {M} (g)$ vanishes precisely if (M, g) is rotationally symmetric with mean-convex boundary.

8 Manifolds with the Mass-to-Capacity Ratio $\le 1$

In this section, prompted by Theorem 7.4, we consider a class of manifolds satisfying a mass-capacity relation

$$\begin{aligned} \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \le 1. \end{aligned}$$

(8.1)

As we will see later in Proposition 8.1, such a class of manifolds includes static metric extensions in the context of the Bartnik quasi-local mass [5].

Theorem 8.1

Let (M, g) be a complete, orientable, asymptotically flat 3-manifold with boundary $\Sigma$, satisfying a mass-capacity relation

$$\begin{aligned} \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \le 1. \end{aligned}$$

Let u be the harmonic function with $u = 0$ at $\Sigma$ and $u \rightarrow 1$ near $\infty$. If $\Sigma$ is connected, $H_2 (M, \Sigma ) = 0$, and g has nonnegative scalar curvature, then

$$\begin{aligned} \frac{1}{ 4 \pi } \int _{\Sigma } H | \nabla u | \ge \bigl ( 2 - \mathfrak {m}\mathfrak {c}_{_{\Sigma }}^{-1} \bigr ) \bigl ( 1 - \mathfrak {m}\mathfrak {c}_{_{\Sigma }}^{-1} \bigr ) . \end{aligned}$$

(8.2)

Moreover, equality holds if and only if (M, g) is isometric to a spatial Schwarzschild manifold outside a rotationally symmetric sphere with nonnegative constant mean curvature.

Proof

For a regular value $t \in [0,1)$, if $u^{(t)}$ denotes the harmonic function outside $\Sigma _t$ with $u^{(t)} = 0$ at $\Sigma _t$ and $u^{(t)} \rightarrow 1$ at $\infty$, then

$$\begin{aligned} u^{(t)} = \frac{ u - t}{1 - t}. \end{aligned}$$

(8.3)

As a result, the capacity $\mathfrak {c}_{_{\Sigma _t} }$ of $\Sigma _t$ is related to that of $\Sigma$ by

$$\begin{aligned} \mathfrak {c}_{_{\Sigma _t}} = \frac{ \mathfrak {c}_{_\Sigma } }{ 1 - t }. \end{aligned}$$

(8.4)

Therefore, by Theorem 7.4,

$$\begin{aligned} \begin{aligned} \left( \frac{1}{16\pi } \int _{\Sigma _t} H^2 \right) ^\frac{1}{2}&\ge 1 - \mathfrak {m}\mathfrak {c}_{_{\Sigma _t}}^{-1} \\&= \ 1 - \mathfrak {m}\mathfrak {c}_{_{\Sigma }}^{-1} (1 - t ). \end{aligned} \end{aligned}$$

(8.5)

Under the assumption $\mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \le 1$, we have

$$\begin{aligned} 1 - \mathfrak {m}\mathfrak {c}_{_{\Sigma }}^{-1} (1 - t ) \ge 0. \end{aligned}$$

Hence, (8.5) is equivalent to

$$\begin{aligned} \begin{aligned} \frac{1}{16\pi } \int _{\Sigma _t} H^2 \ge \bigl [ 1 - \mathfrak {m}\mathfrak {c}_{_{\Sigma }}^{-1} (1 - t ) \bigr ] ^2. \end{aligned} \end{aligned}$$

(8.6)

To proceed, we return to the basic identity (3.7) in Sect. 3. Given any regular values $t_1< t_2 < 1$, by (3.7),

$$\begin{aligned}&4 \pi ( t_2 - t_1) + \int _{\Sigma _{t_1} } H | \nabla u | - \int _{\Sigma _{t_2} } H | \nabla u | \quad \qquad \nonumber \\ {}&\quad \ge \ \int _{t_1}^{t_2} \int _{\Sigma _t} \frac{1}{2} | \mathring{\displaystyle {\mathbb {I}\mathbb {I}}} |^2 + | \nabla u |^{-2} \big | \nabla _{_{\Sigma _t} } | \nabla u | \big |^2 + \frac{3}{4} H^2 + \frac{1}{2} R \quad \qquad \nonumber \\ {}&\quad \ge \ \int _{t_1}^{t_2} \frac{3}{4} \int _{\Sigma _t} H^2. \quad \qquad \end{aligned}$$

(8.7)

Thus, it follows from (8.6) and (8.7) that

$$\begin{aligned} \begin{aligned}&\ 4 \pi ( t_2 - t_1) + \int _{\Sigma _{t_1} } H | \nabla u | - \int _{\Sigma _{t_2} } H | \nabla u | \\ {}&\quad \ge \ 12 \pi \int _{t_1}^{t_2} \bigl [ 1 - \mathfrak {m}\mathfrak {c}_{_{\Sigma }}^{-1} (1 - t ) \bigr ] ^2. \end{aligned} \end{aligned}$$

Letting $t_2 \rightarrow 1$, by Lemma 2.2, we obtain

$$\begin{aligned} \begin{aligned}&\ 4 \pi ( 1 - t_1) + \int _{\Sigma _{t_1} } H | \nabla u | \\ {}&\quad \ge \ 12 \pi \int _{t_1}^{1} \bigl [ 1 - \mathfrak {m}\mathfrak {c}_{_{\Sigma }}^{-1} (1 - t ) \bigr ] ^2 \\ {}&\quad = \ 12 \pi (1 - t_1) - 12 \pi \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} ( 1 - t_1)^2 + 4 \pi ( \mathfrak {m}\mathfrak {c}_{_{\Sigma }}^{-1} ) ^2 ( 1 - t_1)^3, \end{aligned} \end{aligned}$$

or, equivalently

$$\begin{aligned} \begin{aligned}&\ 12 \pi \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} ( 1 - t_1) - 4 \pi ( \mathfrak {m}\mathfrak {c}_{_{\Sigma }}^{-1}) ^2 ( 1 - t_1)^2 \\ {}&\quad \ge \ 8 \pi - \frac{1}{1 - t_1} \int _{\Sigma _{t_1} } H | \nabla u |. \end{aligned} \end{aligned}$$

(8.8)

In particular, at $t_1 = 0$, we have

$$\begin{aligned} \begin{aligned} 12 \pi \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} - 4 \pi ( \mathfrak {m}\mathfrak {c}_{_{\Sigma }}^{-1} ) ^2 \ge 8 \pi - \int _{\Sigma } H | \nabla u |, \end{aligned} \end{aligned}$$

which proves (8.2).

If equality in (8.2) holds, then equality in (8.8) holds with $t_1 = 0$. This necessarily implies equality in (8.6) holds for a.e. $t \in [0,1]$. As a result, at $t = 0$,

$$\begin{aligned} \frac{1}{16\pi } \int _{\Sigma } H^2 = \bigl [ 1 - \mathfrak {m}\mathfrak {c}_{_{\Sigma }}^{-1} \bigr ] ^2. \end{aligned}$$

Since $1 - \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \ge 0$, we conclude by Theorem 7.4 that (M, g) is isometric to a spatial Schwarzschild manifold outside a rotationally symmetric sphere with nonnegative mean curvature.

Suppose $(M, g) = \big ( {\mathbb {R}}^3 {\setminus } \{ | x | < r \}, ( 1 + \frac{\mathfrak {m}}{ 2 | x |} ) ^4 g_{_E} \big )$ with mean-convex boundary $\{ |x| = r \}$, then (8.5)–(8.8) all become equality. Hence, equality in (8.2) holds. This completes the proof. $\square$

Remark 8.1

We compare (8.2) and (3.15). If (M, g) has $\mathfrak {m}= 0$, (8.2) is as the same as (3.15), both of which reduce to $\int _{\Sigma } H | \nabla u | \ge 8 \pi .$ For (M, g) with $\mathfrak {m}\ne 0$, (8.2) improves (3.15) by unveiling the quadratic term $4 \pi ( \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} )^2.$

Remark 8.2

Condition (8.1) is a global condition on the triple $(M, g, \Sigma )$. It has a feature of being inheritable to other surfaces enclosing $\Sigma$. More precisely,

$$\begin{aligned} \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \le 1 \ \Longrightarrow \ \mathfrak {m}\mathfrak {c}_{_S}^{-1} \le 1 \end{aligned}$$

for any other surfaces S in M enclosing $\Sigma$. This follows from the fact $\mathfrak {c}_{_\Sigma } \le \mathfrak {c}_{_S}$, which is a consequence of the variational characterization of the surface capacity.

Remark 8.3

Theorem 8.1 shows a necessary condition of $\mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \le 1$ is $\int _{\Sigma } H | \nabla u | \ge 0$. Therefore, by Remark 8.2,

$$\begin{aligned} \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \le 1 \ \Longrightarrow \ \int _{S} H | \nabla u_{_S} | \ge 0, \end{aligned}$$

for any surfaces S enclosing $\Sigma$. Here $u_{_S}$ denotes the harmonic function in the exterior of S, with $u_{_S} = 0$ at S and $u_{_S} \rightarrow 1$ at $\infty$.

Manifolds (M, g) satisfying $\mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \le 1$ include regions, in a spatial Schwarzschild manifold with positive mass, which are the exterior to a surface enclosing the horizon. That is, if

$$\begin{aligned} (M_\mathfrak {m}, g_\mathfrak {m}) = \left( {\mathbb {R}}^3 \setminus \left\{ | x | < \frac{1}{2} \mathfrak {m}\right\} , \left( 1 + \frac{\mathfrak {m}}{ 2 | x |} \right) ^4 g_{_E} \right) \end{aligned}$$

with $\mathfrak {m}> 0$ and if $\Sigma \subset M_\mathfrak {m}$ is a closed surface bounding some region D with the horizon $\Sigma _{h} = \{ | x | = \frac{1}{2} \mathfrak {m}\},$ then $(M_\mathfrak {m}{\setminus } D, g_\mathfrak {m})$ satisfies

$$\begin{aligned} \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \le 1. \end{aligned}$$

This is because of $\ c_{_{\Sigma _h}} = \mathfrak {m}$ on $(M_\mathfrak {m}, g_\mathfrak {m})$ and $\mathfrak {c}_{_{\Sigma _h} } \le \mathfrak {c}_{_\Sigma }$.

To put the next corollary of Theorem 8.1 in context, we mention a few additional facts on $(M_\mathfrak {m}, g_\mathfrak {m})$. Let $\Sigma _r = \{ | x | = r \} \subset M_\mathfrak {m}$. The mean curvature $H_r$ of $\Sigma _r$ equals

$$\begin{aligned} H_r = k^{-2} ( 2 k^{-1} - 1 ) 2 r^{-1}, \end{aligned}$$

where $k \in (1, 2]$ is the constant determined by $\displaystyle \mathfrak {m}= 2 r ( k - 1 )$. The product $\mathfrak {m}H_r$ satisfies

$$\begin{aligned} \mathfrak {m}H_r = 2 k^{-1} ( 2 k^{-1} - 1 ) 2 (1-k^{-1}). \end{aligned}$$

The capacity $\mathfrak {c}_r$ of $\Sigma _r$ is given by

$$\begin{aligned} \mathfrak {c}_r = r + \frac{\mathfrak {m}}{2}, \quad \text{ and } \text{ hence }\ \, \mathfrak {m}\mathfrak {c}_r^{-1} = 2 ( 1 - k^{-1} ). \end{aligned}$$

As a result, $\mathfrak {m}H_r$ and $\mathfrak {m}\mathfrak {c}_{r}^{-1}$ are related by

$$\begin{aligned} \mathfrak {m}H_r = ( 2 - \mathfrak {m}\mathfrak {c}_r^{-1} )(1 - \mathfrak {m}\mathfrak {c}_r^{-1}) \mathfrak {m}\mathfrak {c}_r^{-1}. \end{aligned}$$

(8.9)

As a function of r, calculation shows

$$\begin{aligned} \max _{ \frac{1}{2} \mathfrak {m}\le r < \infty } \mathfrak {m}H_r = \frac{2}{ 3 \sqrt{ 3} }, \end{aligned}$$

(8.10)

where this maximum is achieved uniquely at

$$\begin{aligned} r_p = \bigg ( 1 + \frac{ \sqrt{3} }{2} \bigg ) \mathfrak {m}, \quad \text{ satisfying } \,\ \left( 1 + \frac{ \mathfrak {m}}{2 r_p} \right) ^2 r_p = 3 \mathfrak {m}. \end{aligned}$$

(8.11)

The sphere $\{ |x| = r_p \}$ is often known as the photon sphere in $(M_\mathfrak {m}, g_\mathfrak {m})$. The mass-to-capacity ratio at $\Sigma _{r_p}$ is given by

$$\begin{aligned} \mathfrak {m}\mathfrak {c}_{r_p}^{-1} = 1 - \frac{1}{ \sqrt{3} }. \end{aligned}$$

(8.12)

The following corollary gives a partial classification or comparison result for manifolds with $\mathfrak {m}\mathfrak {c}_{_\Sigma } \le 1$, depending on the maximum of $\mathfrak {m}H$ at the boundary.

Corollary 8.1

Let (M, g) be a complete, orientable, asymptotically flat 3-manifold with boundary $\Sigma$, with the mass-to-capacity ratio satisfying

$$\begin{aligned} 0 < \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \le 1. \end{aligned}$$

Suppose $\Sigma$ is connected, $H_2 (M, \Sigma ) = 0$, and g has nonnegative scalar curvature. Then

• (i) either $\Sigma$ has vanishing mean curvature, in which case (M, g) must be isometric to a spatial Schwarzschild manifold outside the horizon;

• (ii) or $H_{\textrm{max}} = \max _{\Sigma } H > 0$ and one of the following holds:

  • (a) $\mathfrak {m}H_{\textrm{max}} < \frac{2}{3 \sqrt{3} }$ and

    $$\begin{aligned} \displaystyle \mathfrak {c}_{_\Sigma } \le \mathfrak {c}_{r_1} \quad \text{ or } \quad \mathfrak {c}_{_\Sigma } \ge \mathfrak {c}_{r_2}. \end{aligned}$$

    Here $\mathfrak {c}_{r_i}$ is the capacity of the sphere $\Sigma _{r_i} = \{ |x| = r_i \}$, $i = 1, 2$, in the spatial Schwarzschild manifold

    $$\begin{aligned} (M_\mathfrak {m}, g_\mathfrak {m}) = \left( {\mathbb {R}}^3 \setminus \left\{ | x | < \frac{1}{2} \mathfrak {m}\right\} , \left( 1 + \frac{\mathfrak {m}}{ 2 | x |} \right) ^4 g_{_E} \right) \end{aligned}$$

    which has the same mass as (M, g), and the constants $r_1, r_2$ are chosen so that

    $$\begin{aligned} H_{r_1} = H_{r_2} = H_{\textrm{max}} \quad \text{ and } \quad \frac{1}{2} \mathfrak {m}< r_1< \bigg ( 1 + \frac{1 }{2} \sqrt{3} \bigg )\mathfrak {m} < r_2, \end{aligned}$$

    (8.13)

    where $H_{r_i}$ is the mean curvature of $\Sigma _{r_i}$ in $(M_\mathfrak {m}, g_\mathfrak {m})$. Moreover, $\mathfrak {c}_{_\Sigma } = \mathfrak {c}_{r_i}$ for an $r_i$ if and only if (M, g) is isometric to $(M_\mathfrak {m}, g_\mathfrak {m})$ outside $\Sigma _{r_i}$;

  • (b) $\textstyle \mathfrak {m}H_{\textrm{max}} \ge \frac{2}{3 \sqrt{3} }$ and equality holds if and only if (M, g) is isometric to the spatial Schwarzschild manifold $(M_\mathfrak {m}, g_\mathfrak {m})$ outside the photon sphere $\big \{ |x| = ( 1 + \frac{1 }{2} \sqrt{3} ) \mathfrak {m}\big \}$.

Proof

Let $q = \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \in (0, 1]$. By Theorem 8.1,

$$\begin{aligned} \begin{aligned} \frac{1}{ 4 \pi } \int _{\Sigma } H | \nabla u | \ge&\ \left( 2 - q \right) \left( 1 - q \right) . \end{aligned} \end{aligned}$$

(8.14)

In particular, $\int _\Sigma H | \nabla u | \ge 0$. As $| \nabla u | > 0$ along $\Sigma$, we have $H_{\textrm{max}} \ge 0$, and $H_{\textrm{max}} = 0$ if and only if $H = 0$.

If $H=0$, Theorem 7.4 shows $q \ge 1$. Hence, $q = 1$, and therefore $q = 1 - \left( \frac{1}{ 16 \pi } \int _{\Sigma } H^2 \right) ^\frac{1}{2}$. By the equality case of Theorem 7.4, (M, g) is isometric to a spatial Schwarzschild manifold outside the horizon.

In what follows, we suppose $H_{\textrm{max}} > 0$. Since $\int _{\Sigma } | \nabla u | = 4 \pi \mathfrak {c}_{_\Sigma }$, (8.14) implies

$$\begin{aligned} H_{\textrm{max}} \, \mathfrak {c}_{_\Sigma } \ge \left( 2 - q \right) \left( 1 - q \right) . \end{aligned}$$

(8.15)

As $\mathfrak {m}> 0$, this gives

$$\begin{aligned} \mathfrak {m}\, H_{\textrm{max}} \ge \left( 2 - q \right) \left( 1 - q \right) q. \end{aligned}$$

(8.16)

As a result, either

$$\begin{aligned} \mathfrak {m}\, H_{\textrm{max}} \ge \frac{2}{3 \sqrt{3} } = \max _{ x \in [0, 1] } ( 2 - x ) ( 1 - x ) x, \end{aligned}$$

(8.17)

or

$$\begin{aligned} 0< \mathfrak {m}\, H_{\textrm{max}} < \frac{2}{3 \sqrt{3} }. \end{aligned}$$

(8.18)

If (8.17) holds with equality, then

$$\begin{aligned} H_{\textrm{max}} \mathfrak {c}_{\Sigma } = \frac{1}{ 4 \pi } \int _{\Sigma } H | \nabla u | = \left( 2 - q \right) \left( 1 - q \right) , \end{aligned}$$

with $q = 1 - \frac{1}{\sqrt{3} }$. By Theorem 8.1 and the fact (8.10)–(8.12), (M, g) is isometric to a spatial Schwarzschild manifold with the photon sphere boundary.

Next, we suppose (8.18) holds. Let $r_i$, $i=1,2$, be the constants given in (8.13). It follows from (8.9) and (8.16) that

$$\begin{aligned} \begin{aligned}&( 2 - \mathfrak {m}\mathfrak {c}_{r_i}^{-1} )(1 - \mathfrak {m}\mathfrak {c}_{r_i}^{-1}) \mathfrak {m}\mathfrak {c}_{r_i}^{-1} \\ {}&\quad = \ \mathfrak {m}H_{r_i} = \mathfrak {m}H_{\text {max}} \\ {}&\quad \ge \ ( 2 - q) (1 - q) q. \end{aligned} \end{aligned}$$

Analyzing the function $f(x) = (2 -x)(1-x)x$ and using the assumption $0 < q \le 1$, we conclude

$$\begin{aligned} q \le \mathfrak {m}\mathfrak {c}_{r_2}^{-1} \quad \text {or} \quad q \ge \mathfrak {m}{\mathfrak {c}}_{r_1}^{-1}, \end{aligned}$$

or equivalently

$$\begin{aligned} \mathfrak {c}_{r_2} \le \mathfrak {c}_{_\Sigma } \quad \text {or} \quad \mathfrak {c}_{r_1} \ge \mathfrak {c}_{_\Sigma }. \end{aligned}$$

(8.19)

If $\mathfrak {c}_{_\Sigma } = \mathfrak {c}_{r_i}$ for an $r_i$, then

$$\begin{aligned} H_{\textrm{max}} \mathfrak {c}_{_\Sigma } = \frac{1}{ 4 \pi } \int _{\Sigma } H | \nabla u | = \left( 2 - q \right) \left( 1 - q \right) , \end{aligned}$$

with $q = \mathfrak {m}\mathfrak {c}_{r_i}^{-1}$. By Theorem 8.1, (M, g) is isometric to a spatial Schwarzschild manifold with boundary $\{ |x| = r_i \}$. This completes the proof. $\square$

Remark 8.4

Corollary 8.1 can be applied to manifolds (M, g) with CMC boundary, i.e., $\Sigma$ has constant mean curvature. In this case, it might be interesting to understand the supremum of $\mathfrak {m}H$ over such manifolds.

Remark 8.5

Corollary 8.1 (i) shows $(M_{\mathfrak {m}}, g_{\mathfrak {m}})$, Schwarzschild manifolds outside the horizon, are the unique manifolds with the given topological and curvature assumptions, satisfying $\mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \le 1$ and the boundary being minimal.

On the other hand, we note $(M_{\mathfrak {m}}, g_{\mathfrak {m}})$ is not characterized by the condition $\mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} = 1$. To get other examples with $\mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} = 1$, we can first return to the proof of Theorem 8.1. Suppose we start with an arbitrary (M, g) with boundary $\Sigma$ satisfying $\mathfrak {c}_{_\Sigma } < \mathfrak {m}$. Then, by (8.4), we may consider the value $t_0$ determined by

$$\begin{aligned} \mathfrak {m}= \frac{ \mathfrak {c}_{_\Sigma } }{ 1 - t_0 }. \end{aligned}$$

(8.20)

If $t_0$ is a regular value of u, then the exterior region of $\Sigma _{t_0}$ in (M, g) satisfies $\mathfrak {m}\mathfrak {c}_{_{\Sigma _{t_0} } }^{-1} = 1$. (Otherwise, one can still consider regular values t greater than and close to $t_0$. For these t, the exterior region of $\Sigma _{t}$ satisfies $\mathfrak {m}\mathfrak {c}_{_{\Sigma _{t} } }^{-1} < 1$ and $\mathfrak {m}\mathfrak {c}_{_{\Sigma _{t} } }^{-1}$ can be made arbitrarily close to 1.) To get a precise example, we may consider a rotationally symmetric, asymptotically flat metric g on ${\mathbb {R}}^3$ so that g has nonnegative scalar curvature, positive mass, and $({\mathbb {R}}^3, g)$ has no closed minimal surfaces; see [20, Section 2.4] for this class of metrics. By (4.7), there is a small, rotationally symmetric sphere $\Sigma _r$ in $({\mathbb {R}}^3, g)$ with $\mathfrak {c}_{_\Sigma } < \mathfrak {m}$. Let u be the capacitary function on the exterior of $\Sigma _r$, then u is rotationally symmetric, and the level set $\{ u = t \}$ is a rotationally symmetric sphere for each $t \in (0,1)$. The region exterior to $\{ u = t_0 \}$ then satisfies $\mathfrak {m}\mathfrak {c}_{_{\Sigma _{t_0}}}^{-1} = 1$; moreover, it is not $(M_{\mathfrak {m}}, g_{\mathfrak {m}})$ as $\{ u = t_0 \}$ is not minimal.

Next, we mention some other implications of (8.1) which are corollaries of Theorems 7.2 and  7.3.

Corollary 8.2

Let (M, g) be a complete, orientable, asymptotically flat 3-manifold with boundary $\Sigma$, satisfying

$$\begin{aligned} \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \le 1. \end{aligned}$$

Suppose $\Sigma$ is connected, $H_2 (M, \Sigma ) = 0$, and g has nonnegative scalar curvature. Then

• (i) $\textstyle \mathfrak {m}\le \frac{r_{_\Sigma }}{2} \big [ 1 + ( \frac{1}{16\pi } \int _\Sigma H^2) ^\frac{1}{2} \big ] ,$ where $r_{_\Sigma }$ is the area–radius of $\Sigma$; and

• (ii) $\textstyle \frac{1}{\pi } \int _{\Sigma } | \nabla u |^2 \ge 1$, where u is the harmonic function on (M, g) with $u = 0$ at $\Sigma$ and $u \rightarrow 1$ near $\infty$.

Moreover, equality holds in either inequality if and only if (M, g) is isometric to a spatial Schwarzschild manifold outside the horizon.

Proof

Inequalities in (i), (ii) follow from (7.25) in Theorem 7.2, (7.26) in Theorem 7.3, respectively. The rigidity part follows from the rigidity conclusion in Theorem 7.2 or Theorem 7.3, together with the extra information $\mathfrak {m}= \mathfrak {c}_{_\Sigma }$. $\square$

Remark 8.6

Heuristically, (ii) of Corollary 8.2 suggests the condition

$$\begin{aligned} \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \le 1 \end{aligned}$$

may rule out manifolds having long cylindrical neighborhoods shielding the boundary. The following is a simple example. Suppose $\Sigma$ is a sphere or a torus and $\gamma$ is a metric of nonnegative Gauss curvature on $\Sigma$. Given a constant $L > 0$, consider the product manifold

$$\begin{aligned} (P, g_{_P} ) = ( \Sigma \times [0, L], \gamma + \textrm{d} t^2). \end{aligned}$$

If (M, g) contains a neighborhood U of $\partial M$ so that (U, g) is isometric to $(P, g_{_P})$ with $\partial M = \Sigma \times \{ 0 \}$, then one can consider the harmonic function v on $(P, g_{_P})$ with $v = 0$ on $\Sigma \times \{ 0 \}$ and $v = 1$ on $\Sigma \times \{ L \}$. By the maximum principle, $| \nabla u | \le | \nabla v |$. Hence

$$\begin{aligned} 4 r_{_\Sigma }^2 L^{-2} = \frac{1}{\pi } \int _{\Sigma } | \nabla v |^2 \ge \frac{1}{\pi } \int _{\Sigma } | \nabla u |^2, \end{aligned}$$

(8.21)

where $4 \pi r_{_\Sigma }^2$ is the area of $(\Sigma , \gamma )$. Therefore, if (M, g) satisfies condition (8.1), then (ii) of Corollary 8.2 shows $L \le 2 r_{_\Sigma }$.

One can always construct manifolds satisfying (8.1) by cutting off a compact set in a given asymptotically flat (M, g). For instance, if a triple $(M, g, \Sigma )$ has $\mathfrak {m}> \mathfrak {c}_{_\Sigma }$, then, by (8.4), the exterior of $\Sigma _t$ in (M, g) satisfies $\mathfrak {m}\le \mathfrak {c}_{_{\Sigma _t}}$ for any regular $t \ge 1 - \mathfrak {c}_{_\Sigma } \mathfrak {m}^{-1}$.

The complement of a finite domain enclosing the Schwarzschild horizon in $(M_\mathfrak {m}, g_{\mathfrak {m}})$ is an example of a static extension in the context of Bartnik’s quasi-local mass [5]. Here an asymptotically flat (M, g) is called static (see [12] for instance) if there is a nontrivial function N on (M, g), referred as a static potential, such that

$$\begin{aligned} \left\{ \begin{array}{l} N \text { Ric }= \nabla ^2 N, \\ \Delta N = 0, \\ \end{array} \right. \end{aligned}$$

(8.22)

where $\text{ Ric }$ denotes the Ricci curvature of g. These spaces necessarily have zero scalar curvature.

The next proposition, among other things, shows an asymptotically flat manifold with boundary, admitting a positive static potential, satisfies (8.1).

Proposition 8.1

Let (M, g) be a complete, orientable, asymptotically flat 3-manifold with boundary $\Sigma$. Suppose there is a static potential N that is positive in the interior of (M, g). Then

$$\begin{aligned} \mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \le 1. \end{aligned}$$

If $\Sigma$ is connected and $H_2 (M, \Sigma ) = 0$, then

• (i) $\mathfrak {m}\le \frac{r_{_\Sigma }}{2} \big [ 1 + ( \frac{1}{16\pi } \int _\Sigma H^2 ) ^\frac{1}{2} \big ] ,$ where $r_{_\Sigma }$ is the area-radius of $\Sigma$; and

• (ii) any closed, regular level set $S_t = N^{-1} (t)$ is connected and enclosing $\Sigma$; if N is normalized so that $N \rightarrow 1$ at $\infty$, then, along $S_t$,

  $$\begin{aligned}{} & {} N^2 \le \frac{1}{16 \pi } \int _{S_t} H^2, \\{} & {} 1-N^2 \le \left( \frac{1}{\pi } \int _{S_t} | \nabla N |^2 \right) ^\frac{1}{2} \le ( 1 - N ) \left[ 1 + \left( \frac{1}{16 \pi } \int _{S_t} H^2 \right) ^\frac{1}{2} \right] , \end{aligned}$$

  and

  $$\begin{aligned} N ( 1 - N) ( 1 + N) \le \frac{1}{4\pi } \int _{S_t} H | \nabla N |. \end{aligned}$$

  Moreover, equality holds in any of these inequalities if and only if the exterior of $S_t$ in (M, g) is isometric to a spatial Schwarzschild manifold outside a rotationally symmetric sphere with nonnegative constant mean curvature.

Proof

The static system (8.22) and the assumption $N > 0$ near $\infty$ imply that, upon multiplying N by a constant,

$$\begin{aligned} N = 1 - \frac{\mathfrak {m}}{|x|} + o (|x|^{-1} ), \end{aligned}$$

where $\mathfrak {m}$ is the mass of (M, g) (see [9, 24] for instance). If in addition $N \ge 0$ at $\Sigma$, then $\mathfrak {m}\le \mathfrak {c}_{_\Sigma }$ by the maximum principle.

The mass estimate in (i) follows from (i) of Corollary 8.2.

Suppose $S_t$ is a closed, regular level set of N. By the topological assumption on M and the fact N is harmonic, $S_t$ only has one connected component and it encloses $\Sigma$. Let $E_t$ denote the exterior of $S_t$ in M. The inequalities in (ii), with the rigidity conclusions, follow from applying Theorem 7.4, Corollary 7.1, Theorem 7.3, and Theorem 8.1, respectively, to

$$\begin{aligned} u^{(t)} = \frac{ N - t}{1-t} \end{aligned}$$

on $(E_t, g)$ and using the fact $\displaystyle \mathfrak {c}_{_{S_t}} = \frac{ \mathfrak {m}}{1-t}$. $\square$

In the context of the Bartnik mass [5], asymptotically flat extensions are often assumed to have no closed minimal surface enclosing the boundary to prevent the infimum of the mass over all extensions from being trivially zero. We note here, if an asymptotically flat 3-manifold (M, g) with boundary $\Sigma$ satisfies the mass-to-capacity relation $\mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \le 1,$ then necessarily there are no closed minimal surfaces enclosing $\Sigma$. This is because, if such a minimal surface S exists, then $\mathfrak {m}\ge \mathfrak {c}_{_S}$ by the result of Bray [6]. On the other hand, $\mathfrak {c}_{_S} > \mathfrak {c}_{_\Sigma }$. Hence, $\mathfrak {m}> \mathfrak {c}_{_\Sigma }$, violating (8.1).

Considering this and Proposition 8.1, we think manifolds satisfying condition (8.1) are worthy of further study. The mass-to-capacity ratio condition $\mathfrak {m}\mathfrak {c}_{_\Sigma }^{-1} \le 1$ may serve as an alternative to the no-minimal-surface or outer-minimizing conditions in the formulation of the Bartnik mass.

Appendix A: Regularization and Integration

In this appendix, we give the regularization arguments that can be used to verify the monotonicity of $\Psi (t)$, $\mathcal {A}(t)$ and $\mathcal {B}(t)$ in Sect. 3.

Lemma A.1

Let u be a harmonic function on a compact Riemannian manifold $(\Omega , g)$ with boundary $\partial \Omega$. Suppose $\max _{\Omega } u < 1$. Then

$$\begin{aligned} \begin{aligned} \int _{\partial \Omega } \frac{ | \nabla u | }{ 1 - u } \frac{\partial u}{\partial \zeta } = \int _\Omega \frac{ | \nabla u |^3 }{ ( 1 - u )^2 } + \int _{ \{ \nabla u \ne 0 \} \subset \Omega } \frac{ \nabla ^2 u ( \nabla u, \nabla u ) }{ (1 - u) | \nabla u | } \end{aligned} \end{aligned}$$

(A.1)

and

$$\begin{aligned} \begin{aligned} \int _{\partial \Omega } \frac{ | \nabla u | }{ (1 - u)^3 } \frac{\partial u}{\partial \zeta } = \int _\Omega \frac{ 3 | \nabla u |^3 }{ ( 1 - u )^4 } + \int _{ \{ \nabla u \ne 0 \} \subset \Omega } \frac{ \nabla ^2 u ( \nabla u, \nabla u ) }{ (1 - u)^3 | \nabla u | }. \end{aligned} \end{aligned}$$

(A.2)

Here $\zeta$ denotes the unit normal to $\partial \Omega$ pointing out of $\Omega$.

Proof

Given any constant $\epsilon > 0$, one has

$$\begin{aligned} \text{ div }\left( \frac{ \sqrt{ | \nabla u |^2 + \epsilon } }{ 1 - u } \nabla u \right) = \frac{ \sqrt{ | \nabla u |^2 + \epsilon } }{ ( 1 - u )^2 } | \nabla u |^2 + \frac{1}{ 1 - u} \frac{ \nabla ^2 u ( \nabla u, \nabla u ) }{\sqrt{ | \nabla u |^2 + \epsilon }}. \end{aligned}$$

Therefore,

$$\begin{aligned} \int _{\partial \Omega } \frac{ \sqrt{ | \nabla u |^2 + \epsilon } }{ 1 - u } \frac{\partial u}{\partial \zeta } = \int _\Omega \frac{ \sqrt{ | \nabla u |^2 + \epsilon } }{ ( 1 - u )^2 } | \nabla u |^2 + \int _{ \Omega } \frac{1}{ 1 - u} \frac{ \nabla ^2 u ( \nabla u, \nabla u ) }{\sqrt{ | \nabla u |^2 + \epsilon }}. \end{aligned}$$

(A.3)

For the third term in (A.3), one notes

$$\begin{aligned} \begin{aligned} \int _{ \Omega } \frac{1}{ 1 - u} \frac{ \nabla ^2 u ( \nabla u, \nabla u ) }{\sqrt{ | \nabla u |^2 + \epsilon }}&= \ \int _{ \{ \nabla u \ne 0 \} \subset \Omega } \frac{1}{ 1 - u} \frac{ \nabla ^2 u ( \nabla u, \nabla u ) }{\sqrt{ | \nabla u |^2 + \epsilon }} \\&= \ \int _{ \{ \nabla u \ne 0 \} \subset \Omega } \frac{1}{ 1 - u} \, \nabla ^2 u \left( \frac{\nabla u}{ | \nabla u | }, \frac{ \nabla u }{ | \nabla u | } \right) \frac{ | \nabla u |^2 }{\sqrt{ | \nabla u |^2 + \epsilon }}. \end{aligned} \end{aligned}$$

Thus, taking $\epsilon \rightarrow 0$ in (A.3) proves (A.1).

Similarly, to show (A.2), one has

$$\begin{aligned} \text{ div }\left[ \frac{ \sqrt{ | \nabla u |^2 + \epsilon } }{ (1 - u)^3 } \nabla u \right] = \frac{ 3 \sqrt{ | \nabla u |^2 + \epsilon } }{ ( 1 - u )^4 } | \nabla u |^2 + \frac{1}{ (1 - u)^3 } \frac{ \nabla ^2 u ( \nabla u, \nabla u ) }{\sqrt{ | \nabla u |^2 + \epsilon }}. \end{aligned}$$

Consequently,

$$\begin{aligned} \int _{\partial \Omega } \frac{ \sqrt{ | \nabla u |^2 {+} \epsilon } }{ (1 {-} u)^3 } \frac{\partial u}{\partial \zeta } {=} \int _\Omega \frac{ 3 \sqrt{ | \nabla u |^2 {+} \epsilon } }{ ( 1 {-} u )^4 } | \nabla u |^2 {+} \int _{ \Omega } \frac{1}{ (1 {-} u)^3 } \frac{ \nabla ^2 u ( \nabla u, \nabla u ) }{\sqrt{ | \nabla u |^2 {+} \epsilon }}. \end{aligned}$$

(A.4)

The third term above satisfies

$$\begin{aligned} \begin{aligned}&\int _{ \Omega } \frac{1}{ (1 - u)^3 } \frac{ \nabla ^2 u ( \nabla u, \nabla u ) }{\sqrt{ | \nabla u |^2 + \epsilon }}\\ {}&\quad = \ \int _{ \{ \nabla u \ne 0 \} \subset \Omega } \frac{1}{ (1 - u)^3 } \, \nabla ^2 u \left( \frac{\nabla u}{ | \nabla u | }, \frac{ \nabla u }{ | \nabla u | } \right) \frac{ | \nabla u |^2 }{\sqrt{ | \nabla u |^2 + \epsilon }}. \end{aligned} \end{aligned}$$

Thus, (A.2) follows by taking $\epsilon \rightarrow 0$ in (A.4). $\square$

Lemma A.2

Let u be a harmonic function on a compact, orientable, Riemannian 3-manifold $(\Omega , g)$ with boundary $\partial \Omega$. Suppose $\max _\Omega u < 1$ and u equals a constant on each connected component of $\partial \Omega$. Then

$$\begin{aligned} \begin{aligned} \int _{\partial \Omega } \frac{ H | \nabla u | }{ ( 1 - u )^2} \le \int _{t_1}^{t_2} \frac{1}{(1-t)^2} \left\{ \int _{\Sigma _t} \left[ \frac{2 H | \nabla u | }{1-t} - \frac{1}{2} \left( \frac{ | \nabla ^2 u |^2 }{ | \nabla u |^2 } {+} R \right) \right] {+} 2 \pi \chi (\Sigma _t) \right\} . \end{aligned}\nonumber \\ \end{aligned}$$

(A.5)

Here the mean curvature H of $\partial \Omega$ is taken with respect to the unit normal $\zeta$ pointing out of $\Omega$, the mean curvature H of a regular level set $\Sigma _t$ is taken with respect to $| \nabla u |^{-1} \nabla u$, $\chi (\Sigma _t)$ is the Euler characteristic of $\Sigma _t$, $t_1 = \min _{\Omega } u$, and $t_2 = \max _{\Omega } u$.

Proof

For any constant $\epsilon > 0$, one has

$$\begin{aligned} \begin{aligned} \text{ div }\left[ \frac{ \nabla \sqrt{ | \nabla u |^2 + \epsilon } }{ ( 1 - u )^2 } \right] = \frac{ \Delta \sqrt{ | \nabla u |^2 + \epsilon } }{ ( 1 - u )^2} + \frac{2 \, \nabla ^2 u ( \nabla u, \nabla u ) }{ ( 1 - u )^3 \sqrt{ | \nabla u |^2 + \epsilon } }. \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} \int _{\partial \Omega } \frac{ \partial _\zeta \sqrt{ | \nabla u |^2 + \epsilon } }{ ( 1 - u )^2 } = \int _{\Omega } \frac{ \Delta \sqrt{ | \nabla u |^2 + \epsilon } }{ ( 1 - u )^2} + \int _{\Omega } \frac{2 \, \nabla ^2 u ( \nabla u, \nabla u ) }{ ( 1 - u )^3 \sqrt{ | \nabla u |^2 + \epsilon } }. \end{aligned}$$

(A.6)

As u is constant on each connected component of $\partial \Omega$, direct calculations gives

$$\begin{aligned} \partial _\zeta \sqrt{ | \nabla u |^2 + \epsilon } = - \frac{ | \nabla u |^2}{ \sqrt{ | \nabla u |^2 + \epsilon } } H. \end{aligned}$$

(See Lemma 2.1 in [16] for instance.) Thus,

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \int _{\partial \Omega } \frac{ \partial _\zeta \sqrt{ | \nabla u |^2 + \epsilon } }{ ( 1 - u )^2 } = - \int _{\partial \Omega } \frac{H | \nabla u | }{ ( 1 - u )^2}. \end{aligned}$$

(A.7)

As in the proof of the previous lemma, taking $\epsilon \rightarrow 0$ in the third term in (A.6) gives

$$\begin{aligned} \begin{aligned} \lim _{ \epsilon \rightarrow 0} \int _{\Omega } \frac{2 \, \nabla ^2 u ( \nabla u, \nabla u ) }{ ( 1 - u )^3 \sqrt{ | \nabla u |^2 + \epsilon } }&= \ \int _{ \{ \nabla u \ne 0 \} \subset \Omega } \frac{2 \, \nabla ^2 u ( \nabla u, \nabla u ) }{ ( 1 - u )^3 | \nabla u | } \\&= \ - \int _{t_1}^{t_2} \frac{2}{(1-t)^3 } \int _{\Sigma _t} H | \nabla u |, \end{aligned} \end{aligned}$$

(A.8)

where the second equation follows from the coarea formula and (3.5).

To deal with the second term in (A.6), we follow an argument of Stern [30]. Let $\mathcal {C}$ denote the set of critical values of u in $[t_1, t_2]$. Let W denote an open set of $[ t_1, t_2]$ such that W contains $\mathcal {C}$. Let D be the complement of W in $[ t_1, t_2 ]$.

On $u^{-1} (D)$, $\frac{ \Delta \sqrt{ | \nabla u |^2 + \epsilon } }{ ( 1 - u )^2 |\nabla u | }$ is integrable. By coarea formula,

$$\begin{aligned} \int _{u^{-1} ( D ) } \frac{ \Delta \sqrt{ | \nabla u |^2 + \epsilon } }{ ( 1 - u )^2 } = \int _{ D} \int _{\Sigma _t} \frac{ \Delta \sqrt{ | \nabla u |^2 + \epsilon } }{ ( 1 - u )^2 | \nabla u | }. \end{aligned}$$

Along $\Sigma _t$ which a regular level set of u, by equation (14) in [30],

$$\begin{aligned} \Delta \sqrt{ | \nabla u |^2 + \epsilon } \ge \frac{1}{2 \sqrt{ | \nabla u |^2 + \epsilon } } \big [ | \nabla ^2 u |^2 + (R - 2 K _{_{\Sigma _t} } ) | \nabla u |^2 \big ] , \end{aligned}$$

(A.9)

where $K_{_{\Sigma _t} }$ is the Gauss curvature of $\Sigma _t$. Thus,

$$\begin{aligned} \int _{u^{-1} ( D ) } \frac{ \Delta \sqrt{ | \nabla u |^2 + \epsilon } }{ ( 1 - u )^2 } \ge \int _{ D} \int _{\Sigma _t} \frac{ 1}{ ( 1 - u )^2 | \nabla u | } \frac{\big [ | \nabla ^2 u |^2 + (R - 2 K_{_{\Sigma _t} } ) | \nabla u |^2 \big ] }{ 2 \sqrt{ | \nabla u |^2 + \epsilon } }.\nonumber \\ \end{aligned}$$

(A.10)

With W fixed, letting $\epsilon \rightarrow 0$ in (A.10) gives

$$\begin{aligned} \liminf _{\epsilon \rightarrow 0} \int _{u^{-1} ( D ) } \frac{ \Delta \sqrt{ | \nabla u |^2 + \epsilon } }{ ( 1 - u )^2 }&\ge \int _{ D} \int _{\Sigma _t} \frac{\big [ | \nabla ^2 u |^2 + (R - 2 K_{_{\Sigma _t} } ) | \nabla u |^2 \big ] }{ ( 1 - u )^2 \, 2 | \nabla u |^2 } \nonumber \\ {}&= \int _{ D} \frac{1}{(1-t)^2} \left[ \int _{\Sigma _t} \frac{1}{2} \big ( | \nabla u |^{-2} | \nabla ^2 u |^2 + R \big ) \right. \nonumber \\ {}&\quad \, \left. - 2 \pi \chi (\Sigma _t) \right] , \end{aligned}$$

(A.11)

where one also used the Gauss–Bonnet theorem.

To estimate the integral on $u^{-1} (W)$, one notes

$$\begin{aligned} \begin{aligned} \Delta \sqrt{ | \nabla u |^2 + \epsilon }&= \ \frac{1}{ \sqrt{ | \nabla u |^2 + \epsilon } } \left[ | \nabla ^2 u |^2 + \textrm{ Ric }(\nabla u, \nabla u ) - \frac{1}{ 4 ( |\nabla u |^2 + \epsilon ) } \big | \nabla | \nabla u |^2 \big |^2 \right] \\ {}&\ge \ \frac{1}{ \sqrt{ | \nabla u |^2 + \epsilon } } \textrm{ Ric }(\nabla u, \nabla u ). \end{aligned} \end{aligned}$$

This implies

$$\begin{aligned} \begin{aligned} \int _{u^{-1} ( W ) } \frac{ \Delta \sqrt{ | \nabla u |^2 + \epsilon } }{ ( 1 - u )^2 }&\ge - \max _{\Omega } | \text{ Ric }| \int _{u^{-1} ( W ) } \frac{ | \nabla u | }{ ( 1 - u )^2 } \\&= - \max _{\Omega } | \text{ Ric }| \int _{ W } \int _{\Sigma _t} \frac{ 1 }{ ( 1 - u )^2 }. \end{aligned} \end{aligned}$$

(A.12)

It follows from (A.11) and (A.12) that

$$\begin{aligned} \begin{aligned} \lim _{\epsilon \rightarrow 0} \int _{\Omega } \frac{ \Delta \sqrt{ | \nabla u |^2 + \epsilon } }{ ( 1 - u )^2}&\ge \int _{ D} \frac{1}{(1-t)^2} \left[ \int _{\Sigma _t} \frac{1}{2} \big ( | \nabla u |^{-2} | \nabla ^2 u |^2 + R \big ) - 2 \pi \chi (\Sigma _t) \right] \\ {}&\quad - \max _{\Omega } | \textrm{ Ric }| \int _{ W } \int _{\Sigma _t} \frac{ 1 }{ ( 1 - u )^2 }. \end{aligned} \end{aligned}$$

As $\int _\Omega \frac{ | \nabla u |}{ ( 1 - u)^2} < \infty$, by choosing the measure of W to be arbitrarily small, one has

$$\begin{aligned} \begin{aligned} \lim _{\epsilon \rightarrow 0} \int _{\Omega } \frac{ \Delta \sqrt{ | \nabla u |^2 + \epsilon } }{ ( 1 - u )^2} \ge&\ \int _{t_1}^{t_2} \frac{1}{(1-t)^2} \left[ \int _{\Sigma _t} \frac{1}{2} \big ( | \nabla u |^{-2} | \nabla ^2 u |^2 + R \big ) - 2 \pi \chi (\Sigma _t) \right] .\end{aligned}\nonumber \\ \end{aligned}$$

(A.13)

The lemma now follows from (A.6), (A.7), (A.8) and (A.13). $\square$

Remark A.1

(A.5) may be viewed as a weighted version of the identity (3.4).

Proposition A.1

Let $(\Omega , g)$ be a connected, compact, orientable, Riemannian 3-manifold with boundary $\partial \Omega$. Suppose $\partial \Omega$ is the disjoint union of two nonempty pieces $S_1$ and $S_2$. Let u be a harmonic function on $(\Omega , g)$ such that $u = c_i$ on $S_i$, $i = 1, 2$, where $c_1$, $c_2$ are constants with $c_1< c_2 < 1$. For regular values t, let

$$\begin{aligned} A(t)= & {} 8 \pi - \frac{1}{( 1 - t)} \int _{\Sigma _t} H | \nabla u|, \ \ \ \ \mathcal {A}(t) = \frac{A (t) }{1-t}, \\ B(t)= & {} 4 \pi - \frac{1}{( 1 - t)^2} \int _{\Sigma _t} | \nabla u|^2, \ \ \ \ \mathcal {B} (t) = \frac{B(t) }{1-t}. \end{aligned}$$

Then, for any two regular values $t_1 < t_2$,

$$\begin{aligned} \mathcal {B} (t_2) - \mathcal {B} (t_1) = \int _{t_1}^{t_2} \frac{1}{(1-t)^2} \left[ 3 B(t) - A (t) \right] , \end{aligned}$$

(A.14)

and

$$\begin{aligned} \begin{aligned} \mathcal {A}(t_2) - \mathcal {A}(t_1) \ge&\ \int _{t_1}^{t_2} \frac{1}{(1-t)^2} \left[ 3 B(t) - A (t) + 2 \pi \big ( 2 - \chi (\Sigma _t) \big ) + \psi (t) \right] , \end{aligned}\nonumber \\ \end{aligned}$$

(A.15)

where

$$\begin{aligned} \psi (t) = \int _{\Sigma _t} \left[ \frac{3}{4} \left( H - \frac{ 2 | \nabla u | }{ 1 - u } \right) ^2 + | \nabla u |^{-2}\big | \nabla _{_{\Sigma _t} } | \nabla u | \big |^2 + \frac{1}{2} | \mathring{ \displaystyle {\mathbb {I}\mathbb {I}}} |^2 + \frac{1}{2} R \right] . \end{aligned}$$

As a result, if

• (M, g) is a complete, orientable, asymptotically flat 3-manifold with connected boundary $\Sigma$ and $H_2 (M, \Sigma ) = 0$;

• u is the harmonic function on (M, g) with $u = 0$ at $\Sigma$ and $u \rightarrow 1$ at $\infty$; and

• g has nonnegative scalar curvature,

then $\Sigma _t$ is connected, $3 B(t) - A (t) \ge 0$ by (3.3), and consequently,

$$\begin{aligned} \mathcal {B} (t_2) - \mathcal {B} (t_1) \ge 0 \quad \text{ and } \quad \mathcal {A} (t_2) - \mathcal {A} (t_1) \ge 0. \end{aligned}$$

Proof

Applying (A.2) in Lemma A.1 to $\Omega _{[t_1, t_2]} = \{ x \ | \ t_1 \le u(x) \le t_2 \}$, one has

$$\begin{aligned} \begin{aligned} \int _{\Sigma _{t_2} } \frac{ | \nabla u |^2 }{(1- u )^3} - \int _{\Sigma _{t_1} } \frac{ | \nabla u |^2 }{(1- u )^3}&= \ \int _{\Omega _{ [t_1, t_2] } } \frac{ 3 | \nabla u |^3 }{ ( 1 - u )^4 } + \int _{ \{ \nabla u \ne 0 \} \subset \Omega _{[t_1, t_2]} } \frac{ \nabla ^2 u ( \nabla u, \nabla u ) }{ (1 - u)^3 | \nabla u | } \\&= \ \int _{t_1}^{t_2} \int _{\Sigma _t} \left[ \frac{3 | \nabla u |^2}{(1-t)^4 } - \frac{H | \nabla u |}{ ( 1 - t)^3 } \right] , \end{aligned} \end{aligned}$$

This, combined with $\frac{1}{1-t_2} - \frac{1}{1 - t_1} = \int _{t_1}^{t_2} \frac{1}{( 1 - t)^2 }$, shows

$$\begin{aligned} \begin{aligned} \mathcal {B}(t_2) - \mathcal {B} (t_1)&= \ \int _{t_1}^{t_2} \left[ \frac{4\pi }{(1-t)^2} + \int _{\Sigma _t} \frac{H | \nabla u |}{ ( 1 - t)^3 } - \int _{\Sigma _t} \frac{3 | \nabla u |^2}{(1-t)^4 } \right] \\&= \ \int _{t_1}^{t_2} \frac{1}{(1-t)^2} \left[ 3 B(t) - A (t) \right] , \end{aligned} \end{aligned}$$

(A.16)

which proves (A.14).

Similarly, applying Lemma A.2 to u on $\Omega _{[t_1, t_2]}$ and using (3.6), one has

$$\begin{aligned} \begin{aligned}&\ \frac{1}{(1- t_2)^2} \int _{\Sigma _{t_2} } H |\nabla u | - \frac{1}{(1- t_1)^2} \int _{\Sigma _{t_1} } H |\nabla u | \\ {}&\quad \le \ \int _{t_1}^{t_2} \frac{1}{(1-t)^2} \left[ 2 \pi \chi (\Sigma _t) + \int _{\Sigma _t} \frac{2 H | \nabla u | }{1-t} \right. \\ {}&\qquad \left. - \int _{\Sigma _t} \frac{3}{4} H^2 - \int _{\Sigma _t} \left( | \nabla u |^{-2} \big | \nabla _{_{\Sigma _t} } | \nabla u | \big |^2 + \frac{1}{2} | \mathring{ \displaystyle {\mathbb {I}\mathbb {I}}} |^2 + \frac{1}{2} R \right) \right] \\ {}&\quad = \ \int _{t_1}^{t_2} \frac{1}{(1-t)^2} \left[ 2 \pi \chi (\Sigma _t) - \psi (t) - \frac{1}{1-t} \int _{\Sigma _t} H | \nabla u | + \frac{3}{(1-t)^2 } \int _{\Sigma _t} | \nabla u |^2 \right] . \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} \mathcal {A} (t_2) - \mathcal {A}(t_1) \ge \int _{t_1}^{t_2} \frac{1}{(1-t)^2} \big [ 3 B(t) - A (t) + 2 \pi \big ( 2 - \chi (\Sigma _t) \big ) + \psi (t) \big ] , \end{aligned}$$

which proves (A.15). $\square$

Remark A.2

As a corollary of (A.14) and (A.15),

$$\begin{aligned} \begin{aligned}&\ \left[ \mathcal {A}(t_2) - \mathcal {B}(t_2) \right] - \left[ \mathcal {A}(t_1) - \mathcal {B}(t_1) \right] \\ {}&\quad \ge \ \int _{t_1}^{t_2} \frac{1}{(1-t)^2} \big [ 2 \pi \big ( 2 - \chi (\Sigma _t) \big ) + \psi (t) \big ] . \end{aligned} \end{aligned}$$

(A.17)

This corresponds to the monotonicity of $F(t) = \mathcal {A}(t) - \mathcal {B}(t)$ in [2].