Implications of Some Mass-Capacity Inequalities

Abstract

Applying a family of mass-capacity related inequalities proved in Miao (Peking Math J 2023, https://doi.org/10.1007/s42543-023-00071-7), we obtain sufficient conditions that imply the nonnegativity as well as positive lower bounds of the mass, on a class of manifolds with nonnegative scalar curvature, with or without a singularity.

1 Introduction

A smooth Riemannian 3-manifold (M, g) is called asymptotically flat (AF) if M, outside a compact set, is diffeomorphic to $\mathbb {R}^3$ minus a ball; the associated metric coefficients satisfy

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

for some $\tau > \frac{1}{2}$; and the scalar curvature of g is integrable. Under these AF conditions, the limit, near $\infty$,

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

exists and is called the ADM mass [2] of (M, g). It is a result of Bartnik [3], and of Chruściel [9], that $\mathfrak {m}$ is a geometric invariant, independent on the choice of the coordinates $\{ x_i \}$.

A fundamental result on the mass and the scalar curvature is the Riemannian positive mass theorem (PMT):

Theorem 1.1

([20, 22]) Let (M, g) be 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 $\mathbb {R}^3$.

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

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

where the infimum is taken over all locally Lipschitz functions f that vanishes on $\Sigma$ and tend to 1 at infinity. Equivalently, if $\phi$ denotes the function with

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

then, $\displaystyle \mathfrak {c}_{_\Sigma } = \frac{1}{4 \pi } \int _M | \nabla \phi |^2 = \frac{1}{4\pi } \int _{\Sigma } | \nabla \phi |$, and

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

Regarding the mass and the capacity, if $\Sigma$ is a minimal surface, Bray showed

Theorem 1.2

([4]) Let (M, g) be 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 iff (M, g) is isometric to a spatial Schwarzschild manifold outside the horizon.

In [16, Theorem 7.4], an inequality relating the mass-to-capacity ratio to the Willmore functional of the boundary was obtained:

Theorem 1.3

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

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

(1.1)

Here $\mathfrak {m}$ is the mass of (M, g), $\mathfrak {c}_{_\Sigma }$ is the capacity of $\Sigma$ in (M, g), and H is the mean curvature of $\Sigma$. Moreover, equality in (1.1) holds if and only if (M, g) is isometric to a spatial Schwarzschild manifold outside a rotationally symmetric sphere with nonnegative mean curvature.

As shown in [16], (1.1) implies the 3-dimensional PMT. For instance, assuming M is topologically $\mathbb {R}^3$, applying (1.1) to the exterior of a geodesic sphere $S_r$ with radius r centered at any point $p \in M$, one has

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

(1.2)

Letting $r \rightarrow 0$, one obtains $\mathfrak {m}\ge 0$. Earlier proofs of 3-d PMT via harmonic functions were given by Bray-Kazaras-Khuri-Stern [6] and Agostiniani-Mazzieri-Oronzio [1].

Theorem 1.3 follows from two other results (Corollary 7.1 and Theorem 7.3) in [16]:

Theorem 1.4

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

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

(1.3)

and

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

(1.4)

Here u is the harmonic function with $u = 0$ at $\Sigma$ and $u \rightarrow 1$ near $\infty$. Moreover,

• equality in (1.3) holds if and only if (M, g) is isometric to a spatial Schwarzschild manifold outside a rotationally symmetric sphere with nonnegative mean curvature;

• equality in (1.4) holds if and only if (M, g) is isometric to a spatial Schwarzschild manifold outside a rotationally symmetric sphere.

A corollary of (1.3) (see [16, Theorem 7.2]) is an upper bound on the capacity-to-area-radius ratio, first derived by Bray and the author [7].

Theorem 1.5

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

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

(1.5)

Here $\mathfrak {c}_{_\Sigma }$ is the capacity of $\Sigma$ in (M, g) and $r_{_\Sigma } = \left( \frac{ | \Sigma | }{4 \pi } \right) ^\frac{1}{2}$ 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 mean curvature.

In this paper, we give some other applications of (1.1), (1.3) and (1.4).

First, for later purposes, we remark on the topological assumption “$H_2 (M, \Sigma ) = 0$” in Theorems 1.3 – 1.5 above: the assumption is imposed only to ensure each regular level set of the harmonic function u, vanishing at the boundary and tending to 1 near $\infty$, to be connected in the interior of M (see the paragraph preceding the proof of Theorem 3.1 in [16]); indeed, (1.1), (1.3) and (1.4) (and all other results from [16]) hold if “$H_2 (M, \Sigma ) = 0$" is replaced by assuming

• ($*$) each closed, connected, orientable surface in the interior of M either is the boundary of a bounded domain, or together with $\Sigma$ forms the boundary of a bounded domain.

Now we motivate the main tasks in this paper. Let us first return to the setting of (1.2), in which the surface $S_r$ “closes up nicely” (to bound a geodesic ball). In this setting, by a result of Mondino and Templeton-Browne [18], $\{ S_r \}$ can be perturbed to yield another family of surfaces $\{ \Sigma _r \}$ so that, as $r \rightarrow 0$,

$$\begin{aligned} \int _{\Sigma _r} H^2= & {} 16 \pi - \frac{ 8 \pi }{3} R(p) r^2 \nonumber \\{} & {} + \frac{4\pi }{3}\left[ \frac{1}{9} R(p)^2 - \frac{4 }{15} | \mathring{\text{ Ric }} (p) |^2 - \frac{1}{5} \Delta R (p) \right] r^4 + O (r^5). \end{aligned}$$

(1.6)

Here R denotes the scalar curvature and $\mathring{\text{ Ric }} = \text{ Ric }- \frac{1}{3} R g$ is the traceless part of $\text{ Ric }$, the Ricci tensor. Applying (1.1) to the exterior of these $\Sigma _r$ in (M, g), one obtains

$$\begin{aligned} \frac{ \mathfrak {m}}{ \mathfrak {c}_{_{\Sigma _r} } } \ge \frac{1}{12} R(p) r^2 + \left[ \frac{1}{90} | \mathring{ \text{ Ric }} (p) |^2 - \frac{1}{864} R(p)^2 + \frac{1}{120} \Delta R (p) \right] r^4 + O (r^5).\nonumber \\ \end{aligned}$$

(1.7)

If $R \ge 0$, (1.7) shows the inequality $\mathfrak {m}\ge 0$ as well as the rigidity of $\mathfrak {m}= 0$.

In general, (1.1) suggests that, if it is applied to obtain $\mathfrak {m}\ge 0$ on an (M, g), the manifold boundary $\Sigma$ does not need to admit a “nice fill-in". Rewriting (1.1) as

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

one may seek conditions on metrics g with a “singularity" so that $\mathfrak {m}\ge 0$ while g is allowed to be incomplete.

Similarly, on an (M, g) with two ends, one of which is asymptotically flat (AF), assuming it admits a harmonic function u that tends to 1 at the AF end and tends to 0 at the other end, one may aim to apply (1.4), i.e.

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

to bound $\mathfrak {m}$ via the energy of u on the entire (M, g).

Below we formulate a class of manifolds to carry out the above mentioned tasks. Throughout the paper, let N be a noncompact, connected, orientable 3-manifold. We assume N admits an increasing exhaustion sequence of bounded domains with connected boundary. Precisely, this means there exists a sequence of closed, orientable surfaces $\{ \Sigma _k \}_{k=1}^\infty$ in N such that

• $\Sigma _k$ is connected;

• $\Sigma _k = \partial D_k$ for a precompact domain $D_k \subset N$; and

• ${\bar{D}}_k \subset D_{k+1}$ and $N = \cup _{k=1}^\infty D_k$. Here ${\bar{D}}_k = D_k \cup \Sigma _k$ is the closure of $D_k$ in N.

Fix a point $p \in N$, let $M = N \setminus \{p \}$. On M, let g be a smooth metric that is asymptotically flat near p (Fig. 1). We refer p as the asymptotically flat (AF) $\infty$ of (M, g). Unless otherwise specified, we do not impose assumptions on the behavior of g near $\Sigma _k$ as $k \rightarrow \infty$. In particular, (M, g) does not need to be complete,

Given any closed, connected surface $S \subset M$, we say S encloses p if $S = \partial D_{_S}$ for some precompact domain $D_{_S} \subset N$ such that $p \in D_{_S}$. Let $\mathcal {S}$ denote the set of all such surfaces $S \subset M$ enclosing p. Clearly, $\Sigma _k \in \mathcal {S}$ for large k. Define

$$\begin{aligned} \mathfrak {c}(M,g) = \inf _{ S \in \mathcal {S}} \, \mathfrak {c}_{_S}. \end{aligned}$$

(1.8)

Here $\mathfrak {c}_{_S}$ is the capacity of S in the asymptotically flat $(E_{_S}, g)$, where

$$\begin{aligned} E_{_S} = ( D_{_S} \setminus \{p\} ) \cup S. \end{aligned}$$

As a functional on $\mathcal {S}$, the capacity $\mathfrak {c}_{_S}$ has a monotone property, that is if $S_1, S_2 \in \mathcal {S}$ and $D_{_{S_1} } \subset D_{_{S_2}}$, then $\mathfrak {c}_{_{S_1}} \ge \mathfrak {c}_{_{S_2}}$. Such a property readily implies $\{ \mathfrak {c}_{_{\Sigma _k}} \}$ is monotone non-increasing and

$$\begin{aligned} \mathfrak {c}(M,g) = \lim _{ k \rightarrow \infty } \mathfrak {c}_{_{\Sigma _k}}. \end{aligned}$$

(1.9)

Standard arguments show $\mathfrak {c}(M, g) > 0$ if and only if there exists a harmonic function w on (M, g) such that $0< w < 1$ on M and $w (x) \rightarrow 1$ at $\infty$ (i.e. as $x \rightarrow p$). (See Proposition 3.1 in Sect. 3.)

Fig. 1

On the left is an examples of (M, g) with $\mathfrak {c}(M,g) = 0$; the arrow denotes the AF end; $\{ \Sigma _k \}$ may approach a “singularity" as $k \rightarrow \infty$. On the right is an example of (M, g) with $\mathfrak {c}(M,g) > 0$; besides the AF end, (M, g) has another end with suitable growth

For manifolds (M, g) with $\mathfrak {c}(M,g) = 0$, we seek conditions that imply the AF end of (M, g) has mass $\mathfrak {m}\ge 0$, see Theorem 2.1 and Remark 2.2. For (M, g) with $\mathfrak {c}(M,g) > 0$, we explore for sufficient conditions that bound $\mathfrak {m}$ from below via $\mathfrak {c}(M,g)$, see Theorem 3.1 and Corollary 3.1.

2 Singular Metrics with $\mathfrak {m}\ge 0$

Let N, M and g be given in the definition of $\mathfrak {c}(M,g)$ in (1.8). Given $S \in \mathcal {S}$, let

$$\begin{aligned} W(S) = \int _S H^2. \end{aligned}$$

We want to apply (1.1) to $(E_{_S}, g)$. For this purpose, we assume the background manifold N satisfies $H_2 (N) = 0$. Under this assumption, any closed, connected surface $S'$ in $M = N {\setminus } \{ p \}$ is the boundary of a bounded domain $D \subset N$. If $p \not \in D$, then $D \subset M$; if $p \in D$, then $S'$ is homologous to $S \in \mathcal {S}$. Therefore, condition $(*)$ holds on $E_{_S}$.

The following is a direct corollary of (1.1).

Proposition 2.1

Suppose $H_2 (N) = 0$ and (M, g) has nonnegative scalar curvature. Then

$$\begin{aligned} \mathfrak {c}_{_{S_k}} W(S_k)^\frac{1}{2} \rightarrow 0 \ \text {along a sequence } \{ S_k \} \subset \mathcal {S} \ \ \Rightarrow \ \mathfrak {m}\ge 0. \end{aligned}$$

(2.1)

Proof

If $W(S_k) \le 16 \pi$ for some k, then (1.1) implies $\mathfrak {m}\ge 0$.

Suppose $W(S_k) > 16 \pi$ for every k, then “$\mathfrak {c}_{_{S_k}} W(S_k)^\frac{1}{2} \rightarrow 0$" implies “$\mathfrak {c}_{_{S_k}} \rightarrow 0$". Rewriting (1.1) as

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

(2.2)

and letting $k \rightarrow \infty$, we have $\mathfrak {m}\ge 0$. $\square$

Given $S \in \mathcal {S}$, let $\displaystyle \mathfrak {m}_{_H} (S) = \frac{r_{_S} }{2} \left( 1 - \frac{1}{16\pi } W (S) \right)$ denote the Hawking mass of S ( [11]). Inequality (2.3) in the next Proposition is comparable to the result of Huisken and Ilmanen [13] on the relation between $\mathfrak {m}$ and $\mathfrak {m}_{_H} (S)$.

Proposition 2.2

Suppose $H_2 (N) = 0$ and (M, g) has nonnegative scalar curvature. If a surface $S \in \mathcal {S}$ satisfies $W(S) \ge 16 \pi$, then

$$\begin{aligned} \mathfrak {m}\ge \mathfrak {c}_{_S} \left[ 1 - \left( \frac{1}{16\pi } W(S) \right) ^\frac{1}{2} \right] \ge \mathfrak {m}_{_H} (S). \end{aligned}$$

(2.3)

Proof

If $W(S) \ge 16 \pi$, then (1.5) implies

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

(2.4)

This combined with (1.1) proves (2.3). $\square$

Remark 2.1

Similar to (2.1), a condition of “$r_{_{S_k}} W(S_k) \rightarrow 0$” along $\{ S_k \} \subset \mathcal {S}$ also implies “$\mathfrak {m}\ge 0$”. However, if $\inf _{k} W(S_k) \ge 16 \pi$, then

$$\begin{aligned} ``r_{_{S_k}} W(S_k) \rightarrow 0" \ \Rightarrow \ ``r_{_{S_k}} \rightarrow 0 \ \text {and} \ r_{_{S_k}} W(S_k)^\frac{1}{2} \rightarrow 0" \ \Rightarrow \ ``\mathfrak {c}_{_{S_k}} \rightarrow 0", \end{aligned}$$

where the last step is by (1.5). Combined with (2.4), this implies the assumption of “$\mathfrak {c}_{_{S_k}} W(S_k)^\frac{1}{2} \rightarrow 0$” in Proposition 2.1.

In what follows, let $\{ \Sigma _k \} \subset \mathcal {S}$ be the sequence of surfaces given in the introduction. The numerical value of $\mathfrak {c}_{_{\Sigma _k}}$ depends on g near the AF end. However, a property of “$\mathfrak {c}_{_{\Sigma _k}} \rightarrow 0$" does not. This was shown by Bray and Jauregui [5] in the context of (M, g) having a zero area singularity. Their argument applies to “$\mathfrak {c}_{_{\Sigma _k}} W(\Sigma _k)^\frac{1}{2} \rightarrow 0$". To illustrate this, it is convenient to adopt a notion of relative capacity (see [14] for instance). Given two surfaces $S, {{\tilde{S}}} \in \mathcal {S}$, suppose $S \cap {{\tilde{S}}} = \emptyset$ and $D_{_{{{\tilde{S}}}}} \subset D_{_S}$. The capacity of S relative to ${{\tilde{S}}}$ is

$$\begin{aligned} \mathfrak {c}_{_{(S, {{\tilde{S}}})} }= \frac{1}{4\pi } \int _{D_{_{S}} \setminus D_{_{{{\tilde{S}}} } } } | \nabla v |^2, \end{aligned}$$

(2.5)

where v is the harmonic function on $D_{_{S}} {\setminus } D_{_{{{\tilde{S}}} } }$ with $v = 0$ at S and $v = 1$ at $\tilde{S}$.

Proposition 2.3

Let ${{\tilde{S}}} \in \mathcal {S}$ be a fixed surface. Then, as $k \rightarrow \infty$,

$$\begin{aligned} \mathfrak {c}_{_{\Sigma _k}} W(\Sigma _k)^\frac{1}{2} \rightarrow 0 \ \Longleftrightarrow \ \mathfrak {c}_{_{(\Sigma _k, {{\tilde{S}}}) }} W(\Sigma _k)^\frac{1}{2} \rightarrow 0. \end{aligned}$$

Proof

For large k, let $u_k$, $v_k$ be the harmonic function on $D_{k} {\setminus } \{ p \}$, $D_{k} {\setminus } D_{_{{{\tilde{S}}}}}$, with boundary values $u_k = 0$ at $\Sigma _k$, $u_k \rightarrow 1$ at the AF $\infty$, $v_k = 0$ at $\Sigma _k$, $v_k = 1$ at ${{\tilde{S}}}$, respectively. Let $\beta _k = \min _{{{\tilde{S}}}} u_k$. By the maximum principle, $v_k \ge u_k \ge \beta _k v_k$ on $D_{k} {\setminus } D_{_{{{\tilde{S}}}}}$, which implies $\partial _\nu v_k \ge \partial _\nu u_k \ge \beta _k \partial _\nu v_k$ at $\Sigma _k$. Here $\nu$ denotes the unit normal to $\Sigma _k$ pointing to $\infty$. Since $4 \pi \mathfrak {c}_{_{\Sigma _k}} = \int _{\Sigma _k} \partial _\nu u_k$ and $4 \pi \mathfrak {c}_{_{(\Sigma _k, {{\tilde{S}}})}} = \int _{\Sigma _k } \partial _\nu v_k$, one has

$$\begin{aligned} \beta _k^{-1} \mathfrak {c}_{_{\Sigma _k}} \ge \mathfrak {c}_{_{(\Sigma _k, {{\tilde{S}}} )} } \ge \mathfrak {c}_{_{\Sigma _k}}. \end{aligned}$$

The claim follows by noting that $\beta _k$ has a uniform positive lower bound as $k \rightarrow \infty$. $\square$

As an application of Propositions 2.1 and 2.3, we have

Theorem 2.1

Let N be a noncompact, connected, orientable 3-manifold. Suppose $H_2 (N) = 0$. Let $M = N \setminus \{ p \}$ where p is a point in N. Let g be a smooth metric with nonnegative scalar curvature on M such that g is asymptotically flat near p. Assume there is a precompact domain $D \subset N$ such that $p \in D$ and $( N {\setminus } D, g)$ is isometric to

$$\begin{aligned} ( (0, \delta ] \times \Sigma , {\bar{g}} + h ), \end{aligned}$$

where

• $\delta > 0$ is a constant, $\Sigma$ is a closed, connected, orientable surface;

• ${\bar{g}} = d r^2 + a(r)^2 \sigma$, in which $\sigma$ is a given metric on $\Sigma$ and a(r) is a positive function on $(0, \delta ]$; and

• $\lambda ^{-1} \le | {\bar{g}} + h |_{{\bar{g}}} \le \lambda$ for some constant $\lambda >0$.

Then

$$\begin{aligned} \lim _{r \rightarrow 0} \left( \int _r^\delta \frac{1}{a (x)^2} \, d x \right) ^{-1} \left[ | a'(r ) | + a(r) | {\bar{\nabla }} h |_{\bar{g}} \right] = 0 \ \Longrightarrow \ \mathfrak {m}\ge 0. \end{aligned}$$

(2.6)

Remark 2.2

If $a (r) = r^b$ for a constant $b >0$, then (2.6) translates to

$$\begin{aligned} \lim _{r \rightarrow 0} \, r^{3b -2} \left( 1 + r | {\bar{\nabla }} h |_{{\bar{g}}} \right) = 0 \ \Rightarrow \ \mathfrak {m}\ge 0. \end{aligned}$$

This in particular implies, if g has a conical or $r^b$-horn type singularity modeled on ${\bar{g}} = d r^2 + r^{2b} \sigma$ near $r=0$, then, under a mild asymptotic assumption of

$$\begin{aligned} \lambda ^{-1} \le | {\bar{g}} + h |_{{\bar{g}}} \le \lambda \ \ \text {and} \ \ r | {\bar{\nabla }} h |_{{\bar{g}}} = O (1), \end{aligned}$$

one has “$b > \frac{2}{3} \Rightarrow \mathfrak {m}\ge 0$". (Related results on PMT with isolated singularities can be found in [10, 15, 21]).

Proof of Theorem 2.1

Let $\Sigma _r = \{ r \} \times \Sigma$, $r \in (0, \delta ]$. For $s \in (0, \delta )$, let ${\bar{\mathfrak {c}}}_{_{( \Sigma _s, \Sigma _\delta ) } }$, ${\bar{W}} (\Sigma _s)$ denote the capacity of $\Sigma _s$ relative to $\Sigma _{\delta }$, the Willmore functional of $\Sigma _s$, respectively, with respect to ${\bar{g}}$.

The function $u ( r) = \left( \int _s^\delta a(x)^{-2} \, d x \right) ^{-1} \, \int _s^r a(x)^{-2} \, d x$ is ${\bar{g}}$-harmonic on $[s, \delta ] \times \Sigma$ with $u = 0$ at $\Sigma _s$ and $u = 1$ at $\Sigma _\delta$. This implies

$$\begin{aligned} {\bar{\mathfrak {c}}}_{_{( \Sigma _s, \Sigma _\delta ) } } = \frac{ |\Sigma |_{\sigma } }{4\pi } \left( \int _s^\delta a(x)^{-2} \, d x \right) ^{-1}, \end{aligned}$$

(2.7)

where $| \Sigma |_\sigma$ is the area of $(\Sigma , \sigma )$. The mean curvature ${\bar{H}}$ of $\Sigma _s$ with respect to ${\bar{g}}$ is ${\bar{H}} = 2 a^{-1} a'$. Hence,

$$\begin{aligned} {\bar{W}} (\Sigma _s) = 4 a'(s)^2 | \Sigma |_\sigma . \end{aligned}$$

We compare ${\bar{\mathfrak {c}}}_{_{( \Sigma _s, \Sigma _\delta ) } }$ and $\mathfrak {c}_{_{( \Sigma _s, \Sigma _\delta ) } }$. Let ${\bar{\nabla }}$, $\nabla$ and $d V_{{\bar{g}}}$, $d V_g$ denote the gradient, the volume form with respect to ${\bar{g}}$, g, respectively. Since $\mathfrak {c}_{_{( \Sigma _s, \Sigma _\delta ) } }$ equals the infimum of the g-Dirichlet energy of functions that vanish at $\Sigma _s$ and equal 1 at $\Sigma _\delta$, we have

$$\begin{aligned} \begin{aligned} \mathfrak {c}_{_{( \Sigma _s, \Sigma _\delta ) } } \le&\ \int _{[ s, \delta ] \times \Sigma } | \nabla u |_g^2 \, d V_g \\ \le&\ C \int _{[ s, \delta ] \times \Sigma } | {\bar{\nabla }} u |_{{\bar{g}}}^2 \, d V_{{\bar{g}}} = C \, {\bar{\mathfrak {c}}}_{_{( \Sigma _s, \Sigma _\delta ) } }. \end{aligned} \end{aligned}$$

(2.8)

Here $C > 0$ denotes a constant independent on s and we have used the assumption $\lambda ^{-1} \le | g |_{{\bar{g}}} \le \lambda$.

We also compare ${\bar{W}} (\Sigma _s)$ and $W (\Sigma _s)$. Let ${\bar{\displaystyle {\mathbb{I}\mathbb{I}}}}$ denote the second fundamental form of $\Sigma _s$ with respect to ${\bar{g}}$. Direct calculation shows

$$\begin{aligned} H - {\bar{H}} = | {\bar{\displaystyle {\mathbb{I}\mathbb{I}}}} |_{{\bar{g}}} \, O ( | h |_{{\bar{g}}} ) + O ( | {\bar{\nabla }} h |_{{\bar{g}}} ). \end{aligned}$$

(2.9)

(For instance, see formula (2.33) in [17] and the proof therein.) Therefore,

$$\begin{aligned} \begin{aligned} H^2 =&\ {\bar{H}}^2 + \left[ | {\bar{\displaystyle {\mathbb{I}\mathbb{I}}}} |_{{\bar{g}}} \, O ( | h |_{{\bar{g}}} ) + O ( | {\bar{\nabla }} h |_{{\bar{g}}} ) \right] ^2 + {\bar{H}} \left[ | {\bar{\displaystyle {\mathbb{I}\mathbb{I}}}} |_{{\bar{g}}} \, O ( | h |_{{\bar{g}}} ) + O ( | {\bar{\nabla }} h |_{{\bar{g}}} ) \right] \\ =&\ {\bar{H}}^2 + | {\bar{\displaystyle {\mathbb{I}\mathbb{I}}}} |_{{\bar{g}}}^2 \, O ( | h |_{{\bar{g}}} ) + {\bar{H}} \, O ( | {\bar{\nabla }} h |_{{\bar{g}}} ) + O ( | {\bar{\nabla }} h |_{{\bar{g}}}^2 ). \end{aligned}\nonumber \\ \end{aligned}$$

(2.10)

Let $d \sigma _g$, $d \sigma _{{\bar{g}}}$ denote the area form on $\Sigma _s$ with respect to g, ${\bar{g}}$, respectively. Then

$$\begin{aligned} \begin{aligned} \int _{\Sigma _s} H^2 \, d \sigma _{ g} \le&\ C \int _{\Sigma _s} H^2 \, d \sigma _{ {\bar{g}}} \\ =&\ C \, {\bar{W}} (\Sigma _s) + \left[ | {\bar{\displaystyle {\mathbb{I}\mathbb{I}}}} |_{{\bar{g}}}^2 \, O ( | h |_{{\bar{g}}} ) + {\bar{H}} \, O ( | {\bar{\nabla }} h |_{{\bar{g}}} ) + O ( | {\bar{\nabla }} h |_{{\bar{g}}}^2 ) \right] \, | \Sigma _s |_{{\bar{g}}}. \end{aligned}\nonumber \\ \end{aligned}$$

(2.11)

Plugging in ${\bar{W}} (\Sigma _s) = 4 {a'}^2 | \Sigma |_\sigma$, $| {\bar{\displaystyle {\mathbb{I}\mathbb{I}}}} |^2_{{\bar{g}}} = 2 a^{-2} {a'}^2$, and ${\bar{H}}^2 = 4 a^{-2} {a'}^2$, we have

$$\begin{aligned} \begin{aligned} W (\Sigma _s) \le&\ C \, | \Sigma |_{\sigma } \left[ {a'}^2 + {a'}^2 \, | h |_{{\bar{g}}} + a a' | {\bar{\nabla }} h |_{{\bar{g}}} + a^2 | {\bar{\nabla }} h |_{{\bar{g}}}^2 \right] \\ \le&\ C \, | \Sigma |_{\sigma } \left[ {a'}^2 ( 1 + | h |_{{\bar{g}}}) + a^2 | {\bar{\nabla }} h |_{{\bar{g}}}^2 \right] . \end{aligned} \end{aligned}$$

(2.12)

As $| h |_{{\bar{g}}}$ is bounded by assumption, it follows from (2.7), (2.8) and (2.12) that

$$\begin{aligned} \mathfrak {c}_{_{( \Sigma _s, \Sigma _\delta ) } } W (\Sigma _s)^\frac{1}{2} \le C \left( \int _s^\delta a(x)^{-2} \, d x \right) ^{-1} \left[ |a'| + a | {\bar{\nabla }} h |_{{\bar{g}}} \right] . \end{aligned}$$

(2.13)

(2.6) now follows from Propositions 2.1 and 2.3. $\square$

Remark 2.3

The negative mass Schwarzschild manifolds are known to have an $r^b$-horn type singularity with $b = \frac{2}{3}$ (see [5, 21] for instance). In [5], Bray and Jauregui developed a theory of “zero area singularities” (ZAS) modeled on the singularity of these manifolds. Among other things, they introduced a notion of the mass of ZAS. In [19, Theorem 4.8], Robbins showed the ADM mass of an asymptotically flat 3-manifold with a single ZAS is at least the ZAS mass. The conclusion on the $r^b$-horn type singularity in Remark 2.2 can also be reached via the results on ZAS in [5, 19].

We end this section by applying (1.1) to obtain information of $W( \cdot )$ in the negative mass Schwarzschild manifolds.

Proposition 2.4

Consider a spatial Schwarzschild manifold with negative mass, i.e.

$$\begin{aligned} (M_{\mathfrak {m}}, g_{\mathfrak {m}} ) = \left( (0, \infty ) \times S^2, \frac{1}{ 1 + \frac{2m}{r} } dr^2 + r^2 \sigma _o \right) , \end{aligned}$$

where $(S^2, \sigma _o)$ denotes the standard unit sphere and the mass $\mathfrak {m}= - m$ is negative. Let $\Sigma \subset M_{\mathfrak {m}}$ be any connected, closed surface that is homologous to a slice $\{r \} \times S^2$. Let $r_{max} (\Sigma ) = \max _{_{ x \in \Sigma } } \, r (x).$ Then

$$\begin{aligned} W(\Sigma ) \ge 16 \pi \left( 1 + \frac{2m}{ r_{max} (\Sigma ) } \right) . \end{aligned}$$

(2.14)

In particular, $W (\Sigma ) \rightarrow \infty$ as $r_{max} (\Sigma ) \rightarrow 0$.

Proof

(1.1) implies

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

(2.15)

Let $\Sigma _* = \{ r_{max} (\Sigma ) \} \times \mathbb {S}^2$. As $\Sigma _*$ encloses $\Sigma$,

$$\begin{aligned} \mathfrak {c}_{_{\Sigma _*}} \ge \mathfrak {c}_{_\Sigma }. \end{aligned}$$

(2.16)

Since $\mathfrak {m}= - m < 0$, the above implies

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

(2.17)

Direct calculation gives

$$\begin{aligned} \mathfrak {c}_{_{\Sigma _*} } = \frac{ m }{\left( 1 + \frac{2m}{ r_{max} (\Sigma ) } \right) ^\frac{1}{2} - 1 }. \end{aligned}$$

(2.18)

(2.14) follows from (2.17) and (2.18). $\square$

Proposition 2.4 gives another perspective of the singularity of $(M_{\mathfrak {m}}, g_{\mathfrak {m}})$ via the Willmore functional $W(\cdot )$.

3 Bounding $\mathfrak {m}$ via $\mathfrak {c}(M,g)$

Let (M, g) be given in the definition of $\mathfrak {c}(M,g)$ in (1.8). In this section, we relate $\mathfrak {m}$ and $2 \mathfrak {c}(M,g)$ assuming $\mathfrak {c}(M,g) > 0$. We begin with a characterization of $\mathfrak {c}(M, g) > 0$ which follows from standard arguments on harmonic functions.

Proposition 3.1

Let $\mathfrak {c}(M,g)$ be defined in (1.8). Then $\mathfrak {c}(M, g) > 0$ if and only if there exists a harmonic function w on (M, g) such that $0< w < 1$ on M and $w (x) \rightarrow 1$ at $\infty$ (i.e. as $x \rightarrow p$).

Proof

For each k, let $u_k$ be the harmonic function on $(E_{_{\Sigma _k} }, g)$ with $u_k \rightarrow 1$ as $x \rightarrow \infty$ and $u_k = 0$ at $\Sigma _k$. Given any surface $S \in \mathcal {S}$, by the maximum principle, $\{ u_k \}$ forms an increasing sequence in the exterior of S relative to $\infty$ (i.e. in $D_{_S} \setminus \{ p \}$). Interior elliptic estimates imply $\{ u_k \}$ converges to a harmonic function $u_\infty$ on M uniformly on compact sets in any $C^i$-norm. The limit $u_\infty$ satisfies $0 < u_\infty \le 1$ and $u_\infty \rightarrow 1$ as $x \rightarrow \infty$. By the strong maximum principle, either $u_\infty \equiv 1$ or $0< u_\infty < 1$.

Suppose (M, g) admits a harmonic w with $0< w < 1$ and $w \rightarrow 1$ at $\infty$. Then w is an upper barrier for $\{ u_k \}$, which implies $u_\infty \le w$, and hence $0< u_\infty < 1$. In this case, $\mathfrak {c}(M, g)$ must be positive. Otherwise, if $\mathfrak {c}(M,g) = 0$, then $\lim _{k \rightarrow \infty } \int _{E_{_{\Sigma _k}} } | \nabla u_k |^2 = 0$, which would imply $\int _{K} | \nabla u_\infty |^2 = 0$ on any compact set K in M, and hence $u_\infty \equiv 1$, a contradiction.

Next suppose $\mathfrak {c}(M, g) > 0$. We want to show $0< u_\infty < 1$. If not, $u_\infty \equiv 1$ on M. Pick any surface $S \in \mathcal {S}$, then $\lim _{k \rightarrow \infty } u_k = 1$ at S. Let $\beta _k = \min _{S} u_k$ for large k. Let ${{\tilde{u}}}_k$ be the harmonic function on $E_{_S}$ with ${{\tilde{u}}}_k \rightarrow 1$ at $\infty$ and ${{\tilde{u}}}_k = \beta _k$ at S. By the maximum principle, $u_k \ge {{\tilde{u}}}_k$. Therefore, ${{\tilde{c}}}_k \ge c_{k}$, where $\tilde{c}_k$, $c_k$ are the coefficients in the expansions

$$\begin{aligned} {{\tilde{u}}}_k= & {} 1 - {{\tilde{c}}}_k |x|^{-1} + o ( | x|^{-1} ),\\ u_k= & {} 1 - c_k |x|^{-1} + o ( |x|^{-1} ), \end{aligned}$$

as $x \rightarrow \infty$. Here we have $c_k = \mathfrak {c}_{_{S_k}}$ and

$$\begin{aligned} 4 \pi {{\tilde{c}}}_k = \lim _{r \rightarrow \infty } \int _{|x| = r} \frac{\partial {{\tilde{u}}}_k}{\partial \nu } = \int _{S } \frac{\partial {{\tilde{u}}}_k}{\partial \nu }, \end{aligned}$$

where $\nu$ denotes the corresponding unit normal pointing to $\infty$. Elliptic boundary estimates applied to $w_k = {{\tilde{u}}}_k - \beta _k$ shows

$$\begin{aligned} \lim _{k \rightarrow \infty } \max _{S} | \nabla w_k | = 0. \end{aligned}$$

Consequently, ${{\tilde{c}}}_k \rightarrow 0$ as $k \rightarrow \infty$. Combined with ${{\tilde{c}}}_k \ge \mathfrak {c}_k$, this shows

$$\begin{aligned} \mathfrak {c}(M,g) = \lim _{ k \rightarrow \infty } \mathfrak {c}_{_{S_k}} = 0, \end{aligned}$$

which is a contradiction. Therefore, $0< u_\infty < 1$. This completes the proof. $\square$

Remark 3.1

One may further require w satisfies $\inf _M w = 0$ in Proposition 3.1. To see this, it suffices to examine the proof beginning with assuming $\mathfrak {c}(M,g) > 0$. In this case, we have shown $0< u_\infty < 1$ on M. Suppose $\inf _M u_\infty > 0$, consider $v = ( 1 - \inf _M u_\infty )^{-1} ( u_\infty - \inf _M u_\infty )$. Then $v < u_\infty$ and v also acts as a barrier for $\{ u_k \}$, which implies $u_\infty \le v$, a contradiction. Hence, $\inf _M u_\infty = 0$.

Next, we focus on the case in which the function u tends to zero at “the other end”.

Proposition 3.2

Suppose there is a harmonic function u on (M, g) with $0< u < 1$, $u (x) \rightarrow 1$ at $\infty$ (i.e. as $x \rightarrow p$), and $\lim _{k \rightarrow \infty } \max _{\Sigma _k} u = 0$. Then

$$\begin{aligned} \mathfrak {c}(M, g) = \mathcal {C}, \end{aligned}$$

(3.1)

where $\mathcal {C} > 0$ is the coefficient in the expansion of

$$\begin{aligned} u = 1 - \mathcal {C}|x|^{-1} + o ( | x|^{-1} ) \end{aligned}$$

in the AF end.

Proof

Let $u_k$ be the harmonic function on $(E_{_{\Sigma _k}}, g)$ with $u_k \rightarrow 1$ at $\infty$ and $u_k = 0$ at $\Sigma _k$. Then $u_k \le u$, which implies $\mathcal {C} \le c_k$, where $c_k = \mathfrak {c}_{_{\Sigma _k}}$ is the coeffiicent in

$$\begin{aligned} u_k = 1 - c_k |x|^{-1} + o ( |x|^{-1} ) \end{aligned}$$

as $x \rightarrow \infty$. This shows

$$\begin{aligned} \mathcal {C}\le \lim _{k \rightarrow \infty } \mathfrak {c}_{_{\Sigma _k} } = \mathfrak {c}(M,g). \end{aligned}$$

To show the other direction, consider $\alpha _k = \max _{\Sigma _k} u$. On $E_{_{\Sigma _k}}$, by the maximum principle, $\frac{ 1}{1 - \alpha _k} ( u - \alpha _k ) \le u_k$, which implies $\frac{1}{1 - \alpha _k} C \ge c_k$. As $\alpha _k \rightarrow 0$, this gives

$$\begin{aligned} \mathcal {C}\ge \lim _{ k \rightarrow \infty } ( 1 - \alpha _k ) \mathfrak {c}_{_{\Sigma _k} } = \mathfrak {c}(M, g). \end{aligned}$$

Therefore, $\mathcal {C}= \mathfrak {c}(M, g)$. $\square$

We are now in a position to derive applications of (1.4).

Theorem 3.1

Let N be a noncompact, connected, orientable 3-manifold admitting an exhaustion sequence of precompact domains $D_k$ with connected boundary $\partial D_k$, $k = 1, 2, \ldots$. Suppose $H_2 (N) = 0$. Let $M = N \setminus \{ p \}$ where p is a point in N. Let g be a smooth metric with nonnegative scalar curvature on M such that g is asymptotically flat near p. Assume there is a harmonic function u on (M, g) with $0< u < 1$, $u (x) \rightarrow 1$ as $x \rightarrow p$, and $\displaystyle \lim _{k \rightarrow \infty } \max _{{\partial D_k}} u = 0$. Then

• (i) The limit $\displaystyle \lim _{t \rightarrow 0} \int _{u^{-1} (t) } | \nabla u |^2$ exists (finite or $\infty$), where $t \in (0,1)$ is a regular value of u; and

• (ii) $\displaystyle \mathfrak {m}\ge 2 \mathfrak {c}(M,g) \left[ 1 - \lim _{t \rightarrow 0} \left( \frac{1}{4 \pi } \int _{u^{-1} (t)} | \nabla u |^2 \right) ^\frac{1}{2} \right]$.

Proof

Given a regular value $t \in (0,1)$ of u, let $\Sigma _t = u^{-1} (t)$. $\Sigma _t$ is a closed, orientable surface in $M = N {\setminus } \{ p \}$. Let $\Sigma _t^{(1)}$ denote any connected component of $\Sigma _t$. Since $H_2 (N) = 0$, $\Sigma _t^{(1)}$ is the boundary of a bounded domain $\Omega _1$ in N. If $p \notin \Omega _1$, then u is identically a constant by the maximum principle. Hence, $\Sigma _t^{(1)}$ encloses p. As a result, if there are two connected components of $\Sigma _t$, then both of them enclose p, and thus form the boundary of a bounded domain in M. By the maximum principle, u is a constant, which is a contradiction. Therefore, $\Sigma _t$ is connected. Since t is arbitrary, this in particular shows (1.4) is applicable to $(E_t, g)$, where $E_t = \{ u \ge t \} \subset M$ is the exterior of $\Sigma _t$ with respect to $\infty$.

Applying (1.4) to $(E_t, g)$, we have

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

(3.2)

Here $u_t = \frac{1 }{1-t} ( u - t)$ is the harmonic function on $(E_t, g)$ that tends to 1 at $\infty$ and equals 0 at $\Sigma _t$, $\mathfrak {c}_{_{\Sigma _t} } = \frac{1}{1-t} \mathcal {C}$, and $\mathcal {C}$ is the coefficient in the expansion of

$$\begin{aligned} u = 1 - \mathcal {C}|x|^{-1} + o ( | x|^{-1} ). \end{aligned}$$

It follows from (3.2) that

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

(3.3)

Consider the function

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

In [16, Theorem 3.2 (ii)], we showed $\mathcal {B}(t)$ is monotone nondecreasing in t if g has nonnegative scalar curvature. As a result,

$$\begin{aligned} \lim _{t \rightarrow 0 } \mathcal {B} (t) \ \text {exists}. \end{aligned}$$

Consequently,

$$\begin{aligned} \lim _{t \rightarrow 0 } \int _{\Sigma _t} | \nabla u |^2 \ \text {exists}. \end{aligned}$$

This proves (i). (ii) follows from (3.3), (i) and Proposition 3.2. $\square$

We have not assumed g to be complete on M so far. In particular, (M, g) in Theorem 3.1 could just be the interior of an AF manifold with boundary $\Sigma$ and the function u may simply be the restriction, to the interior, of the harmonic function that tends to 1 at $\infty$ and vanishes at $\Sigma$. In that extreme case, $\lim _{t \rightarrow 0 } \int _{\Sigma _t} | \nabla u |^2 = \int _{\Sigma } | \nabla u |^2$ and (ii) reduces to (1.4).

If g is complete on M, we have the following corollary.

Corollary 3.1

Let N, p, M, g and u be given as in Theorem 3.1. Suppose (M, g) is complete and has Ricci curvature bounded from below. Then

$$\begin{aligned} \mathfrak {m}\ge 2 \mathcal {C}, \end{aligned}$$

(3.4)

where $\mathcal {C}= \mathfrak {c}(M,g)$ is the coefficient in the expansion of

$$\begin{aligned} u = 1 - \mathcal {C}|x|^{-1} + o ( | x|^{-1} ) \end{aligned}$$

as $x \rightarrow p$.

Corollary 3.1 relates to a result of Bray [4]. In [4, Theorem 8], Bray proved that, if (M, g) is a complete asymptotically flat 3-manifold with nonnegative scalar curvature which has multiple AF ends and mass $\mathfrak {m}$ in a chosen end, then

$$\begin{aligned} \mathfrak {m}\ge 2 \mathcal {C}, \end{aligned}$$

where $\mathcal {C}$ is the coefficient in the expansion $u = 1 - \mathcal {C}|x|^{-1} + o ( | x|^{-1} )$ at the chose end, and u is the harmonic function that tends to 1 at the chosen end and approaches 0 at all other AF ends.

Bray’s theorem allows M to have more general topology and more than two ends. Its proof made use of the 3-d PMT. Complete manifolds whose ends are all asymptotically flat necessarily have bounded Ricci curvature. In this sense, Corollary 3.1 provides a partial generalization of Bray’s result.

Proof of Corollary 3.1

Let $\Sigma _t$ be given in the proof of Theorem 3.1. Since (M, g) is complete and has Ricci curvature bounded from below, by the gradient estimate of Cheng and Yau [8], $\max _{\Sigma _t} \, u^{-1} | \nabla u | \le \Lambda$ where $\Lambda$ is a constant independent on t. Combined with $\int _{\Sigma _t} | \nabla u | = 4 \pi \mathcal {C}$, this shows

$$\begin{aligned} \frac{1}{4\pi } \int _{\Sigma _t} | \nabla u |^2 \le \mathcal {C}\, \Lambda t, \end{aligned}$$

which implies

$$\begin{aligned} \lim _{t \rightarrow 0} \int _{\Sigma _t} | \nabla u |^2 = 0. \end{aligned}$$

(3.5)

It follows from (3.5) and Theorem 3.1 (ii) that $\mathfrak {m}\ge 2 \mathcal {C}$. $\square$

Remark 3.2

Let R, $\text{ Ric }$ denote the scalar curvature, Ricci curvature of g. Since

$$\begin{aligned} R \ge 0 \ \text {and} \ \text{ Ric }\ \text {bounded from above} \ \Rightarrow \ \text{ Ric }\ \text {bounded from below}, \end{aligned}$$

Corollary 3.1 also holds if the assumption of “$\text{ Ric }$ bounded from below” is replaced by “$\text{ Ric }$ bounded from above”.

Remark 3.3

As used in Bray’s work [4], the inequality $\mathfrak {m}\ge 2 \mathcal {C}$ has a geometric interpretation that asserts the mass of the conformally deformed metric $u^4 g$, which might not be complete, is nonnegative. Instead of $\mathfrak {m}\ge 2 \mathcal {C}$, a weaker inequality $\mathfrak {m}\ge \mathcal {C}$ was obtained by Hirsch, Tam and the author in [12].