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.
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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
for some \( \tau > \frac{1}{2}\); and the scalar curvature of g is integrable. Under these AF conditions, the limit, near \(\infty \),
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
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
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
then, \( \displaystyle \mathfrak {c}_{_\Sigma } = \frac{1}{4 \pi } \int _M | \nabla \phi |^2 = \frac{1}{4\pi } \int _{\Sigma } | \nabla \phi | \), and
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
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
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
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
and
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
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 \( r \rightarrow 0\),
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
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
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.
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
Here \( \mathfrak {c}_{_S}\) is the capacity of S in the asymptotically flat \((E_{_S}, g)\), where
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
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.)
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
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
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
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
Proof
If \( W(S) \ge 16 \pi \), then (1.5) implies
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
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
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 \(k \rightarrow \infty \),
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
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
where
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\(\delta > 0\) 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
Remark 2.2
If \( a (r) = r^b\) for a constant \( b >0\), then (2.6) translates to
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
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
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,
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
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
(For instance, see formula (2.33) in [17] and the proof therein.) Therefore,
Let \( d \sigma _g\), \( d \sigma _{{\bar{g}}}\) denote the area form on \(\Sigma _s\) with respect to g, \({\bar{g}}\), respectively. Then
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
As \( | h |_{{\bar{g}}} \) is bounded by assumption, it follows from (2.7), (2.8) and (2.12) that
(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.
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
In particular, \( W (\Sigma ) \rightarrow \infty \) as \( r_{max} (\Sigma ) \rightarrow 0\).
Proof
(1.1) implies
Let \( \Sigma _* = \{ r_{max} (\Sigma ) \} \times \mathbb {S}^2\). As \( \Sigma _* \) encloses \( \Sigma \),
Since \( \mathfrak {m}= - m < 0 \), the above implies
Direct calculation gives
(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
as \(x \rightarrow \infty \). Here we have \(c_k = \mathfrak {c}_{_{S_k}} \) and
where \( \nu \) denotes the corresponding unit normal pointing to \(\infty \). Elliptic boundary estimates applied to \(w_k = {{\tilde{u}}}_k - \beta _k\) shows
Consequently, \( {{\tilde{c}}}_k \rightarrow 0\) as \( k \rightarrow \infty \). Combined with \( {{\tilde{c}}}_k \ge \mathfrak {c}_k \), this shows
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
where \(\mathcal {C} > 0 \) is the coefficient in the expansion of
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
as \(x \rightarrow \infty \). This shows
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
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
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
It follows from (3.2) that
Consider the function
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,
Consequently,
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
where \( \mathcal {C}= \mathfrak {c}(M,g)\) is the coefficient in the expansion of
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
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
which implies
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
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].
References
Agostiniani, V., Mazzieri, L., Oronzio, F.: A Green’s function proof of the positive mass theorem. Commun. Math. Phys. 405, 54 (2024). https://doi.org/10.1007/s00220-024-04941-8
Arnowitt, R., Deser, S., Misner, C.W.: Coordinate invariance and energy expressions in general relativity. Phys. Rev. 122(3), 997–1006 (1961)
Bartnik, R.: The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. 39(5), 661–693 (1986)
Bray, H.L.: Proof of the Riemannian Penrose inequality using the positive mass theorem. J. Differ. Geom. 59(2), 177–267 (2001)
Bray, H.L., Jauregui, J.L.: A geometric theory of zero area singularities in general relativity. Asian J. Math. 17(3), 525–559 (2013)
Bray, H.L., Kazaras, D.P., Khuri, M.A., Stern, D.L.: Harmonicfunctions and the mass of 3-dimensional asymptotically flat Riemannian manifolds. J. Geom. Anal. 32(6), 184 (2022)
Bray, H., Miao, P.: On the capacity of surfaces in manifolds with nonnegative scalar curvature. Invent. Math. 172(3), 459–475 (2008)
Cheng, S.-Y., Yau, S.-T.: Differential equations on Riemannian manifolds and their geometric applications. Commun. Pure Appl. Math. 28(3), 333–354 (1975)
Chruściel, P.: Boundary Conditions at Spatial Infinity from a Hamiltonian Point of View. Topological Properties and Global Structure of Space-Time, pp. 49–59. Plenum Press, New York (1986)
Dai, X., Sun, Y., Wang, C.: Positive mass theorem for asymptotically flat manifolds with isolated conical singularities, arXiv:2401.07186
Hawking, S.W.: Gravitational radiation in an expanding universe. J. Math. Phys. 9, 598–604 (1968)
Hirsch, S., Tam, L. -F., Miao, P.: Monotone quantities of p-harmonic functions and their applications, to appear in Pure Appl. Math. Q., arXiv:2211.06939
Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59(3), 353–437 (2001)
Jauregui, J.L.: Scalar curvature and the relative capacity of geodesic balls. Proc. Am. Math. Soc. 149(11), 4907–4921 (2021)
Li, C., Mantoulidis, C.: Positive scalar curvature with skeleton singularities. Math. Ann. 374, 99–131 (2019)
Miao, P.: Mass, capacitary functions, and the mass-to-capacity ratio. Peking Math J (2023). https://doi.org/10.1007/s42543-023-00071-7
Miao, P., Piubello, A.: Mass and Riemannian polyhedra. Adv. Math. 400, 108287 (2022). https://doi.org/10.1016/j.aim.2022.108287
Mondino, A., Templeton-Browne, A.: Some rigidity results for the Hawking mass and a lower bound for the Bartnik capacity. J. Lond. Math. Soc. 106, 1844–1896 (2022). https://doi.org/10.1112/jlms.12612
Robbins, N.: Zero area singularities in general relativity and inverse mean curvature flow. Class. Quantum Grav. 27, 025011 (2010)
Schoen, R., Yau, S.-T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65(1), 45–76 (1979)
Shi, Y.-G., Tam, L.-F.: Scalar curvature and singular metrics. Pacific J. Math. 293(2), 427–470 (2018)
Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80(3), 381–402 (1981)
Acknowledgements
I thank Christos Mantoulidis who introduced me to the work [18]. I also thank Professor Xianzhe Dai for several helpful communications.
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Miao, P. Implications of Some Mass-Capacity Inequalities. J Geom Anal 34, 241 (2024). https://doi.org/10.1007/s12220-024-01686-7
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DOI: https://doi.org/10.1007/s12220-024-01686-7