Zero area singularities

A geometric theory of zero area singularities
in general relativity

Hubert L. Bray Mathematics Department
Duke University, Box 90320
Durham, NC 27708-0320
bray@math.duke.edu
 and  Jeffrey L. Jauregui Department of Mathematics
David Rittenhouse Lab
209 South 33rd Street
Philadelphia, PA 19104
jauregui@math.upenn.edu
January 25, 2012
Abstract.

The Schwarzschild spacetime metric of negative mass is well-known to contain a naked singularity. In a spacelike slice, this singularity of the metric is characterized by the property that nearby surfaces have arbitrarily small area. We develop a theory of such “zero area singularities” in Riemannian manifolds, generalizing far beyond the Schwarzschild case (for example, allowing the singularities to have nontrivial topology). We also define the mass of such singularities. The main result of this paper is a lower bound on the ADM mass of an asymptotically flat manifold of nonnegative scalar curvature in terms of the masses of its singularities, assuming a certain conjecture in conformal geometry. The proof relies on the Riemannian Penrose inequality [bray_RPI]. Equality is attained in the inequality by the Schwarzschild metric of negative mass. An immediate corollary is a version of the positive mass theorem that allows for certain types of incomplete metrics.

The research of the first author was supported in part by NSF grant #DMS-0706794.

1. Introduction: the negative mass Schwarzschild metric

The first metrics one typically encounters in the study of general relativity are the Minkowski spacetime metric and the Schwarzschild spacetime metric, the latter given by

where is the time coordinate, are spatial spherical coordinates, and is some positive number. This represents the exterior region of a non-rotating black hole of mass in vacuum. A spacelike slice of this Lorentzian metric can be obtained by taking a level set of ; under a coordinate transformation , the resulting 3-manifold is isometric to minus the ball of radius about the origin, with the conformally flat metric

(1)

where is the usual flat metric on and . Its boundary is a minimal surface that represents the apparent horizon of the black hole. We refer to (1) as the Schwarzschild metric (of mass ).

Consider instead the metric with . This gives a Riemannian metric on minus a closed ball of radius about the origin that approaches zero near its inner boundary. One may loosely think of this manifold as a slice of a spacetime with a single “black hole of negative mass.” In fact, this metric has a naked singularity, as the singularity on the inner boundary is not enclosed by any apparent horizon. In this paper we introduce a theory of such “zero area singularities” (ZAS), modeled on the “Schwarzschild ZAS metric” (i.e., with ), yet far more general. Some of the problems we address are:

  1. When can such singularities be “resolved”?

  2. What is a good definition of the mass of such a singularity?

  3. Can the ADM mass of an asymptotically flat manifold of nonnegative scalar curvature be estimated in terms of the masses of its singularities?

The third question is motivated by the positive mass theorem [schoen_yau, witten] and Riemannian Penrose inequality [imcf, bray_RPI] (Theorems 1 and 2 below). The former states that, under suitable conditions, the ADM mass of an asymptotically flat 3-manifold is nonnegative, with zero mass occurring only for the flat metric on . The latter improves this to provide a lower bound on the ADM mass in terms of the masses of its “black holes.” Here, the case of equality is attained by the Schwarzschild metric with . See appendix A for details on asymptotic flatness and ADM mass.

The main theorem of this paper is the Riemannian ZAS inequality (see Theorems 31 and 32), introduced by the first author [bray_npms]. It is an analog of the Riemannian Penrose inequality, but for zero area singularities instead of black holes. Specifically, this inequality gives a lower bound for the ADM mass of an asymptotically flat manifold in terms of the masses of its ZAS. Its proof assumes an unproven conjecture (Conjecture 34) regarding the outermost minimal area enclosure of the boundary, with respect to a conformal metric. While the conjecture is known to be true in some cases, proving it remains an open problem. Although we shall write “the” Riemannian ZAS inequality in this paper, we remark that other similar inequalities may be discovered in the future that also deserve this title. Table 1 illustrates how this theorem fits together with the positive mass theorem and Riemannian Penrose inequality.

sign() metric unique case of equality of
Euclidean positive mass theorem
Schwarzschild metric Riemannian penrose inequality
Schwarzschild ZAS metric Riemannian ZAS inequality (Theorem 31)
Table 1. The Schwarzschild metric as a case of equality

For reference, we recall the following theorems. The geometric assumption of nonnegative scalar curvature is equivalent, physically, to the dominant energy condition (for totally geodesic slices of spacetimes).

Theorem 1 (Positive mass theorem [schoen_yau]).

Let be a complete asymptotically flat Riemannian 3-manifold (without boundary) of nonnegative scalar curvature with ADM mass . Then , with equality holding if and only if is isometric to with the flat metric.

Witten gave an alternative proof of Theorem 1 for spin manifolds [witten].

Theorem 2 (Riemannian Penrose inequality, Theorem 19 of [bray_RPI]).

Let be a complete asymptotically flat Riemannian 3-manifold with compact smooth boundary and nonnegative scalar curvature, with ADM mass . Assume that is minimal (i.e. has zero mean curvature), and let be the outermost minimal area enclosure of . Then , where is the area of . Equality holds if and only if is isometric to the Schwarzschild metric of mass outside of .

See appendix A for details on the outermost minimal area enclosure. Theorem 2 was first proved by Huisken and Ilmanen [imcf] with replaced by the area of the largest connected component of .

1.1. Negative mass in the literature

The concept of negative mass in both classical physics and general relativity has appeared frequently in the literature. The following list is a small sample of such articles and is by no means comprehensive.

  • Bondi discusses negative mass in Newtonian mechanics, distinguishing inertial mass, passive gravitational mass, and active gravitational mass [bondi]. He then proceeds to study a two-body problem in general relativity involving bodies with masses of opposite sign.

  • Bonnor considers Newtonian mechanics and general relativity under the assumption that all mass is negative [bonnor]. Included in the discussion are 1) the motion of test particles for a Schwarzschild spacetime of negative mass, 2) Friedmann-Robertson-Walker cosmology with negative mass density, and 3) charged particles of negative mass.

  • More recently, research has turned toward the question of stability of the negative mass Schwarzschild spacetime. Gibbons, Hartnoll, and Ishibashi studied linear gravitational perturbations to this metric and found it to be stable for a certain boundary condition on the perturbations [gibbons]. However, a separate analysis by Gleiser and Dotti reached a different conclusion, indicating the negative mass Schwarzschild spacetime to be perturbatively unstable for all boundary conditions [gleiser]. The papers are mathematically consistent with each other, with differences arising from subtleties pertaining to defining time evolution in a spacetime with a naked singularity. The issue of stability warrants further analysis, although we do not consider it here.

The present paper offers a new perspective on singularities arising from negative mass, extending past the Schwarzschild case. We will restrict our attention to the case of time-symmetric (i.e., totally geodesic), spacelike slices of spacetimes. This setting is a natural starting point, as it was for the positive mass theorem and Penrose inequality.

1.2. Overview of contents

Before providing an overview of the paper, we emphasize that the statements and proofs of the main theorems appear near the end, in section 5.

In section 2 we make precise the notion of zero area singularity. By necessity, this is preceded by a discussion of convergence for sequences of surfaces. Next, we define two well-behaved classes of ZAS: those that are “regular” and “harmonically regular.”

Section 3 introduces the mass of a ZAS: this is a numerical quantity that ultimately gives a lower bound on the ADM mass in the main theorem. Defining the mass for regular ZAS is straightforward; for arbitrary ZAS, formulating a definition requires more care. We discuss connections between the ZAS mass and the Hawking mass. Next, we define the capacity of a ZAS based on the classical notion of harmonic capacity. The important connection between mass and capacity is that if the capacity is positive, then the mass is .

Spherically symmetric metrics with zero area singularities are studied in section 4. In this simple setting, we explicitly compute the mass and capacity. An example is given that shows the concepts of regular and harmonically regular ZAS are distinct. Experts may prefer to skip this section, which is largely computational and detail-oriented.

The main two theorems, comprising two versions of the Riemannian ZAS inequality, are stated and proved (up to an unproven conjecture) in section 5. An immediate corollary is a version of the positive mass theorem for manifolds with certain types of singularities.

After providing one final example, we conclude with a discussion about several related open problems and conjectures. Two appendices follow which are referred to as needed.

1.3. Comments and acknowledgements

In 1997, the first author, just out of graduate school, sat next to Barry Mazur at a conference dinner at Harvard, who, quite characteristically, asked the first author a series of probing questions about his research, which at the time concerned black holes. One of the questions was “Can a black hole have negative mass, and if so, what properties would it have?” Contemplating this natural question marked the beginning of an enjoyable journey leading to this paper.

The first author initiated this work, originally presented at a conference in 2005 under the heading “Negative Point Mass Singularities” [bray_npms]. The second author commenced work on this project as a graduate student, and wrote his thesis on a closely related topic [jauregui]. He would like to thank Mark Stern and Jeffrey Streets for helpful discussions.

2. Definitions and preliminaries

In this paper, the singularities in question will arise as metric singularities on a boundary component of a manifold. To study the behavior of the metric near a singularity, we make extensive use of the idea of nearby surfaces converging to a boundary component.

Throughout this paper will be a smooth, asymptotically flat Riemannian 3-manifold, with compact, smooth, nonempty boundary (see appendix A for details on asymptotic flatness). We do not assume that extends smoothly to . We make no other restrictions on the topology of (e.g., connectedness, orientability, genus).

2.1. Convergence of surfaces

For our purposes, a surface in will always mean a , closed, embedded 2-manifold in the interior of that is the boundary of a bounded open region . (Note that is uniquely determined by .) We say that a surface encloses a surface if . If is a nonempty subcollection of the components of , we say that a surface encloses if is homologous to .

We next define what it means for a surface to be “close to” (with as above). Let be a neighborhood of that is diffeomorphic to for some . This gives a coordinate system on where and . If is a surface that can be parameterized in these coordinates as , then we say it is a “graph over ”; clearly such encloses . Whenever we discuss the convergence of surfaces, it will be implicit that the surfaces are graphs over .

Definition 3.

Let be a sequence of surfaces that are graphs over that can be parameterized as (see above). We say that converges in to if the functions converge to 0 in .

We emphasize that convergence in depends only on the underlying smooth structure of and not on the metric. As an example, in if for any open set containing , there exists such that for all . We shall not deal with convergence stronger than and will explain the significance of convergence in and as necessary.

2.2. Zero area singularities

We now give the definition of zero area singularity. Both the singular and plural will be abbreviated “ZAS.”

Definition 4.

Let be an asymptotically flat metric on . A connected component of is a zero area singularity (ZAS) of if for every sequence of surfaces converging in to , the areas of measured with respect to converge to zero.

Topologically, a ZAS is a boundary surface in , not a point. However, in terms of the metric, it is often convenient to think of a ZAS as a point formed by shrinking the metric to zero. For example, the boundary sphere of the Schwarzschild ZAS metric is a ZAS. Also, most notions of “point singularity” are ZAS (after deleting the point). A depiction of a manifold with ZAS is given in figure LABEL:fig_ex_zas.

ZAS could be defined for manifolds that are not asymptotically flat, but we do not pursue this direction.

Figure 1. A manifold with zero area singularities

In the case that extends continuously to the boundary, we have several equivalent conditions for ZAS:

Proposition 5.

Suppose is a component of to which extends continuously as a symmetric 2-tensor. The following are equivalent:

  1. is a ZAS of .

  2. has zero area measured with respect to (see below).

  3. For each point , has a null eigenvector tangent to at .

  4. There exists a sequence of surfaces converging in to such that converges to zero.

Here, is the area of measured with respect to .

A continuous, symmetric 2-tensor on a surface that is positive semi-definite can be used to compute areas by integrating the 2-form defined locally in coordinates by .

Proof.

The proof is an immediate consequence of the following observations:

  • If extends continuously to , then for any sequence of surfaces converging in to , the areas converge: .

  • If is the restriction of to the tangent bundle of , then at if and only if has an eigenvector with zero eigenvalue at .

In general, it is not necessary that extend continuously to the boundary in the definition of ZAS.

2.3. Resolutions of regular singularities

We now discuss what it means to “resolve” a zero area singularity. An important case of ZAS occurs when a smooth metric on is deformed by a conformal factor that vanishes on the boundary [bray_npms]:

Definition 6.

Let be a ZAS of . Then is regular if there exists a smooth, nonnegative function and a smooth metric , both defined on a neighborhood of , such that

  1. vanishes precisely on ,

  2. on , where is the unit normal to (taken with respect to and pointing into the manifold), and

  3. on .

If such a pair exists, it is called a local resolution of .

The significance of the condition is explained further in Lemma 11 and is crucial in the proof of Proposition 12. As an example, the Schwarzschild ZAS with is a regular ZAS with a local resolution , where is the flat metric and . A graphical depiction of a local resolution is given in figure LABEL:fig_resolution.

Figure 2. A resolution of a regular ZAS

Much of our work utilizes a nicer class of singularities: those for which the resolution function can be chosen to be harmonic.

Definition 7.

A regular ZAS of is said to be harmonically regular if there exists a local resolution such that is harmonic with respect to . Such a pair is called a local harmonic resolution.

In the case of a local harmonic resolution, the condition holds automatically by the maximum principle. We remark that if one local resolution (or local harmonic resolution) exists, then so do infinitely many.

The Schwarzschild ZAS is harmonically regular, since the function is harmonic with respect to the flat metric on . In section 4 we give examples of ZAS that are not regular and ZAS that are regular but not harmonically regular.

If several components of are (harmonically) regular ZAS, then there is a natural notion of a local (harmonic) resolution of the union of these components: in Definition 6, simply replace with .

Since our ultimate goal—the Riemannian ZAS inequality—is a global geometric statement, we require resolutions that are globally defined.

Definition 8.

Suppose all components of are harmonically regular ZAS. Then the pair is a global harmonic resolution of if

  1. is a smooth, asymptotically flat metric on ,

  2. is the - harmonic function on vanishing on and tending to one at infinity, and

  3. on .

For example, the aforementioned resolution of the Schwarzschild ZAS is a global harmonic resolution. In general, if consists of harmonically regular ZAS, it is not clear that a global harmonic resolution exists; however, this is known to be true:

Proposition 9 (Theorem 58 of [jauregui]).

If is a collection of harmonically regular ZAS in , then admits a global harmonic resolution.

3. Mass and capacity of ZAS

In a time-symmetric (i.e., totally geodesic) spacelike slice of a spacetime, we adopt the viewpoint that black holes may be identified with apparent horizons. An apparent horizon is defined to be a connected component of the outermost minimal surface in the spacelike slice. If is the area of an apparent horizon , then its mass (or “black hole mass”) is defined to be . This definition has physical [penrose] and mathematical [imcf, bray_RPI] motivation; it also equals for the apparent horizon in the Schwarzschild metric of mass . We note the black hole mass is also given by the limit of the Hawking masses of a sequence of surfaces converging in to the apparent horizon. Recall the Hawking mass of any surface in is given by

where is the area of with respect to , is the mean curvature of , and is the area form on induced by . The significance of convergence is explained in the proof of Proposition 13 below.

Defining the mass of a ZAS, on the other hand, is not as straightforward, since the metric becomes degenerate and potentially loses some regularity at the boundary. For regular ZAS, it is possible to define mass in terms of a local resolution in such a way as to not depend on the choice of local resolution. In the general case, defining the mass is more involved. We first consider the regular case.

3.1. The mass of regular ZAS

Following [bray_npms], we define the mass of a regular ZAS:

Definition 10.

Let be a local resolution of a ZAS of . Then the regular mass of is defined by the integral

(2)

where is the unit normal to (pointing into the manifold) and is the area form induced by .

The advantages of this definition are that it

  1. is independent of the choice of local resolution (Proposition 12),

  2. depends only on the local geometry of near (Proposition 13),

  3. is related to the Hawking masses of nearby surfaces (Proposition 14),

  4. arises naturally in the proof of the Riemannian ZAS inequality (Theorem 31), and

  5. equals for the Schwarzschild ZAS metric of ADM mass (left to the reader).

Before elaborating on these issues, we take a moment to describe how the masses of regular ZAS add together. If are regular ZAS and , then applying to gives

(3)

This is analogous with the case of black holes: if are apparent horizons with black hole masses , and if the black hole mass of their union is defined to be (c.f. [penrose]), then

Next, we show the regular mass is well-defined. (This was also proved in [bray_npms, robbins].) First, we require the following lemma:

Lemma 11.

If and are two local resolutions of a regular ZAS , then the ratio extends smoothly to as a strictly positive function.

Proof.

Note that and vanish on and have nonzero normal derivative there. The proof follows from considering Taylor series expansions for coordinate expressions of and near . ∎

Proposition 12.

The definition of is independent of the choice of local resolution.

Proof.

Let and be two local resolutions of , defined on a neighborhood of . Then on ,

By Lemma 11, is smooth and positive on . In particular, on . This allows us to compare area elements and unit normals on in the metrics and :

We show the integrals in are the same whether computed for or .

where the cancellation occurs because vanishes on . ∎

The next few results serve to: give alternate characterizations of the regular mass, relate the regular mass to the Hawking mass, and provide motivation for the definition of the mass of an arbitrary ZAS. We have placed an emphasis on the Hawking mass (versus other quasi-local mass functionals) due to its relevance to the Riemannian Penrose inequality [imcf] and its role in the proof of the Riemannian ZAS inequality for the case of a single ZAS [robbins].

Proposition 13.

Let be a subset of consisting of regular ZAS of . If is a sequence of surfaces converging in to , then

In particular, the right side is independent of the choice of sequence, and the left side depends only on the local geometry of near .

Proof.

Let be some local resolution of . Apply formula in appendix B for the change in mean curvature of a hypersurface under a conformal change of the ambient metric. Below, and are the mean curvatures of in the metrics and , respectively.

Now, take of both sides, and use the facts that vanishes on and the convergence of ensures that the mean curvature of in is uniformly bounded as to deduce:

A similar result is now given for the Hawking mass; the proof also appears in [robbins].

Proposition 14.

Let be a subset of consisting of regular ZAS of . If is a sequence of surfaces converging in to , then

(4)

Moreover, there exists a sequence of surfaces converging in to such that

Proof.

The first part is an application of Hölder’s inequality:

(definition of Hawking mass)
(Hölder’s inequality)

Inequality follows by taking and applying Definition 4 and Proposition 13.

We now construct the sequence . We first argue that there exists a local resolution such that on . Let be some local resolution defined in a neighborhood of . Let be a positive -harmonic function with Dirichlet boundary condition given by on . Set and ; we claim is the desired local resolution.

First, note that is a smooth metric on , since is positive and smooth. Next, , and vanishes only on . Now, we compute the normal derivative of on :

by the boundary condition imposed on . Thus, is the desired local resolution. (We remark that if is a local harmonic resolution, then so is . This follows from equation in appendix B and will be used in the proof of Proposition 17.)

Define to be the level set of , which is smooth and well-defined for all sufficiently large. It is clear that converges to in all as .

Since equality is attained in Hölder’s inequality for constant functions, the proof is complete if we show the ratio of the minimum and maximum values of (the mean curvature of , measured in ) tends to 1 as . From equation in appendix B, is given by

(5)

where is the mean curvature of in . By convergence, is bounded as . By convergence, converges to 1 as . In particular, the second term in dominates. Since is by definition constant on , we have proved the claim. ∎

The following corollary of Proposition 14 will be pertinent when discussing the mass of arbitrary ZAS.

Corollary 15.

If is a subset of consisting of regular ZAS of , then

where the supremum is taken over all sequences converging in to .

It is necessary to take the supremum, since evidently underestimates the regular mass in general.

To summarize, we have seen several expressions for the regular mass as the:

  1. explicit formula (2) in terms of any local resolution,

  2. limit of ,

  3. limit of the Hawking masses of a certain sequence of surfaces, and

  4. of the of the Hawking masses of sequences converging to .

3.2. The mass of arbitrary ZAS

For simplicity, we assume from this point on that all components of are ZAS of . We shall define only the mass of (not the mass of each component of ). A good definition of the mass of should depend only on the local geometry near and agree with the regular mass in the case that the components of are regular (or at least harmonically regular). There are three immediate candidates for the definition of the mass of : the supremum over all sequences converging in some to of

  1. (inspired by Corollary 15),

  2. (inspired by Proposition 13), and

  3. , where is viewed as a regular ZAS that “approximates” . (This is explained below; see also figure LABEL:fig_approx.)

The first two candidates manifestly depend only on the local geometry near and agree with the regular mass for regular ZAS (by Corollary 15 and Proposition 13). In fact, the second is greater than or equal to the first (an application of Hölder’s inequality).

To explain the third quantity above, we show that each surface is naturally a collection of ZAS (with respect to a new metric). Let be the region enclosed by , and let be the unique -harmonic function that vanishes on and tends to one at infinity. Then is an asymptotically flat metric on the manifold . Moreover, is a collection of harmonically regular ZAS for this manifold. This construction is demonstrated in figure LABEL:fig_approx; essentially this process approximates any ZAS with a sequence of harmonically regular ZAS. By construction, is a global harmonic resolution of , so the regular mass of is computed in this resolution as

(6)

where the unit normal and area form are taken with respect to .

Figure 3. Approximating a ZAS by harmonically regular ZAS

We adopt the third candidate for our definition of mass; Proposition 17 and Corollary 29 show this definition agrees with the regular mass for harmonically regular ZAS, and Proposition 27 shows it depends only on the local geometry near . The relationship between the third and first two candidates for the definition of mass is unknown; see section 6.4. Another justification for our choice of the definition of mass is that it naturally gives a lower bound on the ADM mass of (see Theorem 32). It is currently unknown how to obtain such a lower bound in terms of the first two candidates for the mass of .

Definition 16.

Let be zero area singularities of . The mass of is

where the supremum is taken over all sequences converging in to and is given by equation .

Note that while the regular mass of a regular ZAS is a negative real number, takes values in . In section 4 we provide examples for which and . The requirement that the sequences converge in is explained in the proof of the following result.

Proposition 17.

If admits a global harmonic resolution, then the two definitions of mass agree. That is, .

Proof.

Let have a global harmonic resolution . Let be a collection of smooth level sets of that converge in to , and let be -harmonic, vanishing on and tending to 1 at infinity. Now we compute a convenient expression for the regular mass of :

(expression for )
( and )
(7) ( on ).

We claim that the limit of the above equals . Let be the (constant) value of on . Let be the closure of the region exterior to . From formula in appendix B, the function is -harmonic in , zero on and 1 at infinity. Also, is -harmonic on , zero on and at infinity. In particular, by the uniqueness of harmonic functions with identical boundary values, we see that

It follows that

(8)

Continuing with equations (7), and taking , we have

(eqn. (8))

since and in . Then by definition of ZAS mass,

for ZAS admitting a global harmonic resolution. Proposition 18 below gives the reverse inequality. ∎

As a consequence of this result, we may interchangeably use the terms “mass” and “regular mass” whenever a global harmonic resolution exists; by Proposition 9, this is the case for all harmonically regular ZAS. Alternatively, Corollary 29 directly proves (using Proposition 17) that for harmonically regular ZAS. The question of whether for merely regular ZAS is not fully resolved; the answer is known to be yes in the spherically symmetric case. At the very least, we have an inequality relating the two definitions:

Proposition 18 (Proposition 56 of [jauregui]).

If consists of regular ZAS, then

We also point out that in regards to the definition of mass, there exists a sequence of surfaces that attains the supremum and for which the limsup may be replaced by a limit.

Proposition 19 (Proposition 55 of [jauregui]).

There exists a sequence of surfaces converging in to such that

The proof is a basic diagonalization argument applied to a maximizing sequence of sequences .

3.3. The capacity of ZAS

We introduce the capacity of a collection of ZAS in this section. This quantity has a relationship with the mass that plays a role in the proof of the Riemannian ZAS inequality (Theorem 32).

Definition 20.

Suppose is a surface in that is a graph over (recall this terminology from section 2.1). Let be the unique -harmonic function on that vanishes on and tends to 1 at infinity. Then the capacity of is defined to be the number

where and are taken with respect to .

The fact that is finite (and moreover the existence of ) follows from asymptotic flatness. The above integral is unchanged if is replaced with . Since is harmonic, we have . Applying Stokes’ theorem and the boundary conditions on , we conclude that

(9)

where is the unit normal to pointing toward infinity. The capacity is also characterized as the minimum of an energy functional:

(10)

where the infimum is taken over all locally Lipschitz functions that vanish on and tend to 1 at infinity; is the unique function attaining the infimum.

Now we recall a classical monotonicity property of capacity.

Lemma 21.

If and are surfaces that are graphs over and is enclosed by , then . Moreover, equality holds if and only if .

Proof.

Say and . Let be the harmonic functions vanishing on respectively, and tending to 1 at infinity. Since can be extended continuously by zero in while remaining locally Lipschitz, we see that gives an upper bound for in yet equals . Therefore .

In the above, if , then by the regularity of these surfaces ( is sufficient), the volume of is positive. The above proof shows . ∎

We are ready to define the capacity of ZAS [bray_npms].

Definition 22.

Assume the components of are ZAS of , and let be a sequence of surfaces converging to in . Define the capacity of as

The limit exists by the monotonicity guaranteed by Lemma 21.

Note that the capacity takes values in . We will often distinguish between the cases of zero capacity and positive capacity. We show now that is well-defined (as done in [robbins]).

Proposition 23.

The capacity of as defined above is independent of the sequence .

Proof.

Let and be two sequences of surfaces converging to in . Then for any , encloses for all sufficiently large. By Lemma 21, for such and . Taking the limit , we have for all . Taking the limit , we have . By the symmetry of the argument, the opposite inequality holds as well. ∎

As an example, we show below that a collection of regular ZAS has zero capacity. Examples of ZAS with positive capacity will be given in section 4.

Proposition 24.

If the components of are regular ZAS, then the capacity of is zero.

Proof.

First, observe that the capacity of is equal to

(11)

where the infimum is taken over all locally Lipschitz functions that vanish on and tend to 1 at infinity. Now, let be some local resolution, and let denote the distance function from with respect to . For small , let be the Lipschitz test function on given by:

It is straightforward to show the energy of in the sense of is of order . ∎

3.4. The relationship between mass and capacity

Recently appearing in the literature is an estimate relating the capacity of the boundary of an asymptotically flat manifold to the boundary geometry [bray_miao]. In somewhat the same spirit, the following result relates the capacity of a ZAS to the Hawking masses of nearby surfaces [robbins]. It was proved using weak inverse mean curvature flow in the sense of Huisken and Ilmanen [imcf].

Theorem 25 (Robbins [robbins]).

Assume has nonnegative scalar curvature. If is a connected ZAS with positive capacity, and if converges in to , then

With no assumption on the scalar curvature, we prove the following sufficient condition for the mass of ZAS to equal .

Theorem 26.

If is a collection of ZAS of positive capacity, then

Proof.

Suppose is any sequence of surfaces converging in to . Applying the definition of and Hölder’s inequality,

The right hand side converges to , since and (by expression for the capacity of a surface and the definition of the capacity of a ZAS). Therefore for arbitrary , so . ∎

We show in section 4 that the converse fails; there exist ZAS of zero capacity yet negative infinite mass.

3.5. The local nature of mass and capacity

A satisfactory definition of the mass of a collection of ZAS ought to only depend on the local geometry near the singularities. Here we establish that the mass and the (sign of) capacity of ZAS are inherently local notions, despite their definitions in terms of global geometry. This section is meant only to illustrate these ideas and is not essential to the main theorems. Most of the proof of the following proposition was given by Robbins in [robbins].

Proposition 27.

Suppose all components of are ZAS of , and let be any neighborhood of . Then 1) the sign of the capacity of and 2) the mass of depend only on the restriction of to .

Proof.

First, assume has zero capacity with respect to . Let be some surface in enclosing such that and the region it bounds are contained in . Let converge to in . By truncating finitely many terms of the sequence, we may assume all are enclosed by . Let be -harmonic, equal to on , and tending to 1 at infinity. Let be the minimum value attained by on . Let and be functions in the region bounded by and that are -harmonic, equal to on , with and . The setup is illustrated in figure LABEL:fig_local_capacity.

Figure 4. Functions in the proof of Proposition 27

By the maximum principle, the following inequalities hold on :

(12)

where is the outward unit normal to with respect to . Integrating the first pair of inequalities over and using expression for capacity, we have

By assumption, as , so . But by the uniqueness of harmonic functions with identical boundary values, , so . Now, is an increasing sequence by the maximum principle, so it must be that . By integrating the first and last inequality in over , we have

(13)