Implications of some mass-capacity inequalities

Applying a family of mass-capacity related inequalities proved in \cite{M22}, 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.


Introduction
A smooth Riemannian 3-manifold (M, g) is called asymptotically flat (AF) if M , outside a compact set, is diffeomorphic to R 3 minus a ball; the associated metric coefficients satisfy for some τ > 1 2 ; and the scalar curvature of g is integrable.Under these AF conditions, the limit, near ∞, m = lim r→∞ 1 16π |x|=r j,k (g jk,j − g jj,k ) x k |x| exists and is called the ADM mass [2] of (M, g).It is a result of Bartnik [3], and of Chruściel [9], that 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 m ≥ 0, and equality holds if and only if (M, g) is isometric to the Euclidean space R 3 .
On an asymptotically flat 3-manifold (M, g) with boundary Σ = ∂M , the capacity (or L 2 -capacity) of Σ is defined by where the infimum is taken over all locally Lipschitz functions f that vanishes on Σ and tend to 1 at infinity.Equivalently, if ϕ denotes the function with ∆ϕ = 0, ϕ| Σ = 1, and ϕ → 0 at ∞, Regarding the mass and the capacity, if Σ 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 Σ = ∂M .Then and equality holds iff (M, g) is isometric to a spatial Schwarzschild manifold outside the horizon.
Here m is the mass of (M, g), c Σ is the capacity of Σ in (M, g), and H is the mean curvature of Σ.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 R 3 , applying (1.1) to the exterior of a geodesic sphere S r with radius r centered at any point p ∈ M , one has .
Here u is the harmonic function with u = 0 at Σ and u → 1 near ∞.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-toarea-radius ratio, first derived by Bray and the author [7].
Theorem 1.5 ([7]).Let (M, g) be a complete, orientable, asymptotically flat 3manifold with one end, with boundary Σ.Suppose Σ is connected and H 2 (M, Σ) = 0.If g has nonnegative scalar curvature, then Here c Σ is the capacity of Σ in (M, g) and is the area-radius of Σ.
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.
First, for later purposes, we remark on the topological assumption "H 2 (M, Σ) = 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 ∞, 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, Σ) = 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 Σ 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 {Σ r } so that, as r → 0, Here R denotes the scalar curvature and Ric = Ric − 1 3 Rg is the traceless part of Ric, the Ricci tensor.Applying (1.1) to the exterior of these Σ r in (M, g), one obtains If R ≥ 0, (1.7) shows the inequality m ≥ 0 as well as the rigidity of m = 0.
In general, (1.1) suggests that, if it is applied to obtain m ≥ 0 on an (M, g), the manifold boundary Σ 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 m ≥ 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 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 On M , let g be a smooth metric that is asymptotically flat near p.We refer p as the asymptotically flat (AF) ∞ of (M, g).Unless otherwise specified, we do not impose assumptions on the behavior of g near Σ k as k → ∞.In particular, (M, g) does not need to be complete, Given any closed, connected surface S ⊂ M , we say S encloses p if S = ∂D S for some precompact domain D S ⊂ N such that p ∈ D S .Let S denote the set of all such surfaces S ⊂ M enclosing p.Clearly, Σ k ∈ S for large k.Define Here c S is the capacity of S in the asymptotically flat (E S , g), where As a functional on S, the capacity c S has a monotone property, that is if Standard arguments show 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) → 1 at ∞ (i.e. as x → p).(See Proposition 3.1 in Section 3.) On the left is an examples of (M, g) with c(M, g) = 0; the arrow denotes the AF end; {Σ k } may approach a "singularity" as k → ∞.On the right is an example of (M, g) with c(M, g) > 0; besides the AF end, (M, g) has another end with suitable growth.
For manifolds (M, g) with c(M, g) = 0, we seek conditions that imply the AF end of (M, g) has mass m ≥ 0, see Theorem 2.1 and Remark 2.2.For (M, g) with c(M, g) > 0, we explore for sufficient conditions that bound m from below via c(M, g), see Theorem 3.1 and Corollary 3.1.

Singular metrics with m ≥ 0
Let N , M and g be given in the definition of c(M, g) in (1.8).Given S ∈ 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 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 and letting k → ∞, we have m ≥ 0. □ W (S) 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 m and m H (S).
Proof.If W (S) ≥ 16π, then (1.5) implies where the last step is by (1.5).Combined with (2.4), this implies the assumption of "c S k W (S k ) In what follows, let {Σ k } ⊂ S be the sequence of surfaces given in the introduction.The numerical value of c Σ k depends on g near the AF end.However, a property of "c Σ k → 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 "c Σ k W (Σ k ) 1 2 → 0".To illustrate this, it is convenient to adopt a notion of relative capacity (see [14] for instance).Given two surfaces S, S ∈ S, suppose S ∩ S = ∅ and D S ⊂ D S .The capacity of S relative to S is where v is the harmonic function on D S \ D S with v = 0 at S and v = 1 at S.
Proposition 2.3.Let S ∈ S be a fixed surface.Then, as k → ∞, Proof.For large k, let u k , v k be the harmonic function on The claim follows by noting that β k has a uniform positive lower bound as k → ∞. □ 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 \ {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 ⊂ N such that p ∈ D and (N \ D, g) is isometric to

The function u
, where |Σ| σ is the area of (Σ, σ).The mean curvature H of Σ s with respect to ḡ is We compare c(Σs,Σ δ ) and c (Σs,Σ δ ) .Let ∇, ∇ and dV ḡ, dV g denote the gradient, the volume form with respect to ḡ, g, respectively.Since c (Σs,Σ δ ) equals the infimum of the g-Dirichlet energy of functions that vanish at Σ s and equal 1 at Σ δ , we have (2.8) Here C > 0 denotes a constant independent on s and we have used the assumption We also compare W (Σ s ) and W (Σ s ).Let ĪI denote the second fundamental form of Σ s with respect to ḡ. Direct calculation shows (2.9) (For instance, see formula (2.33) in [17] and the proof therein.)Therefore, (2.10) Let dσ g , dσ ḡ denote the area form on Σ s with respect to g, ḡ, respectively.Then (2.11) , and H2 = 4a −2 a ′ 2 , we have (2.12) As |h| ḡ is bounded by assumption, it follows from (2.7), (2.8) and (2.12) that (2.13) (2.6) now follows from Propositions 2.1 and 2.3.□ Remark 2.3.The negative mass Schwarzschild manifolds are known to have an r bhorn type singularity with b = 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 3manifold 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 (•) in the negative mass Schwarzschild manifolds.Proposition 2.4.Consider a spatial Schwarzschild manifold with negative mass, i.e.
where (S 2 , σ o ) denotes the standard unit sphere and the mass m = −m is negative.Let Σ ⊂ M m be any connected, closed surface that is homologous to a slice {r} × S 2 .Let r max (Σ) = max x∈Σ r(x).Then .

Bounding m via c(M, g)
Let (M, g) be given in the definition of c(M, g) in (1.8).In this section, we relate m and 2c(M, g) assuming c(M, g) > 0. We begin with a characterization of c(M, g) > 0 which follows from standard arguments on harmonic functions.Proposition 3.1.Let c(M, g) be defined in (1.8).Then 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) → 1 at ∞ (i.e. as x → p).
Proof.For each k, let u k be the harmonic function on (E Σ k , g) with u k → 1 as x → ∞ and u k = 0 at Σ k .Given any surface S ∈ S, by the maximum principle, {u k } forms an increasing sequence in the exterior of S relative to ∞ (i.e. in D S \ {p}).Interior elliptic estimates imply {u k } converges to a harmonic function u ∞ on M uniformly on compact sets in any C i -norm.The limit u ∞ satisfies 0 < u ∞ ≤ 1 and u ∞ → 1 as x → ∞.By the strong maximum principle, either u ∞ ≡ 1 or 0 < u ∞ < 1.
Suppose (M, g) admits a harmonic w with 0 < w < 1 and w → 1 at ∞. Then w is an upper barrier for {u k }, which implies u ∞ ≤ w, and hence 0 < u ∞ < 1.In this case, c(M, g) must be positive.Otherwise, if c(M, g) = 0, then To see this, it suffices to examine the proof beginning with assuming c(M, g) > 0.
In this case, we have shown 0 Then v < u ∞ and v also acts as a barrier for 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) → 1 at ∞ (i.e. as x → p), and lim k→∞ max Σ k u = 0. Then where C > 0 is the coefficient in the expansion of in the AF end.
Proof.Let u k be the harmonic function on (E Σ k , g) with u k → 1 at ∞ and u k = 0 at Σ k .Then u k ≤ u, which implies C ≤ c k , where c k = c Σ k is the coeffiicent in To show the other direction, consider α k = max Σ k u.On E Σ k , by the maximum principle, Therefore, C = c(M, g).□ We are now in a position to derive applications of (1.4).
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) → 1 as x → p, and lim k→∞ max |∇u| 2 exists (finite or ∞), where t ∈ (0, 1) is a regular value of u; and .
Proof.Given a regular value t ∈ (0, 1) of u, let Σ ∈ Ω 1 , then u is identically a constant by the maximum principle.Hence, Σ t encloses p.As a result, if there are two connected components of Σ 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, Σ t is connected.Since t is arbitrary, this in particular shows (1.4) is applicable to (E t , g), where E t = {u ≥ t} ⊂ M is the exterior of Σ t with respect to ∞.
Here u t = 1 1−t (u − t) is the harmonic function on (E t , g) that tends to 1 at ∞ and equals 0 at Σ t , c Σ t = 1 1−t C, and C is the coefficient in the expansion of u .
Consider the function In [ This proves (i).(ii) follows from (3.3), (i) and Proposition 3.2.□ 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 Σ and the function u may simply be the restriction, to the interior, of the harmonic function that tends to 1 at ∞ and vanishes at Σ.In that extreme case, lim t→0 Σt |∇u| 2 = Σ |∇u| 2 and (ii) reduces to (1.4).
If g is complete on M , we have the following corollary.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 m in a chosen end, then m ≥ 2C, where C is the coefficient in the expansion u = 1 − 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 Σ 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 Σt u −1 |∇u| ≤ Λ where Λ is a constant independent on t.Remark 3.3.As used in Bray's work [4], the inequality m ≥ 2C 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 m ≥ 2C, a weaker inequality m ≥ C was obtained by Hirsch, Tam and the author in [12].

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(3.4)m ≥ 2C, where C = c(M, g) is the coefficient in the expansion of u = 1 − C|x| −1 + o(|x| −1 )as x → p.
(3.5)  and Theorem 3.1 (ii) that m ≥ 2C.□ Remark 3.2.Let R, Ric denote the scalar curvature, Ricci curvature of g.Since R ≥ 0 and Ric bounded from above ⇒ Ric bounded from below, Corollary 3.1 also holds if the assumption of "Ric bounded from below" is replaced by "Ric bounded from above".
16, Theorem 3.2 (ii)], we showed B(t) is monotone nondecreasing in t if g has nonnegative scalar curvature.As a result,