Capacity, quasi-local mass, and singular fill-ins

Definition 1.

A Riemannian 3-manifold ( M , g ) is called asymptotically flat (of order τ) if there is a compact set K such that M ∖ K is diffeomorphic to ℝ 3 minus a ball and, with respect to the standard coordinates on ℝ 3 , the metric g satisfies

for some τ ∈ ( 1 2 , 1 ] , and the scalar curvature R ⁢ ( g ) satisfies

for some q > 3 . Here ∂ denotes partial differentiation on ℝ 3 .

Lemma 2.

If ( M 3 , g ) is asymptotically flat of order 1 2 < τ < 1 , then in Euclidean coordinates x near infinity,

for a constant C g ⁢ ( Σ , M ) > 0 .

Proof.

We will write Δ 0 for the Euclidean Laplacian and use the notation u i = ∂ i ⁡ u , etc. Here and below, C will denote positive constant which is independent of x and r. Let ψ = a ⁢ r - 1 - r - 1 - ϵ , with a , ϵ > 0 . Then

with σ i ⁢ j = O 2 ⁢ ( r - τ ) , κ i = O 1 ⁢ ( r - 1 - τ ) . Choosing ϵ < τ , there is R > 0 independent of a so that Δ g ⁢ ψ ≤ 0 on ℝ 3 ∖ B R ⁢ ( 0 ) . Choose a ≫ 1 so that ψ > ϕ on the complement of ℝ 3 ∖ B R ⁢ ( 0 ) in M. We conclude, from the maximum principle, that ϕ ≤ ψ ≤ a ⁢ r - 1 on M. Similarly, ϕ ≥ - a ⁢ r - 1 , giving the required bound on | ϕ | . By the interior Schauder estimates, [13, Theorem 6.2] we have

Extending ϕ to be a smooth function on in a compact set, we may assume that ϕ is defined in ℝ 3 so that f = Δ 0 ⁢ ϕ = O ⁢ ( r - 3 - τ ) by (A.2) because Δ g ⁢ ϕ = 0 near infinity. Let

By the decay rate of f, this is well defined and Δ 0 ⁢ w ≡ 0 .

Since | y | ≥ r 2 for y ∈ B r / 2 ⁢ ( x ) ,

Outside B r / 2 ⁢ ( x ) , | x - y | ≥ r 2 . Hence

To estimate 𝐈𝐈 , let

Note that a ~ is a finite number by the decay rate of f. For r = | x | > 1 ,

By direct integration it is easy to see that

Since | x - y | ≥ r 2 for y ∈ B r / 2 ⁢ ( 0 ) , we have

By (A.3)-(A.4), we have w = a ⁢ r - 1 + O ⁢ ( r - 1 - τ ) . Thus, by the maximum principle, w ≡ ϕ .

The higher order estimates, again, follow from [13, Theorem 6.2]. This completes the proof of the lemma. ∎

Lemma 1 (Filling, cf. [21, Lemma 2.2]).

Suppose ( Ω , g ) ∈ F ̊ ⁢ ( Σ , γ ) is such that:

1. Σ H = ∂ ⁡ Ω ∖ Σ O is nonempty,

2. R ⁢ ( g ) > 0 on Σ H ⊂ ∂ ⁡ Ω , and

3. every component of Σ H is a stable minimal 2 -sphere.

Then for every η > 0 , there exists ( D , h ) ∈ F ̊ ⁢ ( Σ , γ ) with H h > H g - η on its boundary S O , which corresponds to Σ O .

Lemma 2 (Doubling, cf. [21, Lemma 2.3]).

Suppose ( Ω , g ) ∈ F ̊ ⁢ ( Σ , γ ) , and that the set Σ H = ∂ ⁡ Ω ∖ Σ O is nonempty. Let D denote the doubling of Ω across Σ H , so that ∂ ⁡ D = Σ O ∪ Σ O ′ , where Σ O ′ denotes the mirror image of Σ O . For every η > 0 there exists a scalar-flat Riemannian metric h on D such that ( D , h ) satisfies:

1. Σ O with the induced metric from h is isometric to ( Σ , γ ) ,

2. H h > H g on Σ O , and

3. H h ′ > H g - η on Σ O ′ ,

where H g denotes the mean curvature of Σ O in ( Ω , g ) , and H h , H h ′ denote the mean curvatures of Σ O , Σ O ′ in ( D , h ) .

Lemma 3 (cf. [21, Lemma 3.1]).

If ( Ω , g ) ∈ F ̊ ⁢ ( Σ , γ ) ∖ F ⁢ ( Σ , γ ) , then

Proof.

Use the doubling lemma (see Lemma 2) to get ( D , h ) ∈ ℱ ⁢ ( Σ ¯ , γ ¯ ) , where Σ ¯ = Σ ⊔ Σ ′ , Σ ′ ≈ Σ , γ ¯ | Σ = γ , and which has

It will be convenient to split up ( Σ , γ ) into its connected components:

Apply the cutting lemma [21, Lemma 2.1] (which did not need to be generalized) k times to the manifold ( D , h ) to isolate the boundary components ( Σ j , γ j ) and obtain ( Ω j , γ j ) ∈ ℱ ̊ ⁢ ( Σ j , γ j ) , j = 1 , … , k , with boundary mean curvatures H h , j = H h | Σ j > H g | Σ j . Fix j = 1 , … , k . Since Σ j is connected, [21, Proposition 5.1] gives

Together with (B.1):

This implies, by the additivity theorem ([21, Theorem 1.2]), that

The result follows. ∎

Proposition 4 (cf. [21, Proposition 5.1]).

We have Λ ̊ ⁢ ( Σ , γ ) = Λ ⁢ ( Σ , γ ) .

Proof.

Take the supremum over ℱ ̊ ⁢ ( Σ , γ ) ∖ ℱ ⁢ ( Σ , γ ) in Lemma 3. ∎

Proposition 5 (cf. [21, Proposition 3.1]).

If Σ is a finite union of spheres and γ is any Riemannian metric on Σ, then Λ ̊ ⁢ ( Σ , γ ) < ∞ .

Proof.

From Proposition 4, Λ ̊ ⁢ ( Σ , γ ) = Λ ⁢ ( Σ , γ ) . The latter is known to be finite by combining the additivity theorem [21, Theorem 1.2] with the finiteness theorem for single 2-spheres [21, Theorem 1.3]. ∎

Proposition 6 (Rigidity in F ̊ , cf. [21, Proposition 3.4]).

If ( Ω , g ) ∈ F ̊ ⁢ ( Σ , γ ) attains the supremum Λ ̊ ⁢ ( Σ , γ ) , i.e., if

then Σ is connected, ( Ω , g ) ∈ F ⁢ ( Σ , γ ) , and it is isometric to a mean-convex handlebody with flat interior whose genus is that of Σ. If Σ were a sphere, then ( Ω , g ) can be isometrically immersed in ( R 3 , g 0 ) .

Proof.

The fact that ( Ω , g ) ∈ ℱ ⁢ ( Σ , γ ) follows from Lemma 3. The result follows from the rigidity theorem for ℱ , [21, Theorem 1.4]. ∎