Local foliations by critical surfaces of the Hawking energy and small sphere limit

Local foliations of area constrained Willmore surfaces on a 3-dimensional Riemannian manifold were constructed by Lamm, Metzger and Schulze, and Ikoma, Machiodi and Mondino, the leaves of these foliations are, in particular, critical surfaces of the Hawking energy in case they are contained in a totally geodesic spacelike hypersurface. We generalize these foliations to the general case of a non-totally geodesic spacelike hypersurface, constructing a unique local foliation of area constrained critical surfaces of the Hawking energy. A discrepancy when evaluating the so called small sphere limit of the Hawking energy was found by Friedrich. He studied concentrations of area constrained critical surfaces of the Hawking energy and obtained a result that apparently differs from the well established small sphere limit of the Hawking energy of Horowitz and Schmidt, this small sphere limit in principle must be satisfied by any quasi local energy. We independently confirm the discrepancy and explain the reasons for it to happen. We also prove that these surfaces are suitable to evaluate the Hawking energy in the sense of Lamm, Metzger and Schulze, and we find an indication that these surfaces may induce an excess in the energy measured.


Introduction and Results
The search for a quasi local energy is one of the most prominent problems in classical relativity, with many different candidates (for a detailed review of the topic see [26]). From these candidates one of the most famous is the quasi local energy described by Hawking in 1968 [12], the so called Hawking energy, given by the expression where Σ is a closed surface in a 4 dimensional space time, |Σ| is the area of the surface, and θ + θ − is the product of the null expansions θ + and θ − . The Hawking energy is one of the simplest quasi local energies that one can find and fulfils almost all the expected properties of a quasi local energy, however it has the inconvenience that it is not necessarily positive, there are well known examples in flat space of surfaces that give a negative Hawking energy (Hayward defined a generalization of the Hawking energy in [13] to address this problem. Nevertheless, we will consider Hawking's definition). Therefore it is of high importance to know which surfaces are appropriate to evaluate the Hawking energy, for instance, it was shown by Christodoulou and Yau in [3] and by Miao, Wang and Xie in [23] that under some physically reasonable conditions the Hawking energy (in the time symmetric case) is well behaved when evaluated in constant mean curvature spheres.
This paper is divided into two parts, one devoted to studying foliations of area constrained critical surfaces of the Hawking energy, and other devoted to studying an apparent discrepancy of the small sphere limit when approaching a point in spacelike direction.
1.1. Foliations. We will work in the initial data set setting, this means that we consider a smooth 3-dimensional Riemannian manifold (M, g), which will be equipped with a symmetric 2-tensor k, we denote this manifold as a triple (M, g, k). The motivation for considering this setting comes again from general relativity since (M, g, k) can be seen as a spacelike hypersurface with second fundamental form k in a 4-dimensional spacetime. In this setting the Hawking energy can be written for a surface Σ ⊂ M as where H is the mean curvature of the surface Σ and P = tr g Σ k is the trace of the tensor k with respect to the metric induced in Σ, that is P = tr Σ k = tr k − k(ν, ν), where ν is the outward normal to Σ in M .
From a variational point of view studying (2) is equivalent to studying the Hawking functional We are interested in studying area constrained critical surfaces of this functional, then considering a fixed area, we look for surfaces that maximize or minimize the functional. In particular, these are then critical surfaces of the Hawking energy. In case k = 0, the so called time symmetric case (or a totally geodesic hypersurface) the Hawking functional reduces to the Willmore functional and the critical surfaces of this functional subject to the constraint that |Σ| be fixed are the area constrained Willmore surfaces which we call here for simplicity just Willmore surfaces. These surfaces are characterized by the following Euler Lagrange equation with Lagrange parameter λ.
The Willmore surfaces have been extensively studied and in the context of general relativity they were first introduced by Lamm, Metzger and Schulze in [18], where they showed that there exist a unique foliation of Willmore spheres for asymptotically flat manifolds, this is a foliation that covers the whole manifold except a compact region, what we call a foliation at infinity. In their work they claimed that these surfaces are the optimal surfaces for evaluating the Hawking energy, this since if the manifold has nonnegative scalar curvature (that means that the dominant energy condition holds) the Hawking energy is nonnegative on these surfaces and it is monotonically nondecreasing along the foliation. It was also shown in [16] by Koerber that the leaves of the foliation are strict local area preserving maximizers of the Hawking energy.
This foliation by Willmore spheres at infinity has been improved by Eichmair and Koerber in [6] where they used a Lyapunov-Schmidt reduction procedure (a technique that will be also applied in our construction) to obtain the foliation, furthermore, in [7] they studied the center of mass of this foliation. The non-totally geodesic case was also considered by Fridrich in his thesis [9], where he generalized the foliation of [18] for critical surfaces of the Hawking functional and showed that the Hawking energy is monotonically nondecreasing along the foliation. We will see in Theorem 2.2 that under even more general conditions, if the dominant energy condition holds then, the Hawking energy is nonnegative on these surfaces for a large enough radius.
Theorem. Assuming that on an asymptotically flat initial data set (M, g, k) the dominant energy conditions holds. There exist an r 0 > 0 such that for r ≥ r 0 , if Σ r is a critical surface of the Hawking energy with area radius r ( |Σ r | = 4πr 2 ), it is almost centered, the Lagrange parameter λ is positive with λ = O(r −3 ) and also the mean curvature is positive with H = O(r −1 ) then the Hawking energy on Σ r is nonnegative.
This shows that the Hawking functional critical surfaces in the asymptotically flat case have the same desirable properties as the Willmore surfaces and are "optimal" (in the sense of Lamm, Metzger and Schulze) to evaluate the Hawking energy on a spacelike hypersurface.
Here we are more interested in the local behaviour of the surfaces; in this direction, it was shown by Lamm and Metzger in [17] and later by Laurain and Mondino in [20] that Willmore surfaces concentrated around points which are critical points of the scalar curvature, that is points p ∈ M such that ∇Sc p = 0. Furthermore in [19] Lamm, Metzger and Schulze, and in [15] Ikoma, Machiodi and Mondino showed by a means of a Lyapunov-Schmidt reduction procedure that if at a point p ∈ M , ∇Sc p = 0 and ∇ 2 Sc p is not degenerated then around p there is a local foliation of area constrained Willmore surfaces around that point.
The first part of this paper will be devoted to generalizing these local foliations to the general case when k = 0, obtaining the following results.
We also obtained a uniqueness result.
1.2. Small Sphere Limit. For the second part of this paper, we will focus on studying the small sphere limit of the Hawking energy. In general, any quasi local energy must have the right asymptotics when evaluated on large and small spheres. In particular it must satisfy the small sphere limit.
Here we consider a 4-dimensional spacetime M 4 and will denote the geometric quantities on this manifold by an index (·) 4 . Before introducing the small sphere limit we need to define what a light cut is.
Let p ∈ M 4 and let C p be the future null cone of p, that is, the null hypersurface generated by future null geodesics starting at p. Pick any future directed timelike unit vector e 0 at p. We normalize a null vector L at p by L, e 0 = −1. We consider the null geodesics of the vector L and let l be the affine parameter of these null geodesics. We define the light cuts Σ l to be the family of surfaces on C p determined by the level sets of the affine parameter l.
Let p ∈ M 4 and let C p be the future null cone of p, that is, the null hypersurface generated by future null geodesics starting at p. Pick any future directed timelike unit vector e 0 at p. We normalize a null vector L at p by L, e 0 = −1. We consider the null geodesics of the vector L and let l be the affine parameter of these null geodesics. We define the light cuts Σ l to be the family of surfaces on C p determined by the level sets of the affine parameter l.
The small sphere limit tells us that when evaluating the quasi local energy on surfaces approaching a point p, in a spacetime along the light cuts of the null cone of p, the leading term of the quasi local energy should recover the stress energy tensor in spacetimes with matter fields, i.e., lim r→0 If the point is contained in a spacelike hypersurface M ⊂ M 4 then by using the Gauss-Codazzi equations we obtain where everything is evaluated at p, and the right hand side is the energy density of the Einstein constrained equations on M (here Sc and k are the scalar curvature and second fundamental form of M ). The small sphere limit was first introduced by Horowitz and Schmidt for the Hawking energy [14], it must be satisfy by any reasonable notion of quasi local energy as it was shown for the Brown-York energy [1] the Kijowski-Epp-Liu-Yau energy [30], the Wang-Yau [2] and for their higher dimensional versions [28] among others. In particular, when the point p is contained in a spacelike hypersurface M ⊂ M 4 , we have the following expansion for the Hawking energy for cuts on the light cut S l at p. Having this expansion in mind when studying area constrained critical surfaces of the Hawking functional (3) in a spacelike hypersurface (initial data set), it would be natural to think that such surfaces concentrate around points satisfying that (7) ∇(Sc + (tr k) 2 − |k| 2 ) = 0 at p. However, in [10] Friedrich found that this is not the case. In fact a point having a concentration of these surfaces must satisfy ∇(Sc + 3 5 (tr k) 2 + 1 5 |k| 2 ) = 0 at p, this was an unexpected result that we managed to confirm with our results as well (in Theorem 2.7) and we also obtained in the equivalent Theorem 2.10. This result gives the impression that the local expansion of the Hawking energy depends on how you approach the point. Figure 1.2 illustrates the situation.
In section 3, we will study this discrepancy found by Friedrich and see that it comes from purely geometric reasons, in particular, that even if a priori the two ways to approach the point may look similar, the surfaces used are quite different. Finally, in Remark 3.2 we will see that these results suggest that the critical surfaces of the Hawking functional induce an excess in the measure of the Hawking energy. Figure 1. Comparison between approaching a point along cuts on a null cone and along critical surfaces on a spacelike hypersurface.

Preliminaries and setting.
In this section, we work with data (M, g, k) where (M, g) is a smooth 3-dimensional Riemannian manifold which is equipped with a symmetric 2-tensor k. In General relativity, the data (M, g, k) represents a spacelike hypersurface (or an inital data set) with second fundamental form k in a 4-dimensional spacetime. In this setting we don't need any mention for the spacetime. We introduce the following notation: The covariant derivatives will be denoted by ; and the partial derivatives ∂ ∂x i by a comma or by ∂ i . Now we derive the equation that characterizes the area surfaces equations of the Hawking functional.

Lemma 2.1 (First variation). The area constrained Euler Lagrange equation for the Hawking functional (3) is
Here H is the mean curvature of Σ ,B is the traceless part of the second fundamental form B of Σ in M , that isB = B − 1 2 Hg Σ where g Σ is the induced metric on Σ, Ric is the Ricci curvature of M , ∇ Σ , div Σ and ∆ Σ are the covariant derivate, tangential divergence and Laplace Beltrami operator on Σ. Finally λ ∈ R plays the role of a Lagrange parameter.
Proof. Let Σ ⊂ M be a surface and let f : Σ × (− , ) → M be a variation of Σ with f (Σ, s) = Σ s and lapse ∂f ∂s |s=0 = αν. In [18,Section 3], it was shown that the first variation of the Willmore functional (4) is given by 1 4 now let's compute the variation of 1 2 Σ P 2 dµ. In [21], it was shown that the variation of P is given by using this relation and integration by parts we have We are considering area constrained surfaces, which means surfaces whose variation of area is zero. This traduces to the area constraint Σ Hαdµ = 0. Then our surfaces must satisfy the area constraint and Then combining this expression and the area constraint give us the Euler Lagrange equation (8).
Note that this result is equivalent to [10, Lemma 2.8], and it reduces to the Willmore equation (5) in case k = 0.
Friedrich proved in [9] the existence of a foliation of critical surfaces of the Hawking functional in asymptotically Schwarschild manifolds, and also proved that the Hawking energy is monotonically nondecreasing along the foliation. Now we will show that if the dominant energy condition holds, the Hawking energy is nonnegative on these surfaces. This holds in more general conditions that the ones considered by Friedrich (it holds when assuming general asymptotic flatness). First, recall that the dominant energy condition is given by are the energy density and the momentum density of the Einstein constraint equations. In particular, the dominant energy condition implies µ ≥ 0 which also implies Sc + 2 3 (tr k) 2 ≥ 0.

Theorem 2.2.
Assuming that on an asymptotically flat initial data set (M, g, k), where k decays like |k| + |∇k||x| ≤ C|x| − 3 2 − for some constant C > 0 and ∈ (0, 1 2 ) and the dominant energy conditions holds. There exist an r 0 > 0 such that for r ≥ r 0 , if Σ r is a critical surface of the Hawking energy with area radius r ( |Σ r | = 4πr 2 ), it is almost centered (|x| the distance to the origin of any point in Σ r is comparable to r), the Lagrange parameter λ is positive with λ = O(r −3 ) and also the mean curvature is positive with H = O(r −1 ) then the Hawking energy on Σ r is nonnegative.
Proof. According to (2), it is enough to see that Σr H 2 − P 2 dµ ≤ 16π. We proceed similarly as in [18,Theorem 4]. We consider equation (8) We can estimate for some constant C Now using Gauss-Bonnet theorem to replace Sc Σr and subtracting 1 3 (tr k) 2 on both sides we have Now thanks to the dominant energy condition, we have Sc − 2 3 (tr k) 2 ≥ 0 and by the decay conditions of the assumptions, it is direct to see that for r large enough then it follows directly that Σr H 2 − P 2 dµ ≤ 16π.

Remark 2.3.
Note that the foliation constructed in [9] satisfies the conditions of the previous result. This shows that these surfaces have the same desired properties as the Willmore surfaces in the totally geodesic case (k = 0) when evaluating the Hawking energy.
To produce our foliations, we will use the fact that geodesics spheres of small radius around a point p ∈ M form a foliation, and this foliation can be perturbed in a suitable way. The perturbation procedure consists of a normal perturbation to the geodesics spheres and a perturbation of their center. For this procedure, we will consider the setup considered in [25], which is like the one considered in [15,19,29] when k = 0.
Denote by R p the injectivity radius of p and define r p := 1 8 R p . we will also denote B r := {x ∈ R 3 : ||x|| < r} and S 2 , e i are an orthonormal basis of T p M and e τ i their parallel transport to c(τ ) along the geodesic c(tτ ) 0≤t≤1 . Consider also the dilation α r (x) = rx for r > 0. For each τ and 0 < r < r p , the map F τ • α r gives rise to some rescaled normal coordinates centered at c(τ ), in particular, the metric g in these coordinates satisfies that where δ detones the euclidean metric and σ satisfies |σ ij (x)| ≤ |x| 2 , we denote this by g ij (rx) = r 2 (δ ij + O(|x| 2 r 2 )).

As in [19], let Ω
where the expression of the right is evaluated for Σ = F τ (α r (S n ϕ )) at F τ (r(x + ϕ(x)ν)) with respect to g. Note that this is the equation that characterizes the area constrained critical surfaces of the Hawking functional. To find a foliation, we look for some functions τ (r), ϕ(r) and λ(r) such thatΦ(r, τ (r), ϕ(r), λ(r)) = 0 for some r ∈ (0, r 0 ), then our surfaces Σ r = F τ (r) (α r (S n ϕ (r))) are parameterized by r and with some extra work one can see that they form a foliation.
In order to find these functions, we will use the implicit function theorem, but in an auxiliary ) this manifold is useful since its metric is conformal to g in the F τ • α r coordinates and when r = 0, g τ,0 is just the euclidean metric and k τ,0 = 0, allowing us to work with an r arbitrarily small. Furthermore, we define the operator where the right hand side is evaluated on Σ = S n ϕ at x + ϕ(x)ν(x) with respect to g τ,r on B 2 (we denote this by the subindex r, τ ). The convenience of this operator on the auxiliary manifold is that the metric g r,τ is conformal to g in the coordinates F τ • α r with conformal factor r 2 , k r,τ is also conformal to k and then using how the different terms on (16) transform under this conformal transformation (for instance, H r,τ = rH, ν r,τ = rν, P r,τ = rP etc) one obtains the following relation Φ(r, τ, ϕ, λ) =r 3Φ (r, τ, ϕ, λ) (17) and therefore, if we manage to find a surface satisfying Φ(r, τ, ϕ, λ) = 0 we then have an area constrained critical surfaces of the Hawking functional in our original manifold.
Note that operator the (16) can be decomposed into two parts, one that doesn't depend on k that we denote by W 1 , and another that depends on k which we denote by W 2 . Then we have Note that W 1 (r, τ, ϕ, λ) corresponds to the Willmore operator whose local behaviour has been studied in many different papers like in [17], [19] and [15] among others.
From now on, we will denote by A τ (x) a tensor evaluated at F τ (x) and then A τ (0) is the tensor evaluated at the point c(τ ). Also if τ = 0, we omit the superscript i.e., A 0 = A. Now let's see the operator (16) when one considers a geodesic sphere, that is, when ϕ is equal to zero.

Lemma 2.4. Considering the setting of above one has
Where Proof. In [19, Proposition 2.3] it was shown that In the rest of the proof we omit the superindex τ for simplicity. Now considering the rescaling, we have where the right hand side is evaluated on the geodesic sphere F τ (α r (S n )) := Σ using the metric g. Consider a local frame e i ∈ T M i = 1, 2, 3 such that e 3 = ν is the normal to Σ and e i ∈ T Σ for i = 1, 2 are two parallel tangent vectors i.e. ∇ Σ eα e β = 0 for α, β = 1, 2. We use Latin letters as indices to denote the whole frame i, j, r, s, t... and Greek letters α, β just to denote the vectors tangent to Σ. We use the Einstein summation convention.
, where all the terms are evaluated at the point c(τ ). Now introducing these terms in (22) we have and using that for a geodesic sphere, one has H(r, τ, 0, λ) = 2 r − r 2 3 Ric ij x i x j − r 3 4 Ric ij,l x i x j x l + O(r 4 ) (this expression can be found in [29]) where Ric is evaluated at c(τ ), B(r, τ, 0, λ) = r −1 g Σ + O(r 2 ) and ∇ eα e β = −B(e α , e β ) we have We have an analogous result to [19, Lemma 3.2].
In [19, Section 3], it was shown that when r → 0 the linearization of W 1 reduces to which is the linearization of the Willmore operator in Euclidean space. The kernel of this operator is generated by the constant functions and the first spherical harmonics, that is K = Span{1, x 1 , x 2 , x 3 } where x i are coordinate components of a point x ∈ S 2 . Now notice that by our scaling (as seen in Lemma 2.5) the operator W 1ϕr (r, τ, 0, λ) has order O(r 2 ). Therefore, we have Note that a foliation is a concentration of surfaces where the surfaces can be continuously parameterized by r (that is ∀r ∈ I there is a surface S r ) and where the surfaces do not intersect with each other.

Foliation construction.
As mentioned before, if a surface satisfies Φ ϕr (r, τ, ϕ, λ) = 0 then we have an area constrained critical surface of the Hawking functional, then the idea to construct the foliation is to find by means of the implicit function theorem some τ (r), ϕ(r) and λ(r) such that Φ(r, τ (r), ϕ(r), λ(r)) = 0 for all r ∈ (0, r 0 ). To achieve this, we use that we can decompose C 4, 1 2 (S 2 ) as K ⊕ K ⊥ where K is the kernel of −∆ S 2 (−∆ S 2 − 2) on euclidean space and K ⊥ its L 2 orthogonal complement. Then if one manages to show that Φ(r, τ (r), ϕ(r), λ(r)) = 0 holds on K and on K ⊥ the equation holds on C 4, 1 2 (S 2 ), and this is precisely what we are going to show using the implicit function theorem in each of the cases.
and this operator is invertible since our equation is restricted to K ⊥ (the K part is zero). Then by the implicit function theorem, there exist some δ > 0, τ = τ (r), ϕ(x) = ϕ(x, r) and λ = λ(r) such that Φ(r, τ (r), r 2 ϕ(r), λ(r)) = 0 for 0 < r < δ, this means that for each r we have an area constrained critical surface of the Hawking functional. Now let's see that these surfaces form a foliation.
By construction, we have the following parametrization for our surfaces.
where we write ϕ(r) = ϕ(r)(x) for simplicity. To find the lapse function of these surfaces one calculates and this reduces to ∂G ∂r |r=0 = x + ∂τ k ∂r |r=0 e k , then we see that the lapse function is given by and by (30) this is Note that the second term is equal to zero. For the first, term it is not hard to see using (36) and the chain rule that then from (43) and the invertibility of ∇ 2 (Sc + 3 5 (tr k) 2 + 1 5 |k| 2 ) we have In the following, we show that the right hand side of the previous expression is less than one. The solution of the equation (38) is a function of the form where we denote for any tensors A and B, A * B to be any linear combination of contractions of A and B with the correspondent metric. In particular, we have that ϕ 0 is an even function. In [19, Lemma 4.1], it was shown that W 1ϕrr (0, 0, 0, 0) is an even operator which implies thatπ 1 (W 1ϕrr (0, 0, 0, 0)ϕ 0 ) = 0. Unfortunately the operator W 2ϕrr (0, 0, 0, 0) is not even, it has an odd part which is proportional to ∇k * k, then combining this with the expression of ϕ 0 in (45) we obtain the estimate where C depends on n. Then if |(∇ 2 (Sc + 3 5 (tr k) 2 + 1 5 |k| 2 )) −1 | · |k| |∇k| (|k| 2 + |Ric|) is small enough we have | ∂τ ∂r |r=0 | < 1 and in particular a foliation. The leaves of the foliation are normal graphs of the map r 3 ϕ(r) over geodesics spheres of radius r. This implies that the mean curvature of our surfaces can be estimated by the mean curvature of the geodesic sphere and Hess ϕ 0 . Then using that ||ϕ|| C 2 < C with C depending on the value of Ric and k in these coordinates at p we have Then proceeding in the same way as it was done in [15, Lemma 5.1], we find that the Willmore energy of the surfaces satisfy 1 4 Sr and |S r | = 4πr 2 + O(r 4 ), then it is direct to see that there exists an 0 such that and |S r | < 2 0 for any r ∈ (0, δ). Note that the smaller δ is, the smaller 0 can be.

Remark 2.8.
(i) Note that condition (29) is a sufficient but not a necessary condition to have the foliation. The necessary condition is that α = 1 + ∂τ k ∂r |r=0 e k , ν > 0, if this condition is not fulfilled, then we only have a regularly centered concentration of critical surface of the Hawking functional around p.
(ii) Note that any initial data set with a local minimum or maximum for the function Sc + 3 5 (tr k) 2 + 1 5 |k| 2 has a concentration of such surfaces. In particular, any compact initial data set has at least two.
2.3. Uniqueness and nonexistence. Now we prove that a point possessing a foliation of area constrained critical surfaces of the Hawking energy cannot have any other of such foliations. That is, the previously constructed foliation is unique. Theorem 2.9. (i) Assume that at p ∇(Sc + 3 5 (tr k) 2 + 1 5 |k| 2 ) = 0, ∇ 2 (Sc + 3 5 (tr k) 2 + 1 5 |k| 2 ) is nondegenerate and that the foliation F of Theorem 2.7 exists satisfying satisfy H(Σ) < 4π + 2 0 and |Σ| < 2 0 for any Σ ∈ F and the 0 of the theorem. If F 2 is a foliation around p of area constrained critical spheres of the Hawking functional, which satisfy H(Σ) < 4π + 2 and |Σ| < 2 for any Σ ∈ F 2 and some ≤ 0 , then either F is a restriction of F 2 or F 2 is a restriction of F.
Proof. The idea of the proof is to show that the leaves of the foliation can be expressed as normal graphs over geodesic spheres. Once this is done, we obtain the uniqueness of the foliation from the implicit theorems used in Theorem 2.7.
Consider the leaves of the foliation F 2 being parametrized by their area radius that is S r ∈ F 2 where r satisfies |S r | = 4πr 2 , and we consider r so small that the leaves are contained in a small geodesic sphere where we have a decomposition of the metric as in (73). By assumption, the leaves satisfy H(S r ) < 4π + 2 and |S r | < 2 . Therefore, by considering r smaller if necessary, we can apply directly [10, Proposition 3.2, Corollary 3.3], obtaining that the surfaces satisfy where the C's are constants depending on the injectivity radius of p, and of the value of Ric, ∇Ric at p. Note also that by using (46) From (47) and by considering r small enough, we can apply Lemma A.6, obtaining where y denotes the position vector on some normal coordinates centered at a point p 0 . To see that we can express our leaves as graphs over geodesic spheres we need the normal ν to S r , to satisfy on euclidean space that ν, y r = 0, and this is true if we have that || y r − ν|| L ∞ (Sr) is small. For any tangent vector e i to S r and its tangential projection to a sphere of radius r in euclidean space e T i = e i − δ(e i , y r ) y r , we have then by using that δ(e i , y r ) = (δ − g)(e i , y r ) + g(e i , y r − ν) and the decay of the metric g (like in Lemma A.1) we obtain for some constant C. From this inequality and (48), we obtain ||∇( y r − ν)|| L 2 (Sr) < Cr 2 , then using the inequality (76) from Lemma A.2 with p = 2 we obtain || y r − ν|| L 4 (Sr) < Cr 5 2 , now using (49) again we have ||∇( y r − ν)|| L 4 (Sr) < Cr 3 2 . Finally, using the Sobolev inequality (78) for p = 4 we obtain Then for r small enough, we can express S r as a graph over a geodesic sphere of radiusr =r(r) centered on a point p r , then we can also characterize the leaves by this radius and denote them by Sr. Let us change the notation and simply denoter by r. Then we have S r = Fτ (r) (α r (Sφ)) for someφ ∈ C 4, 1 2 (S 2 ) andτ (r) which satisfiesτ (r) → 0 as r → 0 and c(τ ) = exp p (τ i e i ) where we used the notation of (14).
Denoting by S 2 (a) the unit sphere of center a in R 3 , S ϕ (a) := {x + ϕ(x)ν(x) : x ∈ S 2 (a)} and definingS r := α 1/r (F −1 0 (S r )) with euclidean center of mass denoted by x(r), we have that the previous is equivalent to haveS r = Sφ (r) (x(r)) for some smooth functionφ(r) on S 2 (a). Furthermore, by Theorem A.4, we have that our surfaces approach uniformly a round sphere in Euclidean space as r → 0. Hence we have in particular that ||φ(r)|| C 5 → 0 as r → 0, with this note that we have just proved the same result as in [29,Lemma 2.3], then we can apply the two results that follow after that Lemma [29, Corollary 2.1 and Lemma 2.4] to our situation directly. With this, we perturbed the center of our spheres, obtaining a smooth function a(r) with a(r) ∈ R 3 and lim r→0 ||a(r)|| = 0, such that S r = F r(x(r)+a(r)) (α r (S ϕ(r,a(r)) )) for some smooth function ϕ(r, a(r)) on S 2 which satisfies π 1 (ϕ(r, a(r))) = 0 and that ||ϕ(r, a(r))|| C 5 → 0 as r → 0. We want our ϕ to satisfy the same conditions as the one in Theorem 2.7, this to use the uniqueness of the implicit function theorem. Therefore we also want to have that π 0 (ϕ(r, a(r))) = 0. In order to achieve this, we will have to perturb the radius of our spheres.
We then have π(ϕ * (r)) = 0 and as r * x(1 + ϕ * (r)) = rx(1 + ϕ(r, a(r))) for x ∈ S 2 then where τ (r) = r(x(r) + a(r)). As r * → 0 for r → 0 and for r small enough the relation between r and r * is injective, we can write all of the relation of before in terms of r * instead of r, then we write where we also have that τ (r * ) → 0 and ||ϕ * (r * )|| C 5 → 0 for r * → 0.
For (ii), note that we did not use the foliation property in the previous arguments.

Discrepancy of small sphere limits
In this section, we will compare the small sphere limit when approaching a point along a null cone in a spacetime M 4 with the small sphere limit along a spacelike hypersurface M ⊂ M 4 like it was done in Section 2. An index (·) 4 will denote the geometric quantities on the spacetime M 4 . As in Section 2, the quantities in M have no index.
Note that our critical surfaces of Theorems 2.7 and 2.9 are small deformations of geodesic spheres which satisfy that the smaller the radius, the closer the surface is to a geodesic sphere. Therefore, to understand the discrepancy mentioned in Section 1.2, it is a good idea to study the expansion of the Hawking energy on geodesic spheres of small radius. Recalling that the geodesic spheres are parameterized by (54) and that the mean curvature of the geodesic sphere can be expressed as were Ric is evaluated at p. One can proceed as in [8] and find that in the totally geodesic case (k = 0), the following expansion is found where the Hawking energy is evaluated on the geodesic sphere S r of radius r and centered on a point p. We can then compute, as was done in Theorem 2.7 that with this, we then get the general expansion This result would agree with the result found in [10]; therefore this gives us the idea that the problem in this discrepancy lies in the difference between the light cuts spheres and the geodesic spheres. To see this, we will follow [28] and [2] in order to study in more detail the light cuts spheres and try to compare them with the geodesic spheres.

Remark 3.1.
A natural idea would be to consider the small sphere limit evaluating on space time constant mean curvature (STCMC) surfaces, that is, surfaces satisfying H 2 −P 2 = 4r −2 = Constant. The local behaviour of these surfaces was studied in [25], and it was shown that these surfaces are small deformations of geodesic spheres that also satisfy that the smaller the radius, the closer the surface is to a geodesic sphere. Therefore such a small sphere limit would also lead to (58).
Let C p be the future null cone of p, that is the null hypersurface generated by future null geodesics starting at p. Pick any future directed timelike unit vector e 0 at p, then to parameterize the light cuts Σ l of C p we will consider the map such that for each point x ∈ S 2 and l ∈ [0, δ), X lc (x, l) is a null geodesic parameterized by the affine parameter l, with X lc (x, 0) = p and ∂X lc (x,0) ∂l ∈ T p M 4 a null vector which satisfies ∂X lc (x,0) ∂l , e 0 = −1. We define L = ∂X lc ∂l to be the null generator with ∇ 4 L L = 0. We also choose a local coordinate system {u a } a=1,2 on S 2 such that ∂ a = ∂X lc ∂ua , a = 1, 2 form a tangent basis to Σ l . We defineL to be the null normal vector along Σ l such that L , L = −1. With this, we can define Then we have that the null expansions of the null cone are given by the traces θ + = tr σ + and θ − = tr σ − . In this setting and with the help of normal coordinates (y 0 , y i , i = 0, .., 3 with where ν = x i ∂ ∂y i and x ∈ S 2 . We will consider a situation like in figure 1.2, that is supposing that the vector e 0 is a normal vector to a hypersurface M . Using the results obtained in [28] we have then that the induced metric on Σ l is given by where η is the standard metric on the sphere S 2 and Rm 4 is evaluated at p, the area of Σ l is given by with the area of the edge in Minkowski spacetime. In [27], this property was studied for the three definitions of diamonds, in higher dimensions and also in the vacuum case, obtaining different results in each case (not always proportional to the Einstein tensor) which of course diverge because of the geometric differences of the edges.
We state the following result of De Lellis and Müller in the way how was used in [17], a scaled version. Then there exists a conformal map φ : S := S R E (a E ) → Σ ⊂ R 3 with the following properties. Let γ S be the standard metric on S, N the Euclidean normal vector field and h the conformal factor, that is φ * δ |Σ = h 2 γ S . Then the following estimates hold Finally, we state [22, Lemma 3.1 and Lemma 3.2] in our context. Lemma A.5. Let Σ ⊂ M be a surface with extrinsic diameter d such that 2d is smaller than the injectivity radius of M . Then there exists a point p 0 ∈ M with diam(p 0 , Σ) ≤ d and such that in normal coordinates ψ centered at p 0 we have that where y denotes the position vector on ψ(Σ). Lemma A.6. There exist constants C and a 0 ∈ (0, ∞) such that for every closed smooth surface Σ ⊂ M with |Σ| ≤ a 0 and ||B|| 2 L 2 (Σ) ≤ a 0 , there exist a point p 0 ∈ M , normal coordinates ψ : B ρ (p 0 ) → B ρ (0) ⊂ R 3 and in these coordinates we have that where R denotes the area radius of Σ.
Finally, we state the following useful integrals.