Definition of Center of Mass for Isolated Physical Systems and

Unique Foliations by Stable Spheres with

Constant Mean Curvature Gerhard Huisken^{⋆} and Shing-Tung Yau^{⋆⋆}

⋆ Mathematische Fakultät Universität Tübingen Auf der Morgenstelle 10 72076 Tübingen Germany

⋆⋆ Department of Mathematics Harvard University One Oxford Street Cambridge, MA USA In the description of isolated gravitating systems in General Relativity a spacelike timeslice has the structure of a complete Riemannian three-manifold with an asymptotically flat end. Let (N, g) denote an end of such a complete Riemannian three-manifold, i.e. N is diffeomorphic to R^{3}\B_{1}(0), and the metric on N asymptotically approaches the Euclidean metric near infinity:

g_{ij} = (1 + \frac{2m}{r})δ_{ij} + O(r^{−2}),

where r denotes Euclidean distance in R^{3}. The constant m can be interpreted as the total mass of the isolated system in the end N and is refered to as ADM - mass in the physical literature, compare [ADM].

A basic version of the positive mass theorem [SY1,SY2] states that m is nonnegative if N is the end of a complete three-manifold with nonnegative scalar curvature. Moreover, the mass m is strictly positive unless the three-manifold is flat. It has also been established in [B1] that the mass can be geometrically defined independent of a particular coordinate system.

In this paper we define a geometric center of mass at infinity for asymptotically flat ends with strictly positive mass m > 0 by constructing a unique constant mean curvature foliation in the exterior region.

After some auxiliary results in section 1 we introduce and investigate a class of ”round” surfaces in section 2, in which the constant mean curvature surfaces are to be constructed. The existence proof in section 3 uses a heatflow method to deform a coordinate sphere in normal direction, decreasing its isoperimetric ratio until the surfaces approach a constant mean curvature surface. More precisely, given a coordinate sphere F_{0} : S^{2} → N we solve the initial value problem

\begin{aligned}\frac{d}{dt}F (p, t) & = (h − H)(p, t)ν(p, t) p ∈ S^{2}, t > 0 \\ F (p, 0) & = F_{0}(p), \end{aligned}

where H and ν are the mean curvature and the unit normal on the moving hypersurface respectively and h = ∮ Hdμ is the average of the mean curvature. This evolution decreases area while fixing the enclosed volume, and we prove that for sufficiently large initial spheres the solution exists for all t > 0 and converges to a constant mean curvature sphere as t → ∞. In this part of the proof the assumption m > 0 enters crucially when showing that the surfaces do not drift off to infinity during the evolution.

In section 4 we use the positivity of the mass again to prove that the so constructed surfaces form a stable constant mean curvature foliation for an exterior region of N, approaching a family of concentric Euclidean spheres. The foliation can be interpreted as a geometric ”center of mass” for the infinitely far observer. It defines a natural coordinate system near infinity in an intrinsic way, while retaining regularity and decay properties near infinity that were extensively used in previous studies of foliations near infinity, see e.g. [B2] and [CK].

In section 5 we prove uniqueness of the foliation for ends N with positive mass. More precisely, we prove that the leaves of the foliation are the only stable constant mean curvature surfaces outside a small interior region, compare Theorem 5.1.

This work was begun 1986 at the department of mathematics at UC San Diego, compare the annnouncement in [CY], and the authors would like to thank R.Schoen for very stimulating discussions during this time. The activities in San Diego triggered many further investigations into the subject, the authors particularly note an alternative proof for the existence part of our results by R.Ye in [Y].

1. Preliminaries

Let N be a three-dimensional Riemannian manifold diffeomorphic to R^{3}\B_{1}(0) with a Euclidean coordinate system \{y_{α}\}. Greek indices run from 1 to 3 and we write r = (∑_{1}^{3} y_{α}^{2})^{\frac{1}{2}} for the Euclidean distance. We assume that the metric ḡ = {ḡ_{αβ}} is asymptotically flat, i.e. for r ≥ 1 we have

(1.1)  ḡ_{αβ} = (1 + \frac{m}{2r})^{4}δ_{αβ} + P_{αβ}

with constants C_{1}, C_{2}, …, C_{5} such that

(1.2)  |P_{αβ}| ≤ C_{1}r^{−2},  |∂^{l}P_{αβ}| ≤ C_{l+1}r^{−l−2},  1 ≤ l ≤ 4,

where ∂ denotes partial derivatives with respect to the coordinate system {y_{α}}. We set C_{0} = max(1, m, C_{1}, …, C_{5}) and write c for any absolute constant which is independent of the metric ḡ. Also, we will assume throughout that the mass m is strictly positive.

To compute the curvature of (N, ḡ), we begin with the special case where P_{αβ} ≡ 0, i.e. where the metric arises from the Schwarzschild metric.

1.1 Lemma. If ḡ_{αβ} = g_{αβ}^{S} = (1 + \frac{m}{2r})^{4}δ_{αβ}, then the Ricci curvature R_{αβ}^{S} of g_{αβ}^{S} is given by

R_{αβ}^{S} = \frac{m}{r^{3}} φ^{−2} (δ_{αβ} − 3\frac{y^{α}y^{β}}{r^{2}} ) ,

where here and in the following φ = (1 + \frac{m}{2r}).

Thus one eigenvalue of the Ricci curvature ( in radial direction ) is given by −2mr^{−3}φ^{−6} and the other two eigenvalues are mr^{−3}φ^{−6}. The scalar curvature equals zero.

Proof. The metric g_{αβ}^{S} is conformally Euclidean with conformal factor φ^{4}. It is wellknown that then

\begin{aligned}R_{αβ}^{S} = −2φ^{−1}∂_{α}∂_{β}φ & + 6φ^{−2}∂_{α}φ∂_{β}φ \\ & − 2φ^{−1}∂_{α}∂^{α}φδ_{αβ} − 2φ^{−2}|∂φ|^{2}δ_{αβ}, \end{aligned}

and simple computation yields

R_{αβ}^{S} = (\frac{m}{r^{3}} φ^{−1} − φ^{−2} \frac{m^{2}}{2r^{4}} ) δ_{αβ} + (φ^{−2} \frac{3m^{2}}{2r^{6}} − 3φ^{−1} \frac{m}{r^{5}} ) y^{α}y^{β},

as required.

Near infinity the curvature of the more general metric ḡ will be very close to the curvature of g^{S}:

1.2 Lemma. Let ¯∇ and R̄_{αβ} denote covariant differentiation and Ricci curvature respectively with respect to ḡ. Then we have

(1.3)  \begin{aligned}|R̄_{αβ} − R_{αβ}^{S}| & ≤ c\frac{C_{0}}{r^{4}} , \\ | ¯∇_{γ} R̄_{αβ} − ∇_{γ}^{S}R_{αβ}^{S}| & ≤ c\frac{C_{0}}{r^{5}} , \\ | ¯∇_{ρ} ¯∇_{γ} R̄_{αβ} − ∇_{ρ}^{S}∇_{γ}^{S}R_{αβ}^{S}| & ≤ c\frac{C_{0}}{r^{6}} , \end{aligned}

where c is an absolut constant.

Proof. From the decay assumptions for P_{αβ} in (1.2) we see that the Christoffel symbols Γ̄_{βδ}^{α} of ḡ satisfy

|Γ̄_{βδ}^{α} − Γ_{βδ}^{Sα}| ≤ c\frac{C_{0}}{r^{3}}

with corresponding estimates for the next higher derivatives. This implies the estimates above.

Now let M be a two-dimensional hypersurface in (N, ḡ). We write g = {g_{ij}} for the induced metric on M, Latin indices range from 1 to 2. Let ν be the unit (outward) normal on M and denote covariant differentiation in M by ∇. We often work in an orthonormal frame e_{1}, e_{2}, ν adapted to M, such that the second fundamental form A = {h_{ij}} is given by

h_{ij} = −⟨ ¯∇_{e_{i}}e_{j}, ν⟩ = ⟨ ¯∇_{e_{i}}ν, e_{j}⟩.

We let λ_{1} and λ_{2} be the two principle curvatures on M and write

H = g^{ij}h_{ij} = λ_{1} + λ_{2}, |A^{2}| = h^{ij}h_{ij} = λ_{1}^{2} + λ_{2}^{2}

for the mean curvature and the square of the norm of the second fundamental form on M respectively. Furthermore, we let Å be the traceless second fundamental form such that

\begin{matrix}∘ \\ h \end{matrix}_{ij} = h_{ij} − \frac{1}{2}Hg_{ij}, |Å|^{2} = |A|^{2} − \frac{1}{2}H^{2} = \frac{1}{2}(λ_{1} − λ_{2})^{2}.

The curvature of N and the second fundamental form of M are related by the equations of Gauß and Codazzi

(1.4)  \begin{aligned}R_{ijkl} & = R̄_{ijkl} + h_{ik}h_{jl} − h_{il}h_{jk}, \\ R̄_{3ijk} & = ∇_{k}h_{ij} − ∇_{j}h_{ik}, \end{aligned}

where the index 3 indicates the ν direction. Next we state Simons’ identity for the Laplacian of the second fundamental form, see e.g. [SSY], and an important consequence of the Codazzi equations, see [Hu1]:

1.3 Lemma. i) The second fundamental form satisfies the identity

\begin{matrix}Δh_{ij} = & ∇_{i}∇_{j}H + Hh_{il}h_{lj} − |A|^{2}h_{ij} + HR̄_{3i3j} − h_{ij}R̄_{3l3l} \\ + h_{jl}R̄_{lkik} + h_{il}R̄_{lkjk} − 2h_{lk}R̄_{likj} + ¯∇_{j}R̄_{3lil} + ¯∇_{l}R̄_{3ijl} \end{matrix}

ii) If w_{i} = R̄_{3lil} denotes the projection of R̄ic(ν, ·) onto M, we have for any η > 0 the inequality

|∇A|^{2} ≥ (\frac{3}{4} − η) |∇H|^{2} − (\frac{1}{4}η^{−1} − 1) |w|^{2}.

We will also need a formula for the change of the second fundamental form of M after a conformal change of the ambient metric:

1.4 Lemma. Let M be a hypersurface in (N, ḡ) with principle curvatures λ_{1} and λ_{2}. If ĝ_{αβ} = ψ^{2}ḡ_{αβ} describes a conformal change of the ambient metric, then the eigendirections of the second fundamental form of M remain fixed and the new principle curvatures λ̃_{1} and λ̃_{2} are given by

λ̃_{i} = ψ^{−1}λ_{i} + ψ^{−2}∂_{ν}ψ.

In particular, the difference between the eigenvalues is conformally invariant:

(λ̃_{1} − λ̃_{2}) = ψ^{−1}(λ_{1} − λ_{2}).

Proof. The relation ĝ_{αβ} = ψ^{2}ḡ_{αβ} implies that the Christoffel symbol Γ̂ is given by

Γ̂_{βγ}^{α} = Γ_{βγ}^{α} + ψ^{−1} (δ_{β}^{α}∂_{γ}ψ + δ_{γ}^{α}∂_{β}ψ − ḡ_{βγ}ḡ^{αρ}∂_{ρ}ψ)

and the result is then an immediate consequence of the definition of the second fundamental form above.

2. Spherical Hypersurfaces

The constant mean curvature surfaces constructed in section 3 will be very close to some Euclidean coordinate sphere. In this section we define a class of round surfaces in N and give a detailed description of their geometric properties. In a first proposition we show that exterior surfaces where the traceless part of the second fundamental form is small are close to some Euclidean sphere.

2.1 Proposition. Suppose M is a hypersurface in (N, ḡ) such that r(y) ≥ \frac{1}{10}max_{M}r =: r_{1} for all y ∈ M and such that for some sonstants K_{1}, K_{2}

|Å| ≤ K_{1}r_{1}^{−3}, |∇Å| ≤ K_{2}r_{1}^{−4}.

Then there is an absolute constant c such that the curvature A^{e} of M with respect to the Euclidean metric satisfies

|Å^{e}| ≤ c(K_{1} + C_{0})r_{1}^{−3}, |∇^{e}A^{e}| ≤ c(K_{2} + C_{0})r_{1}^{−4},

provided r_{1} ≥ c(C_{0} + K_{1}). Moreover, there is a number r_{0} ∈ R and a vector a⃗ ∈ R^{3} such that

(2.1)  \begin{aligned}|λ_{i}^{e} − r_{0}^{−1}| & ≤ c (K_{1} + K_{2} + C_{0})r_{1}^{−3},  i = 1, 2, \\ |(y − a⃗) − r_{0}ν_{e}| & ≤ c (K_{1} + K_{2} + C_{0})r_{1}^{−1}, \\ |ν_{e} − r_{0}^{−1}(y − a⃗)| & ≤ c (K_{1} + K_{2} + C_{0})r_{1}^{−2}. \end{aligned}

Here y and ν_{e} are the position vector and the unit normal of M in R^{3} respectively.

Proof. Since the Christoffelsymbols of the metric g and the Schwarzschild metric g^{S} are close together, see e.g. Lemma 1.2, the second fundamental form A^{S} of M with respect to the Schwarzschild metric g^{S} satisfies

|Å^{s}| ≤ c(K_{1} + C_{0})r_{1}^{−3}, |∇^{s}Å^{s}| ≤ c(K_{2} + C_{0})r_{1}^{−4},

provided r_{1} ≥ c(C_{0} + K_{1}). Since g^{s} is conformally Euclidean, we can now apply Lemma 1.4 to obtain

|Å^{e}| ≤ c(K_{1} + C_{0})r_{1}^{−3}, |∇^{e}Å^{e}| ≤ c(K_{2} + C_{0})r_{1}^{−4},

provided r_{1} ≥ c(K_{1} + C_{0}). In the second estimate we also used the fact that the derivative of the conformal factor can be bounded by cC_{0}r_{1}^{−2}. In a crucial step we now conclude from Lemma1.3(ii) that in Euclidean space

|∇^{e}Å^{e}|^{2} = |∇^{e}A^{e}|^{2} − \frac{1}{2}|∇H|^{2} ≥ \frac{1}{3}|∇^{e}A^{e}|^{2},

such that here the gradient of the full second fundamental form satisfies |∇^{e}A^{e}| ≤ c(K_{2} + C_{0})r_{1}^{−4}, proving the first part of the proposition. To find the sphere S_{r_{0}}(a⃗) close to M, first observe that at the point of M where max_{M}r = r_{1} is attained, we have λ_{i}^{e} ≥ r_{1}^{−1}. Moreover, since r_{1} ≤ 10r and in view of the gradient estimate for the curvature there is a number r_{0} such that

|λ_{i}^{e} − r_{0}^{−1}| ≤ c(K_{1} + K_{2} + C_{0})r_{1}^{−3}, i = 1, 2.

Now consider the Gauß-Weingarten relation

\frac{∂}{∂x_{i}} ν_{e} = h_{ij}^{e}g^{jk} \frac{∂y}{∂x_{k}}

for the exterior Euclidean unit normal and the position vector y. In view of the above estimates we have

|\frac{∂}{∂x_{i}} (ν_{e} − r_{0}^{−1}y)| ≤ c(K_{1} + K_{2} + C_{0})r_{1}^{−3}.

Clearly we can then choose a vector a⃗ such that the estimates and the proposition are valid provided r_{1} ≥ c(K_{1} + C_{0}) and also r ≥ \frac{1}{10}r_{1}.

For σ ≥ 1 and B_{1}, B_{2}, B_{3} nonnegative numbers we now define a set B_{σ}(B_{1}, B_{2}, B_{3}) of round surfaces in (N, ḡ) by setting

B_{σ} := {M ⊂ N| σ − B_{1} ≤ r ≤ σ + B_{1}, |Å| ≤ B_{2}σ^{−3}, |∇Å| ≤ B_{3}σ^{−4}}.

If σ is sufficiently large compared to B_{1}, B_{2}, B_{3} we may now use the information on the position of the surface in Proposition 2.1 to accurately compute the mean curvature of M in N.

2.2 Proposition. Let M be a hypersurface in B_{σ}(B_{1}, B_{2}, B_{3}). Suppose σ ≥ c(B_{1} + B_{2} + C_{0}) is such that all assumptions of Proposition 2.1 are satisfied and let r_{0} and a⃗ be as in that proposition. Then there is an absolute constant c such that the mean curvature of M satisfies

|H − \frac{2}{r_{0}} + \frac{4m}{r_{0}^{2}} − \frac{6m⟨a⃗, ν_{e}⟩_{e}}{r_{0}^{3}} | ≤ c (C_{0}^{2} + B_{2} + B_{3})σ^{−3}

provided σ ≥ c (B_{1}^{2} + B_{2} + C_{0}).

Proof. As a first step we compute from Lemma 1.4 the principal curvatures of S_{r_{0}}(a⃗) with repect to the Schwarzschild metric g_{αβ}^{S} = φ^{4}δ_{αβ}, φ = (1 + \frac{m}{2r}). We get

λ_{i}^{S} = φ^{−2}λ_{i}^{e} − \frac{m}{r^{3}} φ^{−3}⟨y, ν_{e}⟩_{e},

where as before y is the position vector, r = r(y) is the Euclidean distance and ⟨ , ⟩_{e} is the Euclidean inner product. In view of the estimates in Proposition 2.1 we have

|λ_{i}^{S} − \frac{1}{r_{0}} φ^{−2} + \frac{mr_{0}}{r^{3}} φ^{−3} + \frac{m}{r^{3}} φ^{−3}⟨a⃗, ν⟩_{e}| ≤ c (B_{2} + B_{3} + C_{0})σ^{−3}.

Observe that

|λ_{i}^{S} − λ_{i}| = \frac{1}{r_{0}} |φ^{−2} − (1 − \frac{m}{r})| + \frac{1}{r}|φ^{−3} − (1 − \frac{3m}{2r})| ≤ c C_{0}σ^{−3},

such that we get the estimate

|λ_{i} − \frac{1}{r_{0}} + \frac{m}{rr_{0}} + \frac{mr_{0}}{r^{3}} + \frac{m⟨a⃗, ν⟩_{e}}{r^{3}} | ≤ c C_{0}σ^{−3}

when σ ≥ cB_{1}. Now notice that

|r − (r_{0} + ⟨a⃗, ν⟩_{e})| ≤ \frac{|a⃗|^{2}}{2r_{0}} .

Then we obtain for sufficiently large σ ≥ cB_{1}^{2} that

|λ_{i} − \frac{1}{r_{0}} + \frac{2m}{r_{0}^{2}} − \frac{3m⟨a⃗, ν⟩_{e}}{r_{0}^{3}} | ≤ c (C_{0}^{2} + B_{2} + B_{3})σ^{−3},

as required.

3. Existence of constant mean curvature surfaces

We want to show here that for sufficiently large σ, related to each coordinate sphere S_{σ}(0) there is a constant mean curvature surface which is round in the sense of Proposition 2.1. To accomplish this we take S_{σ}(0) as an initial surface and evolve it in direction of its unit normal in N with speed given by (H − h), where h = ∮ h dμ is the mean value of the mean curvature. This flow keeps the volume of the evolving surfaces with respect to some fixed reference slice constant and decreases area at the same time. It was first studied for hypersurfaces in R^{n+1} in [Hu1], and later for surfaces in Lorentzian ambient manifolds in [EH1].

To be precise let S_{σ}(0) in (N, g) be given by a map

F_{0}^{σ}: S^{2} → N, F_{0}^{σ}(S^{2}) = S_{σ}(0).

Then we want to find a one-parameter family of maps F_{t}^{σ} = F^{σ}(·, t) such that the initial value problem

(σ)  \begin{aligned}\frac{d}{dt}F^{σ}(p, t) & = (h − H)ν(p, t),  t ≥ 0, p ∈ S^{2} \\ F^{σ}(0) & = F_{0}^{σ} \end{aligned}

is satisfied. Here H(p, t), ν(p, t) are the mean curvature and the exterior unit normal at F^{σ}(p, t) of the surface M_{t} = F_{t}^{σ}(S^{2}). We will omit the superscript σ in the following whenever the meaning is clear from the context.

3.1 Theorem. If the metric g of N is as described in section 1 and if m > 0, then there is σ_{0} depending only on C_{0} and m such that for all σ ≥ σ_{0} the initial value problem (σ) has a unique smooth solution for all times t ≥ 0. As t → ∞, the surfaces converge exponentially fast to a hypersurface M^{σ} of constant mean curvature H_{σ}. There are constants C_{1} and C_{2} depending only on C_{0} and m but not on σ such that |r −σ| ≤ C_{1} and also |H_{σ} − \frac{2}{σ} + \frac{4m}{σ^{2}}| ≤ C_{2}σ^{−3}.

Problem (σ) is a quasilinear parabolic system and it is well known that a shorttime solution exists:

3.2 Lemma. If the initial data F_{0} are smooth, then the initial value problem (σ) has a unique smooth solution M_{t} on some time interval 0 ≤ t < ϵ.

For general initial data solutions to (σ) may develop singularities, e.g. neck-pinching. For the proof of Theorem 3.1 we will show that M_{t} remains round during the evolution, the first goal is

3.3 Theorem. There are constants σ_{0}, B_{1}, B_{2}, B_{3} depending only on m and C_{0} such that for all σ ≥ σ_{0} the solution of (σ) remains in B_{σ}(B_{1}, B_{2}, B_{3}) as long as it exists.

We will in the following always assume that σ, B_{1}, B_{2}, B_{3} are chosen such that M_{0} is strictly inside B_{σ}(B_{1}, B_{2}, B_{3}). In the next two propositions we control the position of M_{t} during the evolution by estimates which are independent of B_{1}.

3.4 Proposition. Suppose M_{t} is a smooth solution of the initial value problem (σ) which is contained in B_{σ}(B_{1}, B_{2}, B_{3}) for all t ∈ [0, T]. Suppose that σ ≥ c (C_{0} + B_{1} + B_{2}) is such that Proposition 2.1 applies and let r_{0}(t) be as in that result. Then there is an absolute constant c such that

(3.1)  |r_{0}(t) − σ| ≤ c (C_{0} + B_{2} + B_{3})

holds uniformly in [0,T], provided σ ≥ c (C_{0} + B_{1} + B_{2}).

Proof. At time t = 0 we have r_{0} = σ and we only have to prove (2.1) under the assumption that σ/2 ≤ r_{0}(t) ≤ 2σ. To see that r_{0}(t) doesn’t change much, observe that in view of equation (σ) the volume V_{N} of the shell W between M_{t} and a fixed reference surface in N, e.g. the surface S_{σ/2}(0) in N is constant and equal to πσ^{3} up to terms of order σ^{2}. If σ ≥ c (C_{0} +B_{1} +B_{2}) is such that Proposition 2.1 is applicable and also r ≥ \frac{3}{4}σ everywhere then we get for the Euclidean volume V_{e} of W that

|V_{e} − V_{N}| ≤ ∫_{W} (1 − \sqrt{det g})dx ≤ cC_{0}σ^{2}.

But from the estimates in Proposition 2.1 we get that V_{e} equals \frac{4}{3}πr_{0}^{3} − \frac{1}{3}πσ^{3} up to terms of order σ^{2}, such that

|\frac{4}{3}πσ^{3} − \frac{4}{3}πr_{0}^{3}| ≤ c (C_{0} + B_{2} + B_{3})σ^{2},

which implies the desired estimate for σ ≥ c (C_{0} + B_{1} + B_{2}).

3.5 Proposition. Suppose that the solution M_{t} of (σ) is contained in B_{σ}(B_{1}, B_{2}, B_{3}) for t ∈ [0, T]. Then there is an absolute constant c such that

max_{M_{t}} r ≤ σ + c(m^{−1} + 1)(C_{0}^{2} + B_{2} + B_{3})

holds for all t ∈ [0, T] provided σ ≥ c (C_{0}^{2} + B_{1}^{2} + B_{2} + B_{3}).

Proof. Suppose that σ ≥ c (C_{0} + B_{1}^{2} + B_{2}) is such that the assumptions of Propositions 2.1 and 2.2 hold, and let a⃗ and r_{0} be the approximate centre and the approximate radius of M_{t} established there. Now let D > 0 and assume that the condition max_{M_{t}} r < σ +D is violated for the first time at (y_{0}, t_{0}), t_{0} > 0. At that point r attains its maximum and thus ⟨y_{0}, ν_{e}⟩_{e} = r. Moreover, since t_{0} is the first time where r reaches σ+D, at (y_{0}, t_{0}) the speed of M_{t} with respect to the Euclidean unit normal is non-negative, i.e. (h − H)⟨ν, ν_{e}⟩_{e} ≥ 0. From the estimates in Proposition 2.1 it follows that ⟨ν, ν_{e}⟩_{e} ≥ 1/2 and also ⟨a⃗, ν_{e}⟩_{e} ≥ \frac{1}{2}|a⃗| at this point provided σ ≥ c (C_{0} + B_{2} + B_{3}). So we have H ≤ h at (y_{0}, t_{0}), but on the other hand we get from Proposition 2.2 that there

(3.2)  \begin{aligned}H − h & ≥ \frac{3m|a⃗|}{r_{0}^{3}} − \frac{6m}{r_{0}^{3}} ∮_{M_{t_{0}}} ⟨a⃗, ν_{e}⟩_{e}dμ_{t} − c (C_{0}^{2} + B_{2} + B_{3})σ^{−3} \\ & ≥ (2m|a⃗| − c (C_{0}^{2} + B_{2} + B_{3}))σ^{−3}, \end{aligned}

provided σ ≥ c (C_{0}^{2} + B_{1}^{2} + B_{2} + B_{3}). Since at (y_{0}, t_{0}) the radius satisfies r = σ + D and in view of Proposition 2.1 |r − (r_{0} + |a⃗|)| ≤ c (C_{0} + B_{2} + B_{3})σ^{−1}, we see from Proposition 3.4 that

|a⃗| ≥ D − c (C_{0} + B_{2} + B_{3}) at (y_{0}, t_{0}).

Thus (2.2) yields a contradiction if D is larger than c (m^{−1} +1)(C_{0}^{2} +B_{2} +B_{3}), completing the proof.

To obtain in the next step a priori estimates for |Å|^{2}, we need evolution equations for the metric and the second fundamental form on M_{t}. Computing as in [Hu1] and [Hu2] we obtain from (σ):

3.6 Lemma. We have the evolution equations

(i)  \frac{d}{dt}g_{ij} = 2(h − H)h_{ij},
(ii)  \frac{d}{dt}ν = ∇H,
(iii)  \frac{d}{dt}μ = H(h − H)μ,
(iv)  \frac{d}{dt}h_{ij} = ∇_{i}∇_{j}H + (h − H)h_{il}h_{lj} − (h − H)R̄_{3i3j},

where μ is the measure on M_{t}.

Using Simons’ identity, Lemma 1.3(i), we can then easily obtain the following additional equations, computing exactly as in [Hu1].

3.7 Lemma. The second fundamental form satisfies the evolution equations

(i)  \frac{d}{dt}h_{ij} = Δh_{ij} − 2Hh_{il}h_{lj} + hh_{il}h_{lj} + |A|^{2}h_{ij} − hR̄_{3i3j} + h_{ij}R̄_{3l3l}
− h_{jl}R̄_{lmim} − h_{il}R̄_{lmjm} + 2h_{lm}R̄_{limj} − ¯∇_{j}R̄_{3lil} − ¯∇_{l}R̄_{3ijl},
(ii)  \frac{d}{dt}H = ΔH + (H − h)(|A|^{2} + R̄ic(ν, ν)),

(iii) \frac{d}{dt}|A|^{2} = Δ|A|^{2} − 2|∇A|^{2} + 2|A|^{4} − 2h tr(A^{3}) − 2hh_{ij}R̄_{3i3j} + 2|A|^{2}R̄ic(ν, ν)
− 4(h_{ij}h_{jl}R̄_{lmim} − h_{ij}h_{lm}R̄_{limj}) − 2h_{ij}( ¯∇_{j}R̄_{3lil} − ¯∇_{l}R̄_{3ijl}).

We also need some technical estimates for the expressions in the last Lemma.

3.8 Lemma. Let M_{t} be a smooth solution of (σ) contained in B_{σ}(B_{1}, B_{2}, B_{3}) for t ∈ [0, T]. Then there is an absolute constant c such that for σ ≥ c (C_{0}^{2} + B_{1}^{2} + B_{2})

(i)  |λ_{i} − \frac{1}{σ} | + |H − h| ≤ c (C_{0}^{2} + B_{2} + B_{3})σ^{−2},
(ii)  \frac{2h}{H^{3}} (|A|^{4} − H tr(A^{3})) ≤ −\frac{1}{2}|Å|^{2},
(iii)  \frac{h}{H^{2}} |\begin{matrix}∘ \\ h \end{matrix}_{ij}R̄_{i3j3}| ≤ cC_{0}|Å|σ^{−3},
(iv)  |h_{ij}h_{jl}R̄_{lmim} − h_{ij}h_{lm}R̄_{limj}| ≤ cC_{0}|Å|^{2}σ^{−3},
(v)  |h_{ij}( ¯∇_{j}R̄_{3lil} + ¯∇_{l}R̄_{3ijl})| ≤ cC_{0}|Å|σ^{−5}.

Proof. The first inequality is an easy consequence of Proposition 2.2 and Proposition 3.4. To obtain the second inequality observe that

|A|^{4} − H tr(A^{3}) = −λ_{1}λ_{2}(λ_{1} − λ_{2})^{2} = −2λ_{1}λ_{2}|Å|^{2},

which implies the desired estimate in view of (i). Now recall that the Riemann curvature tensor on a three-dimensional manifold can be expressed by the Ricci curvature:

(3.3)  R̄_{αβγδ} = (R̄_{αγ}g_{βδ} − R̄_{αδ}g_{βγ} − R̄_{βγ}g_{αδ} + R̄_{βδ}g_{αγ}) − \frac{1}{2}R̄(g_{αγ}g_{βδ} − g_{αδ}g_{βγ}).

It follows that

\begin{matrix}∘ \\ h \end{matrix}_{ij}R̄_{i3j3} = \frac{1}{2}(λ_{1} − λ_{2})(R̄_{11} − R̄_{22}),

but from Lemma 1.1 and Lemma 1.2, using also the estimates in Proposition 2.1, we get |R̄_{11} − R̄_{22}| ≤ cC_{0}σ^{−4} for σ ≥ c (C_{0} + B_{1} + B_{2} + B_{3}) as desired. The fourth estimate follows from the identity

h_{ij}h_{jl}R̄_{lmim} − h_{ij}h_{lm}R̄_{limj} = (λ_{1} − λ_{2})^{2}R̄_{1212},

formula (3.3) above and Lemma 1.1. Finally, using (3.3) again we find

h_{ij}( ¯∇_{j}R̄_{3lil} + ¯∇R̄_{3ijl}) = \frac{1}{2}(λ_{1} − λ_{2})( ¯∇_{1}R̄_{31} − ¯∇_{2}R̄_{32}),

and the conclusion follows as in (iii) from Lemma 1.1 and Lemma 1.2 and from Proposition 2.1.

We are now ready to prove an a priori estimate for the difference between the principal curvatures.

3.9 Proposition. Suppose that the solution M_{t} of (σ) is conatined in B_{σ}(B_{1}, B_{2}, B_{3}) for t ∈ [0, T]. Then there is an absolute constant c such that for σ ≥ c (C_{0}^{2} + B_{1}^{2} + B_{2} + B_{3}) we have the estimate

|Å|^{2} ≤ c C_{0}^{2}σ^{−6}

everywhere in [0, T].

Proof. Consider the function f = |Å|^{2}/H^{2}. Computing as in [Hu1] we obtain from Lemma 3.7(ii),(iii) that

\frac{d}{dt}f =Δf + \frac{2}{H} ⟨∇_{l}H, ∇_{l}f⟩ − \frac{2}{H^{4}} |∇_{i}Hh_{kl} − ∇_{i}h_{kl}H|^{2} + \frac{2h}{H^{3}} (|A|^{4} − H tr(A^{3}))

+ \frac{2h}{H}fR̄ic(ν, ν) − \frac{2h}{H^{2}} \begin{matrix}∘ \\ h \end{matrix}_{ij}R̄_{i3j3}

− \frac{1}{H^{2}} [4(h_{ij}h_{jl}R̄_{lmim} − h_{ij}h_{lm}R̄_{limj}) + 2h_{ij}( ¯∇_{j}R̄_{3lil} + ¯∇_{l}R̄_{3ijl})].

Using now all the estimates in Lemma 3.8 and the fact that R̄ic(ν, ν) is negative for σ ≥ c (C_{0} + B_{2} + B_{3}) by Lemmata 1.1, 1.2 and Proposition 2.1, we conclude that

\frac{d}{dt}f ≤ Δf + \frac{2}{H} ⟨∇_{l}H, ∇_{l}f⟩ − \frac{1}{2}|Å|^{2} + cC_{0}|Å|σ^{−3},

provided σ ≥ c (C_{0}^{2} + B_{1}^{2} + B_{2} + B_{3}) is so large that Propositions 2.1, 2.2, 3.4, 3.5 and Lemma 3.8 all apply. Now suppose that f reaches a value Dσ^{−4} for the first time at (y_{0}, t_{0}), where D is some positive number larger than the initial values of fσ^{4}. Then at (y_{0}, t_{0}) we have \frac{d}{dt}f ≥ 0, Δf ≤ 0 and ∇f = 0 such that

0 ≤ −\frac{1}{2}|Å|^{2} + cC_{0}|Å|σ^{−3} at (y_{0}, t_{0}).

Since at that point f = |Å|^{2}/H^{2} = Dσ^{−4} and since σ^{−1} ≤ H ≤ 3σ^{−1} we get

0 ≤ −\frac{1}{2}Dσ^{−6} + cC_{0}D^{1/2}σ^{−6},

which is a contradiction for D ≥ cC_{0}^{2}. Hence f ≤ cC_{0}^{2}σ^{−4}, completing the proof of the proposition.

In order to prove an a priori estimate for the gradient of the curvature, we introduce the following notation: We write A ∗ B for any linear combination of contractions of A and B with the metric g_{ij}. Then the evolution equation for the curvature gradient has the following form.

3.10 Lemma. The gradient of the second fundamental form satisfies

\frac{d}{dt}|∇Å|^{2} =Δ|Å|^{2} − 2|∇^{2}Å|^{2} + ∇A ∗ ∇Å ∗ A ∗ Å + ∇Å ∗ Å ∗ A ∗ R̄ic

+ ∇Å ∗ ∇Å ∗ A ∗ A + ∇Å ∗ ∇Å ∗ R̄ic + ∇Å ∗ Å ∗ ¯∇R̄ic

− 2h∇_{k}\begin{matrix}∘ \\ h \end{matrix}_{ij}∇_{k}(R̄_{3i3j} − \frac{1}{2}R̄ic(ν, ν)g_{ij}) − 2∇_{k}\begin{matrix}∘ \\ h \end{matrix}_{ij}∇_{k}( ¯∇_{j}R̄_{3lil} + ¯∇_{l}R̄_{3ijl}),

where for simplicity we also wrote A for the tensor h g_{ij}.

Proof. Since (d/dt)g_{ij} = 2(h − H)h_{ij}, the time derivative of the Christoffel symbols is of the form A ∗ ∇A and thus

\begin{aligned}\frac{d}{dt}|∇Å|^{2} & = 2(\frac{d}{dt}∇_{k}\begin{matrix}∘ \\ h \end{matrix}_{ij})∇_{k}\begin{matrix}∘ \\ h \end{matrix}_{ij} \\ & = 2∇_{k}(\frac{d}{dt}\begin{matrix}∘ \\ h \end{matrix}_{ij})∇_{k}\begin{matrix}∘ \\ h \end{matrix}_{ij} + ∇A ∗ ∇Å ∗ Å ∗ A. \end{aligned}

From Lemma 3.7(i),(ii) we compute after some long but easy calculations

\frac{d}{dt}\begin{matrix}∘ \\ h \end{matrix}_{ij} =Δ\begin{matrix}∘ \\ h \end{matrix}_{ij} + (h − 2H)\begin{matrix}∘ \\ h \end{matrix}_{il}\begin{matrix}∘ \\ h \end{matrix}_{lj} − \frac{1}{2}H^{2}\begin{matrix}∘ \\ h \end{matrix}_{ij} + |Å|^{2}(\begin{matrix}∘ \\ h \end{matrix}_{ij} + \frac{1}{2}hg_{ij})

+ \begin{matrix}∘ \\ h \end{matrix}_{ij}R̄ic(ν, ν) − h(R̄_{3i3j} − \frac{1}{2}R̄ic(ν, ν))g_{ij}

− \begin{matrix}∘ \\ h \end{matrix}_{jl}R̄_{lmim} − \begin{matrix}∘ \\ h \end{matrix}_{il}R̄_{lmjm} + 2\begin{matrix}∘ \\ h \end{matrix}_{lm}R̄_{limj} − ¯∇_{j}R̄_{3lil} − ¯∇_{l}R̄_{3ijl},

such that

\begin{matrix}\frac{d}{dt}\begin{matrix}∘ \\ h \end{matrix}_{ij} = & Δ\begin{matrix}∘ \\ h \end{matrix}_{ij} + Å ∗ A ∗ A + Å ∗ R̄ic \\ − h(R̄_{3i3j} − \frac{1}{2}R̄ic(ν, ν))g_{ij} − ¯∇_{j}R̄_{3lil} − ¯∇_{l}R̄_{3ijl}. \end{matrix}

Hence we obtain

\frac{d}{dt}|∇Å|^{2} =2∇_{k}(Δ\begin{matrix}∘ \\ h \end{matrix}_{ij})∇_{k}\begin{matrix}∘ \\ h \end{matrix}_{ij} + ∇Å ∗ ∇A ∗ Å ∗ A + ∇Å ∗ ∇Å ∗ A ∗ A + ∇Å ∗ ∇Å ∗ R̄ic

+ ∇Å ∗ Å ∗ ¯∇R̄ic + ∇Å ∗ Å ∗ A ∗ R̄ic

− 2h∇_{k}\begin{matrix}∘ \\ h \end{matrix}_{ij}∇_{k}(R̄_{3i3j} − \frac{1}{2}R̄ic(ν, ν))g_{ij} − 2∇_{k}\begin{matrix}∘ \\ h \end{matrix}_{ij}∇_{k}( ¯∇_{j}R̄_{3lil} + ¯∇_{l}R̄_{3ijl}).

Now observe that after changing the order of derivatives we get

\begin{matrix}2∇_{k}(Δ\begin{matrix}∘ \\ h \end{matrix}_{ij}) & ∇_{k}\begin{matrix}∘ \\ h \end{matrix}_{ij} = 2Δ(∇_{k}\begin{matrix}∘ \\ h \end{matrix}_{ij})∇_{k}\begin{matrix}∘ \\ h \end{matrix}_{ij} \\ = Δ|Å|^{2} − 2|∇^{2}Å|^{2} + ∇Å ∗ ∇Å ∗ R̄ic + ∇Å ∗ Å ∗ ¯∇R̄ic + ∇Å ∗ Å ∗ A ∗ R̄ic, \end{matrix}

which implies the result. 3.11 Corollary. If M_{t} is a solution of (σ) contained in B_{σ}(B_{1}, B_{2}, B_{3}), then there is an absolute constant c such that

\frac{d}{dt}|∇Å|^{2} ≤ Δ|∇Å|^{2} + c|∇Å|^{2}σ^{−2} + cC_{0}|∇Å|σ^{−6}

provided σ ≥ c (C_{0}^{2} + B_{1}^{2} + B_{2}). Proof. First observe that in view of Lemma 1.3(ii)

\begin{aligned}|∇A|^{2} = |∇Å|^{2} + \frac{1}{2}|∇H|^{2} & ≤ 5|∇Å|^{2} + 4 ∑_{i} |R̄ic(ν, e_{i})|^{2} \\ & ≤ 5|∇Å|^{2} + c (C_{0}^{2} + B_{1}^{2})σ^{−8}, \end{aligned}

where we computed R̄ic(ν, ν) as in the proof of Proposition 2.3. Assuming then that σ ≥ c (C_{0}^{2} + B_{1}^{2} + B_{2} + B_{3}) is so large that Proposition 3.8 applies, we get

\begin{aligned}|∇A ∗ ∇Å ∗ A ∗ Å| & ≤ cB_{2}(C_{0} + B_{1})|∇Å|σ^{−8}, \\ |∇Å ∗ Å ∗ A ∗ R̄ic| & ≤ cC_{0}B_{2}|∇Å|σ^{−7}, \\ |∇Å ∗ ∇Å ∗ A ∗ A| & ≤ c|∇Å|^{2}σ^{−2}, \\ |∇Å ∗ ∇Å ∗ R̄ic| & ≤ cC_{0}|∇Å|^{2}σ^{−3}, \\ |∇Å ∗ Å ∗ ¯∇R̄ic| & ≤ cC_{0}B_{2}|Å|σ^{−7}. \end{aligned}

Furthermore, we can argue exactly as in the proof of Proposition 3.8(iii) and 3.8(v) to obtain the estimates

|2h∇_{k}\begin{matrix}∘ \\ h \end{matrix}_{ij}∇_{k}(R̄_{3i3j} − \frac{1}{2}R̄ic(ν, ν)g_{ij})| ≤ cC_{0}|∇Å|σ^{−6}

and

|2∇_{k}\begin{matrix}∘ \\ h \end{matrix}_{ij}∇_{k}( ¯∇_{j}R̄_{3lil} + ¯∇R̄_{3ijl})| ≤ cC_{0}|∇Å|σ^{−6}.

This implies the estimate of the Corollary for σ ≥ c (C_{0}^{2} + B_{1}^{2} + B_{2} + B_{3}).

We are now ready to prove the a priori estimate for the gradient of the curvature.

3.12 Proposition. Suppose that the solution M_{t} of (σ) is contained in B_{σ}(B_{1}, B_{2}, B_{3}) for t ∈ [0, T]. Then there is an absolute constant c such that for σ ≥ c (C_{0}^{2} + B_{1}^{2} + B_{2} + B_{3}) the estimate

|∇Å|^{2} ≤ cC_{0}^{2}σ^{−8}

holds everywhere in [0, T].

Proof. First observe that by Lemma 3.7 (iii) and (ii), as well as by Lemma (3.8) (iii), (iv) and (v) we have the evolution equation

\frac{d}{dt}|Å|^{2} =△|Å|^{2} − 2|∇Å|^{2} + (λ_{1} − λ_{2})^{2}(λ_{1}^{2} + λ_{2}^{2} − λ_{1}h − λ_{2}h)

+ 2|Å|^{2}R̄ic(ν, ν) − 2h\begin{matrix}∘ \\ h \end{matrix}_{ij}R̄_{3i3j}

− 4(h_{ij}h_{jl}R̄_{lmim} − h_{ij}h_{lm}R̄_{limj}) − 2h_{ij}( ¯∇_{j}R̄_{3lil} − ¯∇_{l}R̄_{3ijl}) ≤ △|Å|^{2} − 2|∇Å|^{2} + cC_{0}|Å|σ^{−5}.

The last inequality follows for σ ≥ c(C_{0}^{2} + B_{1}^{2} + B_{2}) from Lemma 3.8. Now let C_{1} be sufficiently large compared to the absolute constant c_{1} in Corollary 3.11. Then

\begin{matrix}\frac{d}{dt}(|∇Å|^{2} + C_{1}|Å|^{2}σ^{−2}) ≤ & △(|∇Å|^{2} + C_{1}|Å|^{2}σ^{−2}) \\ − c_{1}|∇Å|^{2}σ^{−2} + cC_{0}|∇Å|σ^{−6} + cC_{0}^{2}σ^{−10}, \end{matrix}

where we also used Proposition 3.9 for σ ≥ c(C_{0}^{2} + B_{1}^{2} + B_{2} + B_{3}). Using the parabolic maximum principle as before the desired estimate now follows from Proposition 3.9.

Summarising our a priori estimates in Propositions 3.5, 3.9 and 3.12, we have shown that there is a uniform absolute constant c_{2} such that

max_{M_{t}} |r − σ| ≤c_{2}(m^{−1} + 1)(C_{0}^{2} + B_{2} + B_{3}),
max_{M_{t}} |Å| ≤c_{2}C_{0}σ^{−3},
max_{M_{t}} |∇Å| ≤c_{2}C_{0}σ^{−4},

provided that σ ≥ c_{2}(C_{0}^{2} +B_{1}^{2} +B_{2} +B_{3}). Thus we may first choose B_{2} > c_{2}C_{0}, B_{3} > c_{2}C_{0}, then choose B_{1} larger than c_{2}(m^{−1}+1)(C_{0}^{2}+B_{2}+B_{3}) and finally σ_{0} ≥ c_{2}(C_{0}^{2}+B_{1}^{2}+B_{2}+B_{3}) to ensure that the solution M_{t} of (σ) remains strictly inside B_{σ}(B_{1}, B_{2}, B_{3}), σ ≥ σ_{0}, as long as it exists. This completes the proof of Theorem 3.3.

Once we have obtained Theorem 3.3, higher derivative estimates and longtime existence are easily obtained, see e.g. [Hu1] and [Hu2]. We have only to show that the solution surfaces M_{t} converge exponentially fast to a constant mean curvature limiting surface M_{∞}. For that purpose we need the following lower bound for the first eigenvalue of the Laplace-Beltrami operator on M_{t}.

3.13 Lemma. Suppose M is a hypersurface of N in B_{σ}(B_{1}, B_{2}, B_{3}). Then the lowest eigenvalue μ_{Lap} of the negative of the Laplace-Beltrami operator on M satisfies the inequality

μ_{Lap} ≥ \frac{2}{σ^{2}} − \frac{4m}{σ^{3}} − cC_{0}B_{1}σ^{−4},

provided σ ≥ σ_{0} ≥ c(C_{0}^{2} + B_{1}^{2} + B_{2} + B_{3}).

Proof. On a two-dimensional surface the Laplace-Beltrami operator has its lowest eigenvalue bounded by

μ_{Lap} ≥ 2K,

where K is a lower bound for the Gauß curvature on M. But by Gauß’ equation we have

K ≥ σ̄_{12} + λ_{1}λ_{2},

where σ_{12} is the sectional curvature of N in tangential direction and the assertion follows from Propositions 2.1, 2.2 and also Lemmata 1.1 and 1.2.

To prove now exponential convergence to a constant mean curvature surface, we compute from Lemma 3.6(iii) and from Lemma 3.7(ii) that

\begin{matrix}\frac{d}{dt} ∫_{M_{t}}(H − h)^{2} & dμ = ∫_{M_{t}} 2(H − h)\frac{d}{dt}(H − h)dμ − ∫_{M_{t}}(H − h)^{3}Hdμ \\ = & 2 ∫_{M_{t}}(H − h)(△H + (H − h)(|A|^{2} + R̄ic(ν, ν)))dμ − ∫_{M_{t}}(H − h)^{3}Hdμ, \end{matrix}

where we also used that ∫_{M_{t}} H − h dμ = 0. After integration by parts we obtain from Lemma 3.13

\begin{aligned}\frac{d}{dt} ∫_{M_{t}}(H − h)^{2} & dμ ≤ −(\frac{4}{σ^{2}} − \frac{8m}{σ^{3}} − cC_{0}B_{1}σ^{−4}) ∫_{M_{t}}(H − h)^{2}dμ \\ & + 2 ∫_{M_{t}}(H − h)^{2}(|A|^{2} + R̄ic(ν, ν))dμ − ∫_{M_{t}}(H − h)^{3}H dμ. \end{aligned}

From Lemmata 1.1, 1.2 and Propositions 2.1 and 2.2 we know

|A|^{2} + R̄ic(ν, ν) ≤ \frac{2}{σ^{2}} − \frac{10m}{σ^{3}} + cC_{0}B_{1}σ^{−4},

such that for σ ≥ σ_{0} ≥ cC_{0}B_{1} sufficiently large

\frac{d}{dt} ∫_{M_{t}}(H − h)^{2}dμ ≤ −\frac{(12 − ε)m}{σ^{3}} ∫_{M_{t}}(H − h)^{2}dμ − ∫_{M_{t}}(H − h)^{3}H dμ.

Now notice that in view of Lemma 3.6(iii)

\frac{d}{dt} ∫_{M_{t}} dμ = − ∫_{M_{t}}(H − h)^{2}dμ,

such that

∫_{0}^{∞} ∫_{M_{t}}(H − h)^{2}dμ dt ≤ |M_{0}|

is uniformly bounded. Thus, in view of our higher derivative estimates max_{M_{t}} |H −h| tends to zero as t → ∞, in particular there is t_{0} such that the estimate max |H(H −h)| ≤ 2mσ^{−3} holds for t > t_{0}. Therefore we conclude that

\frac{d}{dt} ∫_{M_{t}}(H − h)^{2}dμ ≤ −\frac{(12 − ε)m}{σ^{3}} ∫_{M_{t}}(H − h)^{2}dμ,

which implies exponential decay of this L^{2}-integral. Exponential convergence in higher norms now follows from standard interpolation inequalities, completing the proof of Theorem 3.1.

3.14 Remark i) Instead of the centered sphere S_{σ}(0) we could have used any sphere S_{σ}(a⃗) as initial surface for the flow with this proof, as long as |a⃗| is sufficiently small compared to σ. With some additional work using the estimates in section 5 it is also possible to prove that round initial spheres displaced by a distance just slightly smaller than σ drift back to the center under the flow.

(ii) If the mass is negative, any sphere which is ’off center’ initially, will drift further away under this flow, compare the expression for the mean curvature in Proposition 2.2.

(iii) Although we have only presented the three-dimensional case here, the existence proof clearly carries over to asymptotically flat manifolds with positive mass in higher dimensions.

4. The Foliation

In this section we prove that the constant mean curvature surfaces constructed in the previous section are stable and form a proper foliation of N\B_{σ_{0}}(0), provided σ_{0} is sufficiently large. We then prove that this foliation is at least locally unique and has a common center of gravity as the radius of the surfaces tends to infinity. Global uniqueness results will be obtained in section 5.

For σ ≥ σ_{0} let M_{σ} be the constant mean curvature surface constructed in Theorem 3.1 and denote by H_{σ} its mean curvature. We say that M is stable if volume preserving variations of M in N do not decrease the area and we say that M is strictly stable if the second variation operator on M,

Lu = −△u − (|A|^{2} + R̄ic(ν, ν))u

has only strictly positive eigenvalues when restricted to functions u with ∫ u dμ = 0. Then we have the following result on the existence of a stable foliation by constant mean curvature surfaces.

4.1 Theorem. There is σ_{0} depending only on m > 0 and C_{0} such that the constant mean curvature surfaces M_{σ} constructed in Theorem 3.1, σ ≥ σ_{0}, form a proper foliation of N\B_{σ_{0}}(0). Each M_{σ} is strictly stable, the lowest eigenvalue of the stability operator L on volume preserving deformations is of order 6mσ^{−3}. Moreover, given constants B_{1}, B_{2}, B_{3} one can choose σ ≥ σ_{0} ≥ c(C_{0}^{2} + B_{1}^{2} + B_{2} + B_{3}) such that M_{σ} is the only surface with constant mean curvature H_{σ} contained in B_{σ}(B_{1}, B_{2}, B_{3}).

Proof. Suppose that B_{1}, B_{2}, B_{3} and σ_{0} ≥ c(C_{0}^{2}+B_{1}^{2}+B_{2}+B_{3}) are such that our previous results, in particular Theorem 3.3, Propositions 2.1,2.2 and Lemma 3.13 all apply. Then we have

(4.1)  |(|A|^{2} + R̄ic(ν, ν)) − \frac{2}{σ^{2}} + \frac{10m}{σ^{3}} |≤ c(C_{0} + B_{1} + B_{2} + B_{3})σ^{−4}

on M_{σ}, such that in view of Lemma 3.13 the lowest eigenvalue of the operator L restricted to functions u with ∫ u dμ = 0 is of order 6mσ^{−3}. This proves the stability of M_{σ} and it is well known that then the M_{σ} form a foliation. To obtain stronger control of the foliation, we examine the operator L now more closely.

Let μ_{0} be the only negative eigenvalue of L with eigenfunction h_{0}, i.e.

(4.2)  △h_{0} + (|A|^{2} + R̄ic(ν, ν))h_{0} = −μ_{0}h_{0}.

Multiplying with h_{0} and integrating we see from (4.1) that

μ_{0} ≥ −\frac{2}{σ^{2}} + \frac{10m}{σ^{3}} − c(C_{0} + B_{1} + B_{2} + B_{3})σ^{−4}.

On the other hand μ_{0} is characterised as

(4.3)  μ_{0} = inf_{∥f∥_{2}=1} ∫ |∇f|^{2} − (|A|^{2} + R̄ic(ν, ν))f^{2} dμ.

Choosing f ≡ const. as a comparison function we get the reverse inequality

(4.4)  μ_{0} ≤ −\frac{2}{σ^{2}} + \frac{10m}{σ^{3}} + c(C_{0} + B_{1} + B_{2} + B_{3})σ^{−4}.

To show that h_{0} is close to a constant function, let h̄_{0} = ∮ h_{0} dμ be the mean value of h_{0} on M_{σ}. After multiplying (4.2) with (h_{0} − h̄_{0}) and integration by parts we obtain

(4.5)  \begin{aligned}∫ |∇h|^{2}dμ = & ∫ (μ_{0} + |A|^{2} + R̄ic(ν, ν))(h_{0} − h̄_{0})^{2}dμ \\ & + h̄_{0} ∫ (|A|^{2} + R̄ic(ν, ν))(h_{0} − h̄_{0})dμ. \end{aligned}

Since by Lemma 3.13

∫ |∇h_{0}|^{2}dμ = ∫ |∇(h_{0} − h̄_{0})|^{2}dμ ≥ \frac{3}{2}σ^{−2} ∫ (h_{0} − h̄_{0})^{2}dμ,

we obtain from (4.3)–(4.5)

∫ (h_{0} − h̄_{0})^{2}dμ ≤ C_{1}|h̄_{0}|σ^{−2} ∫ |h_{0} − h̄_{0}|dμ,

where C_{1} depends on C_{0}, B_{1}, B_{2} and B_{3}. So we conclude that

∫ (h_{0} − h̄_{0})^{2}dμ ≤ cC_{1}^{2}σ^{−4} ∫ |h̄_{0}|^{2}dμ.

Standard linear estimates then show that also

(4.6)  sup |h_{0} − h̄_{0}| ≤ cC_{1}σ^{−2}|h̄_{0}|,

in particular h_{0} doesn’t change sign for σ ≥ σ_{0} large enough. Now let μ_{1} be the next eigenvalue of L_{σ} with corresponding eigenfunction h_{1}. To see now that the mean value h̄_{1} = ∮ h_{1} dμ is very small, observe that h_{1} is orthogonal to h_{0} and therefore

0 = ∫ h_{0}h_{1}dμ = ∫ h_{1}(h_{0} − h̄_{0})dμ + h̄_{0} ∫ h_{1}dμ.

Hence we get from (4.6)

|∫ h_{1}dμ| ≤ cC_{1}σ^{−2} ∫ |h_{1}|dμ.

Now multiply the equation

△h_{1} + (|A|^{2} + R̄ic(ν, ν))h_{1} = −μ_{1}h_{1}

with (h_{1} − h̄_{1}) and integrate. Then, since the lowest eigenvalue of L restricted to functions with zero mean value is 6mσ^{−3} up to terms of order σ^{−4}, we derive from (4.1)

\begin{matrix}(6 − ε)mσ^{−3} ∫ (h_{1} − h̄_{1})^{2}dμ ≤ & μ_{1} ∫ (h_{1} − h̄_{1})^{2}dμ + h̄_{1} ∫ (|A|^{2} + R̄ic(ν, ν))(h_{1} − h̄_{1})dμ \\ ≤ & μ_{1} ∫ (h_{1} − h̄_{1})^{2}dμ + cC_{1}σ^{−4}|h̄_{1}| ∫ |h_{1} − h̄_{1}|dμ \\ ≤ & μ_{1} ∫ (h_{1} − h̄_{1})^{2}dμ + cC_{1}σ^{−6} ∫ (h_{1} − h̄_{1})^{2}dμ, \end{matrix}

such that μ_{1} ≥ (6 − ε)mσ^{−3} provided σ ≥ σ_{0} ≥ c(C_{0}^{2} + B_{1}^{2} + B_{2} + B_{3}) is large. Thus we have now shown that the operator L is invertible with a bound for the inverse operator L^{−1} given by cm^{−1}σ^{−3}.

To complete the proof of Theorem 4.1 it remains only to show that the surface M_{σ} constructed in Theorem 3.1 is the only constant mean curvature surface with mean curvature H_{σ} in the neighbourhood of M_{σ} given by B_{σ}(B_{1}, B_{2}, B_{3}).

Consider the smooth operator H : C^{3}(S^{2}, R^{3}) → C^{1}(S^{2}) which assigns to each embedding F : S^{2} → R^{3} the mean curvature H(F) of F (S^{2}). Given a variation vectorfield V on a constant mean curvature surface M it is well known that the derivative of H at M in direction V is given by

dH(F) · V = −△⟨V, ν⟩ − (|A|^{2} + R̄ic(ν, ν))⟨V, ν⟩ = L⟨V, ν⟩,

where F and L denote the imbedding of M and the second variation operator on M respectively. The tangential component of V doesn’t yield any contribution since the mean curvature of M is constant.

Suppose now there is another surface M_{σ′} in B_{σ}(B_{1}, B_{2}, B_{3}) with constant mean curvature H_{σ}. If F_{0}, F_{1} : S^{2} → R^{3} denote the embedding of M_{σ} and M_{σ′} respectively, then we see from Proposition 2.1 that there is a vector a⃗ such that the second surface can be represented by a normal variation of the form

F_{1}(p) = F_{0}(p) + u(p)ν(p), p ∈ S^{2},

where u =< a⃗, ν > +G, and G satisfies the estimate

(4.7)  σ|G| + σ^{2}|∇G| + σ^{3}|∇^{2}G| + σ^{4}|∇^{3}G| ≤ c(C_{0} + B_{1} + B_{2} + B_{3}).

If we then consider the one-parameter family of surfaces M_{t} given by

F_{t} = F_{0} + t u ν,

we see that all interpolated surfaces are still in B_{σ}(B_{1}, B_{2}, B_{3}) such that the operator L satisfies the eigenvalue estimates established above on all these surfaces. Since H(F_{0}) = H(F_{1}) ≡ H_{σ} we get from Taylor’s theorem that the variation vectorfield V = F_{1} − F_{0} satisfies both

(4.8)  ∥dH(F_{0}) · ∨∥ ≤ sup_{t∈[0,1]} ∥d^{2}H(F (t))(∨, ∨)∥

and

(4.9)  ∥dH(F_{0}) · ∨∥ ≤ sup_{t∈[0,1]} ∥(dH(F (t)) − dH(F_{0})) · ∨∥ .

We have already seen that L ist invertible with ∥L^{−1}∥ ≤ c m^{−1}σ^{3}. On the other hand it is clear from Proposition 2.1 and (4.7) that the second variation of the mean curvature, which involves cubic terms in the second fundamental form and higher derivatives of the metric, satisfies ∥d^{2}H(∨, ∨)∥ ≤ c C_{0}σ^{−3}∥ ∨ ∥^{2}. Hence we see from (4.8) that there is an absolute constant c_{3} such that ∥ ∨ ∥ ≤ c_{3}C_{0}m^{−1} implies ∨ ≡ 0. But in view of (4.7) this means that |a⃗| ≤ c_{3}C_{0}m^{−1} implies F_{0} ≡ F_{1} provided σ ≥ σ_{0} ≥ c(C_{0} + B_{1} + B_{2} + B_{3}) is sufficiently large. We will now use the inequality (4.9) to show that the bound |a⃗| ≤ c_{3}C_{0}m^{−1} is true for suitably large σ ≥ σ_{0}. First observe that for all t ∈ [0, 1]

\begin{aligned}dH(F (t)) · ∨ & = dH(F (t)) · ∨^{⊥} + dH(F (t)) · ∨^{⊤} \\ & = L_{t}⟨∨, ν⟩ + ∨^{⊤}(H^{t}). \end{aligned}

Here ⊥ and ⊤ denote the tangential and vertical component of a vector with respect to F (t)(S^{2}), L_{t} is the second variation operator on F (t)(S^{2}) and ∨^{⊤}(H^{t}) is the derivative of the mean curvature on F (t)(S^{2}) in direction ∨^{⊤}. For a surface in B_{σ}(B_{1}, B_{2}, B_{3}) the second term on the RHS can be estimated by

∥ ∨^{⊤} (H^{t})∥ ≤ c B_{3}σ^{−4}.

Since

L_{t}⟨∨, ν⟩ = −Δ⟨∨, ν⟩ − (|A|^{2} + R̄ic(ν, ν))⟨∨, ν⟩

and since ∨ = a⃗ + G , we get from (4.1) and (4.7) that

∥(dH(F (t)) − dH(F (0)))∨∥ ≤ cσ^{−4}(C_{0} + B_{1} + B_{2} + B_{3}).

Since on the other hand ⟨a⃗, ν⟩ is an approximate eigenfunction of L with eigenvalue close to 6mσ^{−3} we have ∥dH(F_{0}) · ∨∥ ≥ mσ^{−3}|a⃗|. Applying now inequality (4.9) we see that for σ ≥ σ_{0} ≥ c(B_{1} + B_{2} + B_{3} + C_{0})m^{−1} we must have |a⃗| ≤ c_{3}C_{0}m^{−1} and thus F_{0} ≡ F_{1}. This completes the proof of Theorem 4.1.

We will now show that the surfaces M_{σ} have a joint ‘center of gravity’ as σ tends to infinity.

4.2 Theorem. There is a vector a⃗ ∈ R^{3} depending only on N such that the center of gravity of M_{σ} with respect to R^{3} converges to a⃗ as σ → ∞.

Proof. It will be convenient in this proof to label the surfaces M_{σ} by their mean curvature: We write τ(σ) = H_{σ} and M(τ) for the surface M_{σ} with H_{σ} = τ. M(τ) is uniquely determined by Theorem 4.1 and we write

g⃗_{τ} = ∫_{M(τ)} F (τ)dμ_{e}/ ∫_{M(τ)} dμ_{e}

for the center of gravity of M(τ) with respect to the Euclidean metric. Here F (τ) is the embedding of M(τ) and we want to show that −\frac{d}{dτ} g⃗_{τ} is small. So we have to study the solution u of the equation

−Lu = Δu + (|A|^{2} + R̄ic(ν, ν))u =1

describing the lapse function of the foliation on M_{τ}. Let h_{0} be the first eigenfunction of L as before and normalize h_{0} such that ∮ h_{0}dμ = h̄_{0} = ∮ udμ = ū. Then

Δ(u − h_{0}) + (|A|^{2} + R̄ic(ν, ν))(u − h_{0}) =1 − μ_{0}h_{0}.

Now let σ be such that τ(σ) = τ, and multiply both sides with (u − h_{0}). Since the lowest eigenvalue of L on functions of zero mean value is larger than cmσ^{−3}, we derive after integration by parts

3mσ_{1}^{−3} ∫_{M(τ)}(u − h_{0})^{2}dμ ≤−μ_{0} ∫_{M(τ)}(h_{0} − h̄_{0})(u − h_{0})dμ.

Since σ_{0}, B_{1}, B_{2}, B_{3} have been chosen uniformly in σ at this stage, we see from (4.6) that there is a constant C_{4} depending only on m^{−1} and C_{0} such that

|h_{0} − h̄_{0}| ≤ C_{4}σ^{−2}|h̄_{0}|.

Thus, using also the fact that |h_{0}| ≤ cσ^{2} we obtain

∫_{M(τ)}(u − h_{0})^{2}dμ ≤ C_{4}σ^{2}

and standard linear estimates then show that also sup |h_{0} − u| ≤ C_{4}. Hence we finally have

sup |u − ū| ≤ sup(|u − h_{0}| + |h_{0} − h̄_{0}|) ≤ C_{4}

for some constant C_{4} depending only on m^{−1} and C_{0}. In view of Proposition 2.1 the normal of M_{τ} with respect to the metric g differs from the Euclidean unit normal by a term of order σ^{−2}, hence we easily see that

|\frac{d}{dτ} g⃗_{τ} | ≤ C.

Furthermore, since |\frac{dτ}{dσ} | ≤ cσ^{−2} , we conclude that

|\frac{d}{dσ} g⃗_{τ(σ)}| ≤ cσ^{−2},

giving the desired conclusion.

4.3 Remark The estimates established here show that the foliation can be used to define a natural polar coordinate system at infinity: A radial coordinate r̃(σ) is given by |M_{σ}| = 4πr̃_{σ}^{2}, and angular coordinates can be defined since the estimates ensure convergence of the unit normal on M_{σ} for σ → ∞ to points on the Euclidean sphere at infinity.

5. Global Uniqueness

In this section we will show that the foliation {M_{σ}} constructed in Theorem 3.1 and Theorem 4.1 is unique. In fact, we show that the individual surfaces M_{σ} are the only stable constant mean curvature surfaces outside a small interior region of N.

5.1 Theorem. Let m > 0 and 1/2 < q ≤ 1 be given. Then there is H_{0} > 0 depending only on m^{−1}, q and C_{0} such that for all 0 < H < H_{0} the leaf M_{H} of the foliation constructed in Theorem 4.1 is the only surface of constant mean curvature H, which is stable and contained in N\B_{H^{−q}}(0).

Note that no curvature bound for the hypersurface is assumed in this result, and the interior radius H^{−q} is much smaller than the diameter of M_{H} if q < 1. The exponent q > 1/2 seems to be optimal from a technical point of view, but it is an open question whether stable constant mean curvature surfaces are actually completely unique outside a fixed compact set. The stability assumption is only used to derive curvature estimates and can be discarded if only surfaces are considered where the principal curvatures are comparable to \frac{1}{2}H.

Using suitable testfunctions for the stability operator we will first show that any stable constant mean curvature surface satisfies an estimate on an L^{2} - curvature integral. It is then shown that it is ”round” in the sense of section 2 and has to lie in some B_{σ}(B_{1}, B_{2}, B_{3}) where the uniqueness result in Theorem 4.1 can be applied. Let S in the following be a stable hypersurface of constant mean curvature H which is contained in N\B_{σ_{0}}(0). We begin with a technical lemma.

5.2 Lemma. Let X = x_{i} \frac{∂}{∂x_{i}} be the Euclidean coordinate vectorfield and let as before r = (∑ x_{1}^{2})^{1/2}. Then for σ_{0} ≥ cC_{0} we have the estimate

(i)  ∫_{S} ⟨X, ν⟩^{2}r^{−4}dμ ≤ H^{2}|S|.

Moreover, for each α ≥ α_{0} > 2 and σ_{0} ≥ c C_{0}(α_{0} − 2)^{−1} sufficiently large we have the estimate

(ii)  ∫_{S} r^{−α}dμ ≤ c(α_{0})σ_{0}^{(2−α)}H^{2}|S|.

Proof. Since S has constant mean curvature H, the divergence formula takes the form

∫_{S} div_{S}Y dμ = H ∫_{S} ⟨Y, ν⟩ dμ.

Now choose the vectorfield Y = Xr^{−α} , α ≥ 2 and observe that

div_{S}Y = r^{−α}div_{S}X − αr^{−α−2}|X^{τ} |^{2},
|div_{S}X − 2| ≤ cC_{0}r^{−1}.

We then obtain

(5.1) |(2 − α) ∫_{S} r^{−α}dμ + α ∫_{S}⟨X, ν⟩^{2}r^{−α−2}dμ − H ∫_{S}⟨X, ν⟩r^{−α}dμ| ≤ cC_{0} ∫_{S} r^{−α−1}dμ.

Choosing first α = 2 this yields

(5.2)  ∫_{S}⟨x, ν⟩^{2}r^{−4}dμ ≤ \frac{1}{4}H^{2}|S| + cC_{0} ∫_{S} r^{−3}dμ.

Using (5.1) now with α = 3 we derive

∫_{S} r^{−3}dμ ≤ 4σ_{0}^{−1} ∫_{S}⟨x, ν⟩^{2}r^{−4}dμ + σ_{0}^{−1}H^{2}|S| + cC_{0}σ_{0}^{−1} ∫_{S} r^{−3}dμ.

Thus, choosing σ ≥ cC_{0} sufficiently large, and combining this inequality with (5.2) we obtain i). Now let α ≥ α_{0} > 2 be arbitrary in (5.1) to conclude

∫_{S} r^{−α}dμ ≤ c(α_{0} − 2)^{−1}σ_{0}^{(2−α)} (∫_{S}⟨x, ν⟩^{2}r^{−4}dμ + H^{2}|S| + cC_{0} ∫_{S} r^{−3}dμ) .

The desired estimate then follows from i).

In a first step we now use the stability assumption to prove an integral estimate for the difference of the principal curvatures on S.

5.3 Proposition. Suppose that S is stable. Then we have for σ_{0} ≥ cC_{0} the estimate ∫_{S}(λ_{1} − λ_{2})^{2}dμ ≤ cC_{0}σ_{0}^{−1}H^{2}|S| ≤ C̃σ_{0}^{−1}.

Proof. Since S is stable, we have

(5.3)  ∫_{S} |∇f|^{2}dμ − ∫_{S}(|A|^{2} + R̄ic(ν, ν))f^{2}dμ ≥ 0

for any function f with ∫_{S} fdμ = 0. We follow an idea in [CY] and choose ψ to be a conformal map of degree 1 which maps S onto the standard sphere S^{2} in Euclidean space R^{3}. Using the conformal group of S^{2} it can be arranged that each component ψ_{i} of ψ satisfies ∫ ψ_{i}dμ = 0, see [CY] and [LY]. The Dirichlet integral in (5.3) is equal to 8π/3 for each f = ψ_{i}. Since ∑ ψ_{i}^{2} ≡ 1 we conclude that

(5.4)  8π ≥ ∫_{S} |A|^{2} + R̄ic(ν, ν)dμ.

Again following [CY] we can now use the Gauß equation to obtain

(5.5)  |A|^{2} + R̄ic(ν, ν) = \frac{1}{2}(|A|^{2} − \frac{1}{2}H^{2}) + \frac{3}{4}H^{2} + \frac{1}{2}R̄ − K,

where K is the Gauß curvature of S and R̄ is the scalar curvature of N. Combining (5.4) and (5.5) we derive in view of |A|^{2} − \frac{1}{2}H^{2} = \frac{1}{2}(λ_{1} − λ_{2})^{2} that

(5.6)  ∫_{S}(λ_{1} − λ_{2})^{2}dμ + 3 ∫_{S} H^{2}dμ ≤ 48π − 2 ∫_{S} R̄dμ.

Since |R̄| ≤ c C_{0}r^{−3}, Lemma 5.2.(ii) then implies that

(5.7)  H^{2}|S| ≤ C,

with a uniform constant C provided σ_{0} ≥ c C_{0} is large enough. We may now combine (5.4) and (5.5) slightly differently to compute

∫_{S}(λ_{1} − λ_{2})^{2}dμ = ∫_{S} 2|A|^{2} − H^{2}dμ
= ∫_{S} |A|^{2} + R̄ic(ν, ν)dμ + ∫_{S} R̄ − 3R̄ic(ν, ν) − 2Kdμ
≤ 8π + ∫_{S} R̄ − 3R̄ic(ν, ν)dμ − 8π
= ∫_{S} R̄ − 3R̄ic(ν, ν)dμ
≤ c C_{0} ∫_{S} r^{−3}dμ.

The desired estimate then follows from Lemma 5.2 (ii) and (5.7). To proceed further, we use the second variation inequality (5.3) as in [SSY] with the testfunction

f = u − ∮_{S} udμ ≡ u − ū, u = |\begin{matrix}∘ \\ A \end{matrix}|.

With this choice of f we derive

(5.8)  \begin{aligned}∫_{S}(|A|^{2} & + R̄ic(ν, ν))|\begin{matrix}∘ \\ A \end{matrix}|^{2}dμ \\ & ≤ ∫_{S}(2uū − ū^{2})(|A|^{2} + R̄ic(ν, ν))dμ + ∫_{S} |∇|\begin{matrix}∘ \\ A \end{matrix}||^{2} dμ. \end{aligned}

Since the mean curvature is constant we may proceed as in [SSY; 1.34] to derive the following inequality, where we also use the fact that |R̄ic| ≤ c C_{0}r^{−3} and | ¯∇R̄ic| ≤ c C_{0}r^{−4}.

(5.9)  \begin{aligned}|\begin{matrix}∘ \\ A \end{matrix}|Δ|\begin{matrix}∘ \\ A \end{matrix}| & ≥ (H^{2}2 − |A|^{2})|\begin{matrix}∘ \\ A \end{matrix}|^{2} + \frac{1}{1 + ε} |∇|\begin{matrix}∘ \\ A \end{matrix}||^{2} − \frac{c}{1 + ε}C_{0}r^{−6} \\ & − c C_{0}|\begin{matrix}∘ \\ A \end{matrix}|r^{−4} − c C_{0}r^{−3}|\begin{matrix}∘ \\ A \end{matrix}|^{2} − c C_{0}H_{σ}|\begin{matrix}∘ \\ A \end{matrix}|r^{−3} \end{aligned} .

Here ε > 0 is arbitrary and integration yields

(5.10) \begin{aligned}(1 + \frac{1}{1 + ε}) ∫_{S} |∇|\begin{matrix}∘ \\ A \end{matrix}||^{2} dμ & ≤ ∫_{S}(|A|^{2} − H_{σ}^{2})|\begin{matrix}∘ \\ A \end{matrix}|^{2}dμ \\ & + c C_{0} ∫_{S}(|\begin{matrix}∘ \\ A \end{matrix}|^{2} + H_{σ}|\begin{matrix}∘ \\ A \end{matrix}| + |\begin{matrix}∘ \\ A \end{matrix}|^{−1} + ε^{−1}r^{−3})r^{−3}dμ. \end{aligned}

Multiply (5.8) with 2/(1 + ε) and add the last inequality to derive

(\frac{1 − ε}{1 + ε}) ∫_{S} |A|^{2}|\begin{matrix}∘ \\ A \end{matrix}|^{2}dμ + \frac{ε}{1 + ε} ∫_{S} |∇|\begin{matrix}∘ \\ A \end{matrix}||^{2} dμ

≤ ∫_{S}(2uū − ū^{2})(|A|^{2} + R̄ic(ν, ν))dμ − ∫_{S} H^{2}|\begin{matrix}∘ \\ A \end{matrix}|^{2}dμ

+ c C_{0} ∫_{S}(|\begin{matrix}∘ \\ A \end{matrix}|^{2} + H|\begin{matrix}∘ \\ A \end{matrix}| + |\begin{matrix}∘ \\ A \end{matrix}|r^{−1} + ε^{−1}r^{−3})r^{−3}dμ.

Now notice that |A|^{2} = |\begin{matrix}∘ \\ A \end{matrix}|^{2} + \frac{1}{2}H^{2} and 2uū ≤ ū^{2} + |\begin{matrix}∘ \\ A \end{matrix}|^{2} such that

(\frac{1 − ε}{1 + ε}) ∫_{S} |\begin{matrix}∘ \\ A \end{matrix}|^{4}dμ + \frac{(1 − ε)}{(1 + ε)}H^{2} ∫ |\begin{matrix}∘ \\ A \end{matrix}|^{2}dμ + \frac{ε}{1 + ε} ∫_{S} |∇|\begin{matrix}∘ \\ A \end{matrix}||^{2} dμ
≤ ∫_{S}(2uū − ū^{−2}(|\begin{matrix}∘ \\ A \end{matrix}|^{2} + R̄ic(ν, ν))dμ

+ c C_{0} ∫_{S}(|\begin{matrix}∘ \\ A \end{matrix}|^{2} + H|\begin{matrix}∘ \\ A \end{matrix}| + |\begin{matrix}∘ \\ a \end{matrix}|r^{−1} + ε^{−1}r^{−3})r^{−3}dμ.

Since |R̄ic(ν, ν)| ≤ c C_{0}r^{−3} , the RHS can be estimated by

2ū ∫_{S} |\begin{matrix}∘ \\ A \end{matrix}|^{3}dμ + c C_{0} ∫_{S}(|\begin{matrix}∘ \\ A \end{matrix}|^{2} + H|\begin{matrix}∘ \\ A \end{matrix}| + |\begin{matrix}∘ \\ A \end{matrix}|r^{−1} + ε^{−1}r^{−3})r^{−3}dμ.

By Hölder’s inequality and Proposition 5.3

ū^{2} = |S|^{−2} (∫_{S} |\begin{matrix}∘ \\ A \end{matrix}|dμ)^{2} ≤ |S|^{−1} ∫ |\begin{matrix}∘ \\ A \end{matrix}|^{2}dμ ≤ C C_{0}σ_{0}^{−1}H^{2},

such that

2ū ∫_{S} |\begin{matrix}∘ \\ A \end{matrix}|^{3}dμ ≤ \frac{1}{2} ∫_{S} |\begin{matrix}∘ \\ A \end{matrix}|^{4}dμ + c C_{0}σ_{0}^{−1}H^{2} ∫_{S} |\begin{matrix}∘ \\ A \end{matrix}|dμ.

Choosing then σ ≥ c C_{0} sufficiently large and 0 < ε < \frac{1}{2} suffienctly small we conclude

\begin{matrix}∫ |\begin{matrix}∘ \\ A \end{matrix}|^{4}dμ & + H^{2} ∫_{S} |\begin{matrix}∘ \\ A \end{matrix}|^{2}dμ + ∫_{S} |∇|\begin{matrix}∘ \\ A \end{matrix}||^{2} dμ ≤ \\ cC_{0} ∫_{S}(|\begin{matrix}∘ \\ A \end{matrix}|^{2} + H|\begin{matrix}∘ \\ A \end{matrix}| + |\begin{matrix}∘ \\ A \end{matrix}|r^{−1} + r^{−3})r^{−3}dμ. \end{matrix}

Using now again Hölder’s inequality, Lemma 5.2 and Proposition 5.3 we obtain the estimate

(5.11)  ∫_{S} |\begin{matrix}∘ \\ A \end{matrix}|^{4}dμ + H^{2} ∫_{S} |\begin{matrix}∘ \\ A \end{matrix}|^{2}dμ + ∫_{S} |∇|\begin{matrix}∘ \\ A \end{matrix}||^{2} dμ ≤ c C_{0}σ_{0}^{−4}.

At this stage we need a Sobolev inequality on S.

5.4 Proposition. There is an absolute constant c such that provided σ_{0} ≥ c C_{0} sufficiently large we have for all Lipschitz functions g on S the Sobolev inequality

(∫_{S} |g|^{2}dμ)^{\frac{1}{2}} ≤ c ∫_{S} |∇g| + H|g|dμ.

Proof. It is wellknown that the above inequality holds for hypersurfaces S immersed in Euclidean space if H is replaced by the mean curvature H^{e} of S with respect to the Euclidean metric. From Lemma 1.4 and (5.6) it is clear that the difference between H^{e} and H can be bounded by c C_{0}r^{−2}. Since the metric g_{αβ} and the Euclidean metric are uniformly equivalent on N this shows that

(∫_{S} |g|^{2}dμ)^{\frac{1}{2}} ≤ c ∫_{S} |∇g| + H|g| + C_{0}r^{−2}|g|dμ

holds with some absolute constant c. Using now Hölder’s inequality and Lemma 5.2(ii) we estimate

\begin{aligned}∫_{S} C_{0}r^{−2}|g|dμ & ≤ (∫_{S} |g|^{2}dμ)^{\frac{1}{2}} · (∫_{S} r^{−4}dμ)^{\frac{1}{2}} \\ & ≤ c C_{0}σ_{0}^{−1}H|S|^{\frac{1}{2}} · (∫_{S} |g|^{2}dμ)^{\frac{1}{2}} . \end{aligned}

In view of (5.7) this term can be absorbed on the LHS for σ_{0} ≥ c C_{0} sufficiently large, completing the proof of the Sobolev inequality.

5.5 Corollary. There is an absolute constant c such that provided σ_{0} ≥ c C_{0} we have the estimate

c^{−1}H^{−2} ≤ |S| ≤ cH^{−2}.

Proof. The second inequality was shown in Proposition 5.3 and the first inequality follows from Proposition 5.4 if we set g ≡ 1. Notice also that after replacing g by g^{p}, p ≥ 1 in Proposition 5.4 we obtain from Hölder’s inequality that

(∫_{S} |g|^{2p}dμ)^{1/p} ≤ cp^{2}|suppg|^{1/p} · ∫_{S} |∇g|^{2} + H^{2}|g|^{2}dμ

for all p ≥ 1. We are now ready to derive a pointwise estimate for u = |\begin{matrix}∘ \\ A \end{matrix}|. As a first step we prove:

5.6 Lemma. For all δ > 0 there is a constant C(δ, m, C_{0}) such that

|\begin{matrix}∘ \\ A \end{matrix}| ≤ C(δ, m, C_{0})σ_{0}^{−2−δ}H^{−2δ}.

In particular there is ε > 0 depending on q > \frac{1}{2} such that

|\begin{matrix}∘ \\ A \end{matrix}| ≤ C(q, m, C_{0})H^{1+ε}.

Proof. The second inequality follows from the first with σ_{0} ≥ H^{−q}. Let u_{k} = max(u − k, 0) and A(k) = {x ∈ S|u(x) > k}. We multiply inequality (5.9) by u_{k}^{p}, p ≥ 1 and integrate by parts. This yields

\begin{aligned}(\frac{2 + ε}{1 + ε}) & ∫_{A(k)} |∇|\begin{matrix}∘ \\ A \end{matrix}||^{2} u_{k}^{p}dμ + p ∫_{A(k)} u_{k}^{p−1}u |∇|\begin{matrix}∘ \\ A \end{matrix}||^{2} dμ + H^{2} ∫_{A(k)} u^{2}u_{k}^{p}dμ \\ & ≤ ∫_{A(k)} |\begin{matrix}∘ \\ A \end{matrix}|^{4}u_{k}^{p}dμ + c C_{0} ∫_{A(k)} (u^{2} + ur^{−1} + uH + ε^{−1}r^{−3}) r^{−3}u_{k}^{p}dμ. \end{aligned}

Using then |\begin{matrix}∘ \\ A \end{matrix}| = u, r ≥ σ_{0}, and the Schwarz inequality, we derive

∫_{A(k)} |∇u|^{2}u_{k}^{p}dμ + p ∫_{A(k)} uu_{k}^{p−1}|∇u|^{2}dμ + H^{2} ∫_{A(k)} u^{2}u_{k}^{p}dμ
≤ ∫_{A(k)} u^{4}u_{k}^{p}dμ + C ∫_{A(k)} u_{k}^{p}(σ_{0}^{−6} + uσ_{0}^{−4})dμ
≤ ∫_{A(k)} u^{4}u_{k}^{p}dμ + Cσ_{0}^{−5} ∫_{A(k)} u_{k}^{p}dμ.

Now we set f = u_{k}^{p/2+1}, such that the above inequality becomes

\begin{matrix}\frac{1}{p} ∫_{A(k)} |∇f|^{2}dμ & + H^{2} ∫_{A(k)} u^{2}u_{k}^{p}dμ \\ ≤ & c ∫_{A(k)} u^{4}u_{k}^{p}dμ + Cσ_{0}^{−4}u_{k}^{p}dμ. \end{matrix}

The Sobolev inequality established above shows that for 1 < η < ∞

\begin{aligned}( ∫_{A(k)} |f|^{2η}dμ)^{1/η} & ≤ cη^{2}|A(k)|^{1/η} ∫_{A(k)} |∇f|^{2} + H^{2}f^{2}dμ \\ & ≤ c(p, q)|A(k)|^{1/η} ∫_{A(k)} u^{4}u_{k}^{p} + Cσ_{0}^{−5}u_{k}^{p}dμ, \end{aligned}

such that by the Hölder inequality

∫_{A(k)} f^{2}dμ ≤ c(p, q)|A(k)| ∫_{A(k)} u^{4}u_{k}^{p} + Cσ_{0}^{−5}u_{k}^{p}dμ.

To estimate the two terms on the RHS of this inequality, observe that for any ρ ≥ 1 we have from the Sobolev inequality and estimate (5.11) that

\begin{aligned}( ∫_{A(k)} u_{k}^{2ρ}dμ)^{1/ρ} & ≤ cρ^{2}|A(k)|^{1/ρ}( ∫_{A(k)} |∇u|^{2} + H^{2}|u|^{2}dμ) \\ & ≤ cρ^{2}|A(k)|^{1/ρ}Cσ_{0}^{−4}. \end{aligned}

Applying this to the first term on the RHS above we get

∫_{A(k)} u^{4}u_{k}^{p}dμ ≤ ∫_{A(k)} u^{p+4}dμ ≤ Cp^{p+4}σ_{0}^{−2p−8}|A(k)|

and from the second term

∫_{A(k)} u_{k}^{p}dμ ≤ Cp^{p}σ_{0}^{−2p}|A(k)|.

We now fix η = 2 to obtain

∫_{A(k)} f^{2}dμ ≤ C(p)σ_{0}^{−2p−5}|A(k)|^{2},

and since f^{2} = u_{k}^{p+2} we get for h > k ≥ 0 the iteration inequality

|h − k|^{p+2}|A(h)| ≤ C|A(k)|^{2}σ_{0}^{−2p−5}.

By a result due to Stampacchia ([St], Lemma 4.1) this implies that |A(d)| = 0 for

d^{p+2} ≤ Cσ_{0}^{−2p−5}|A(0)|

and hence in view of the area estimate for S

u ≤ d ≤ C(p, m, C_{0})σ_{0}^{(−2p−5)/(p+2)}H^{−2/(p+2)}.

This yields the desired estimate.

Notice that the inequality just established implies convexity of S and strong pinching of the second fundamental form. We can now bound the derivative of the second fundamental form similarly as in Proposition 3.12.

5.7 Lemma. There is a constant c depending on q such that

|∇Å| ≤ CH^{\frac{3+ε}{2}} .

Proof. Using the decay properties of the metric and its derivatives we compute as in Corollary 3.11

Δ|∇Å|^{2} ≥ −c|∇Å|^{2}σ_{0}^{−2} − cC_{0}|∇Å|σ_{0}^{−5}

Note that the decay of the RHS is slightly worse than in Corollary 3.11 since we do not know yet that S is ”centered” and the normal does not necessarily point in an eigendirection of the ambient Ricci curvature. From here we derive as in Proposition 3.12

△(|∇Å|^{2} + C_{1}|Å|^{2}σ_{0}^{−2}) ≥ c_{1}|∇Å|^{2}σ_{0}^{−2} − cC_{0}|∇Å|σ_{0}^{−5} − cC_{0}^{2}|Å|σ_{0}^{−6}.

From |Å| ≤ CH^{1+ε} and σ_{0} ≥ H^{−q}, q > \frac{1}{2} together with the maximum principle we get the desired estimate.

Proof of Theorem 5.1 In view of Lemma 5.6 and 5.7 the surface S is close to a round Euclidean sphere: Since σ_{0} ≥ H^{−q} with q > \frac{1}{2} it follows that the mean curvature H^{e} of S in R^{3} differs from H by no more than CH^{1+ε}, and proceeding as in Proposition 2.1 we see that there is a vector a⃗ in R^{3} and r_{0} > 0 such that S satisfies

(5.12)  \begin{aligned}|λ_{i}^{e} − r_{0}^{−1}| & ≤ CH^{1+ε},  i = 1, 2, \\ |(y − a⃗) − r_{0}ν_{e}| & ≤ CH^{−1+ε}, \\ |ν_{e} − r_{0}^{−1}(y − \overrightarrow{a})| & ≤ CH^{ε}. \end{aligned}

In the next step we have to show that the displacement a⃗ is small. For this purpose we think of a variation in direction of −a⃗ which preserves the enclosed volume and decreases area if S is off center. We begin with the identity

(5.13)  0 = H ∫_{S}⟨b⃗, ν_{e}⟩_{e}dμ_{e} = ∫_{S} H⟨b⃗, ν_{e}⟩_{e}dμ_{e},

where b⃗ = −a⃗/|a⃗| points in direction of the displacement of the surface. In view of Lemmata 1.2 and 1.4 the mean curvature H can be expressed in terms of ψ = (1 + \frac{m}{2r})^{2} and the Euclidean mean curvature H^{e} up to terms of order r^{−3}:

|H − ψ^{−1}H^{e} − 2ψ^{−2}∂_{ν_{e}}ψ| ≤ Cr^{−3}.

So we derive from (5.13) that

(5.14)  \begin{aligned}| ∫_{S} (H^{e}ψ^{−1} & + 2ψ^{−2}∂_{ν_{e}}ψ)⟨b⃗, ν_{e}⟩_{e}dμ_{e}| \\ & ≤ C ∫_{S} r^{−3}dμ_{e} ≤ Cσ_{0}^{−1}, \end{aligned}

where we also used Lemma 5.1 for the last estimate. Using the first variation formula with respect to the Euclidean metric we compute

(5.15)  ∫_{S} H^{e}ψ^{−1}⟨b⃗, ν_{e}⟩_{e}dμ_{e} = ∫_{S} div_{S}(ψ^{−1}b⃗)dμ_{e} = − ∫_{S} ψ^{−2}⟨b⃗, ∇^{e}ψ⟩dμ_{e}.

Using now ⟨∇^{e}ψ, b⃗⟩ = ⟨Dψ, b⃗⟩ − ⟨b⃗, ν_{e}⟩∂_{ν_{e}}ψ together with the fact that |Dψ| ≤ Cr^{−2} we conclude that

(5.16)  | ∫_{S} 3⟨b⃗, ν_{e}⟩∂_{ν_{e}}ψdμ_{e} − ∫_{S}⟨b⃗, Dψ⟩dμ_{e}| ≤ Cσ_{0}^{−1}.

At this stage we utilize that S is close to a Euclidean sphere as described in (5.12) together with |Dψ| ≤ Cr^{−2}: We may compute the integrals in (5.16) on the round Euclidean sphere S_{e} := S_{r_{0}}^{2}(a⃗), introducing an error no larger than CH^{ε}. Hence, writing ⟨b⃗, ν_{e}⟩ = −cosφ, 0 ≤ φ ≤ π and using

\frac{∂}{∂x^{i}} ψ = −m(1 + \frac{m}{2r})x^{i}r^{−3}

as well as |x⃗ − a⃗ − r_{0}ν_{e}| ≤ CH^{−1+ε} we conclude that up to terms of order r^{−2}H^{ε}

⟨b⃗, Dψ⟩ = \frac{m}{r^{3}} |a⃗| + \frac{m}{r^{3}} r_{0}cosφ

and

⟨b⃗, ν_{e}⟩∂_{ν_{e}}ψ = \frac{m}{r^{3}} |a⃗|cos^{2}φ + \frac{m}{r^{3}} r_{0}cosφ.

So we derive from (5.16) that

(5.17)  |m ∫_{S_{e}} \frac{3|a⃗|}{r^{3}}cos^{2}φ + \frac{2r_{0}}{r^{3}} cosφ − \frac{|a⃗|}{r^{3}}dμ_{e}| ≤ CH^{ε}.

It remains to evaluate the integrals. Using polar coordinates on S_{e} we derive

∫_{S_{e}} \frac{cos^{2}φ}{r^{3}}dμ_{e} = 2πr_{0}^{2} ∫_{0}^{π} sinφ cos^{2}φ r^{−3}dφ,

where r = \sqrt{|a⃗|^{2} + r_{0}^{2} + 2r_{0}|a⃗|cosφ}, such that \frac{d}{dφ}r = −r_{0}|a⃗|r^{−1}sinφ. We have r_{0} > |a⃗| and the integral can be explicitely evaluated to yield

(5.18)  ∫_{S_{e}} \frac{cos^{2}φ}{r^{3}}dμ_{e} = −\frac{8}{3}π \frac{1}{r_{0}} + 4π \frac{r_{0}}{r_{0}^{2} − |a⃗|^{2}} .

Similarly we compute

(5.19)  ∫_{S_{e}} \frac{cosφ}{r^{3}}dμ_{e} = −4π \frac{|a⃗|}{r_{0}^{2} − |a⃗|^{2}}

and

(5.20)  ∫_{S_{e}} \frac{1}{r^{3}} dμ_{e} = 4π \frac{r_{0}}{r_{0}^{2} − |a⃗|^{2}} .

Inserting then (5.18), (5.19) and (5.20) into (5.17) we arrive at the crucial inequality

(5.21)  8πm\frac{|a⃗|}{r_{0}} ≤ CH^{ε},

where C and ε only depend on q, m, C_{0}. If we then fix H_{1} so small that |a⃗| < \frac{1}{2}r_{0} for all H < H_{1}, the minimum radius and the maximum radius on the surface S are compatible. The case of surfaces which are far displaced, i.e. |a⃗| > r_{0} can be treated similarly. The behaviour of such exterior surfaces under deformations has some interesting properties which will be treated elsewhere.

In particular we can now replace σ_{0}^{−1} by H in all our previous estimates. For instance we now get from Simons’ identity the inequality

△|Å| ≥ −CH^{4},

and an iteration as in the proof of Lemma 5.6 now yields the improved estimate

(5.22)  |Å| ≤ CH^{2},

leading also to an improved estimate for the derivative of the curvature as in Lemma 5.7:

(5.23)  |∇Å| ≤ CH^{3}.

It follows that the roundness estimates on the surface S in (5.12) can be improved to

\begin{aligned}|λ_{i}^{e} − r_{0}^{−1}| & ≤ CH^{2}, i = 1, 2, \\ |(y − a⃗) − r_{0}ν_{e}| & ≤ C, \\ |ν_{e} − r_{0}^{−1}(y − \overrightarrow{a})| & ≤ CH. \end{aligned}

Carrying then out the computations for the displacement |a⃗| above with these more precise estimates in mind we derive with ε = 1 as in (5.21)

8πm\frac{|a⃗|}{r_{0}} ≤ CH

which implies a uniform bound for |a⃗| depending only on the given data q, m, C_{0} provided 0 < H < H_{1}. Having thus obtained a uniform bound on the position of the surface keeping it close to a coordinate sphere it follows that the off diagonal parts of the Ricci curvature on S satisfy the strong decay estimates stated in Lemma 3.8, Proposition 3.9 and Corollary 3.11. We can therefore proceed as in the parabolic case to finally get the estimates of Proposition 3.9 and Proposition 3.12:

|Å|^{2} ≤ C^{2}H^{6}, |∇Å|^{2} ≤ C^{2}H^{8}.

Notice here that (5.22) and (5.23) guarantee the first two estimates in Lemma 3.8.

The last two estimates above on the traceless part of the second fundamental form then show that for σ_{0} sufficiently large, or equivalently, for 0 < H < H_{0}, H_{0} = H_{0}(q, m, C_{0}) sufficiently small, the surface S is contained in some B_{σ}(B_{1}, B_{2}, B_{3}), and Theorem 4.1 yields that S has to coincide with the leaf of the foliation with mean curvature H. This completes the proof of Theorem 5.1.

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