Curvature flows in semi-Riemannian manifolds
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS
CLAUS GERHARDT
Abstract. We prove that the limit hypersurfaces of converging curva-
ture flows are stable, if the initial velocity has a weak sign, and give a
survey of the existence and regularity results.
Contents
1. Introduction 1
2. Notations and preliminary results 2
3. Evolution equations for some geometric quantities 4
4. Essential parabolic flow equations 9
5. Stability of the limit hypersurfaces 15
6. Existence results 25
7. The inverse mean curvature flow 39
8. The IMCF in ARW spaces 41
Transition from big crunch to big bang 45
References 47
1. Introduction
In this paper we want to give a survey of the existence and regularity results
for extrinsic curvature flows in semi-Riemannian manifolds, i.e., Riemannian or
Lorentzian ambient spaces, with an emphasis on flows in Lorentzian spaces. In
order to treat both cases simultaneously terminology like spacelike, timelike,
etc., that only makes sense in a Lorentzian setting should be ignored in the
Riemannian case.
The general stability result for the limit hypersurfaces of converging curva-
ture flows in Section 5 is new. The regularity result in Theorem 6.5—especially
the time independent Cm+2,α-estimates—for converging curvature flows that
are graphs is interesting too.
Date: October 23, 2018.
2000 Mathematics Subject Classification. 35J60, 53C21, 53C44, 53C50, 58J05.
Key words and phrases. semi-Riemannian manifold, mass, stable solutions, cosmological
spacetime, general relativity, curvature flows, ARW spacetime.
This research was supported by the Deutsche Forschungsgemeinschaft.
http://arxiv.org/abs/0704.0236v4
2 CLAUS GERHARDT
2. Notations and preliminary results
The main objective of this section is to state the equations of Gauß, Co-
dazzi, and Weingarten for hypersurfaces. In view of the subtle but important
difference that is to be seen in the Gauß equation depending on the nature
of the ambient space—Riemannian or Lorentzian—, which we already men-
tioned in the introduction, we shall formulate the governing equations of a
hypersurface M in a semi-Riemannian (n+1)-dimensional space N , which is
either Riemannian or Lorentzian. Geometric quantities in N will be denoted
by (ḡαβ), (R̄αβγδ), etc., and those in M by (gij), (Rijkl), etc. Greek indices
range from 0 to n and Latin from 1 to n; the summation convention is always
used. Generic coordinate systems in N resp. M will be denoted by (xα) resp.
(ξi). Covariant differentiation will simply be indicated by indices, only in case
of possible ambiguity they will be preceded by a semicolon, i.e. for a function
u in N , (uα) will be the gradient and (uαβ) the Hessian, but e.g., the covariant
derivative of the curvature tensor will be abbreviated by R̄αβγδ;ǫ. We also point
out that
(2.1) R̄αβγδ;i = R̄αβγδ;ǫx
with obvious generalizations to other quantities.
Let M be a spacelike hypersurface, i.e. the induced metric is Riemannian,
with a differentiable normal ν. We define the signature of ν, σ = σ(ν), by
(2.2) σ = ḡαβν
ανβ = 〈ν, ν〉.
In case N is Lorentzian, σ = −1, and ν is timelike.
In local coordinates, (xα) and (ξi), the geometric quantities of the spacelike
hypersurface M are connected through the following equations
(2.3) xαij = −σhijνα
the so-called Gauß formula. Here, and also in the sequel, a covariant derivative
is always a full tensor, i.e.
(2.4) xαij = x
,ij − Γ kijxαk + Γ̄αβγx
The comma indicates ordinary partial derivatives.
In this implicit definition the second fundamental form (hij) is taken with
respect to −σν.
The second equation is the Weingarten equation
(2.5) ναi = h
where we remember that ναi is a full tensor.
Finally, we have the Codazzi equation
(2.6) hij;k − hik;j = R̄αβγδναxβi x
and the Gauß equation
(2.7) Rijkl = σ{hikhjl − hilhjk}+ R̄αβγδxαi x
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 3
Here, the signature of ν comes into play.
2.1. Definition. (i) Let F ∈ C0(Γ̄ ) ∩ C2,α(Γ ) be a strictly monotone cur-
vature function, where Γ ⊂ Rn is a convex, open, symmetric cone containing
the positive cone, such that
(2.8) F |∂Γ = 0 ∧ F |Γ > 0.
Let N be semi-Riemannian. A spacelike, orientable1 hypersurface M ⊂ N
is called admissible, if its principal curvatures with respect to a chosen normal
lie in Γ . This definition also applies to subsets of M .
(ii) Let M be an admissible hypersurface and f a function defined in a
neighbourhood of M . M is said to be an upper barrier for the pair (F, f), if
(2.9) F |M ≥ f
(iii) Similarly, a spacelike, orientable hypersurfaceM is called a lower barrier
for the pair (F, f), if at the points Σ ⊂M , where M is admissible, there holds
(2.10) F |Σ ≤ f.
Σ may be empty.
(iv) If we consider the mean curvature function, F = H , then we suppose F
to be defined in Rn and any spacelike, orientable hypersurface is admissible.
One of the assumptions that are used when proving a priori estimates is that
there exists a strictly convex function χ ∈ C2(Ω̄) in a given domain Ω. We
shall state sufficient geometric conditions guaranteeing the existence of such a
function. The lemma below will be valid in Lorentzian as well as Riemannian
manifolds, but we formulate and prove it only for the Lorentzian case.
2.2. Lemma. Let N be globally hyperbolic, S0 a Cauchy hypersurface, (xα)
a special coordinate system associated with S0, and Ω̄ ⊂ N be compact. Then,
there exists a strictly convex function χ ∈ C2(Ω̄) provided the level hypersur-
faces {x0 = const} that intersect Ω̄ are strictly convex.
Proof. For greater clarity set t = x0, i.e., t is a globally defined time function.
Let x = x(ξ) be a local representation for {t = const}, and ti, tij be the
covariant derivatives of t with respect to the induced metric, and tα, tαβ be the
covariant derivatives in N , then
(2.11) 0 = tij = tαβx
j + tαx
and therefore,
(2.12) tαβx
j = −tαx
ij = −h̄ijtανα.
Here, (να) is past directed, i.e., the right-hand side in (2.12) is positive definite
in Ω̄, since (tα) is also past directed.
1A hypersurface is said to be orientable, if it has a continuous normal field.
4 CLAUS GERHARDT
Choose λ > 0 and define χ = eλt, so that
(2.13) χαβ = λ
2eλttαtβ + λe
λttαβ .
Let p ∈ Ω be arbitrary, S = {t = t(p)} be the level hypersurface through p,
and (ηα) ∈ Tp(N). Then, we conclude
(2.14) e−λtχαβη
αηβ = λ2|η0|2 + λtijηiηj + 2λt0jη0ηi,
where tij now represents the left-hand side in (2.12), and we infer further
(2.15)
e−λtχαβη
αηβ ≥ 1
λ2|η|02 + [λǫ− cǫ]σijηiηj
λ{−|η0|2 + σijηiηj}
for some ǫ > 0, and where λ is supposed to be large. Therefore, we have in Ω̄
(2.16) χαβ ≥ cḡαβ , c > 0,
i.e., χ is strictly convex. �
3. Evolution equations for some geometric quantities
Curvature flows are used for different purposes, they can be merely vehicles
to approximate a stationary solution, in which case the flow is driven not
only by a curvature function but also by the corresponding right-hand side,
an external force, if you like, or the flow is a pure curvature flow driven only
by a curvature function, and it is used to analyze the topology of the initial
hypersurface, if the ambient space is Riemannian, or the singularities of the
ambient space, in the Lorentzian case.
In this section we are treating very general curvature flows2 in a semi-
Riemannian manifold N = Nn+1, though we only have the Riemannian or
Lorentzian case in mind, such that the flow can be either a pure curvature flow
or may also be driven by an external force. The nature of the ambient space,
i.e., the signature of its metric, is expressed by a parameter σ = ±1, such that
σ = 1 corresponds to the Riemannian and σ = −1 the Lorentzian case. The
parameter σ can also be viewed as the signature of the normal of the spacelike
hypersurfaces, namely,
(3.1) σ = 〈ν, ν〉.
Properties like spacelike, achronal, etc., however, only make sense, when N
is Lorentzian and should be ignored otherwise.
We consider a strictly monotone, symmetric, and concave curvature F ∈
C4,α(Γ ), homogeneous of degree 1, a function 0 < f ∈ C4,α(Ω), where Ω ⊂ N
is an open set, and a real function Φ ∈ C4,α(R+) satisfying
(3.2) Φ̇ > 0 and Φ̈ ≤ 0.
For notational reasons, let us abbreviate
(3.3) f̃ = Φ(f).
2We emphasize that we are only considering flows driven by the extrinsic curvature not
by the intrinsic curvature.
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 5
Important examples of functions Φ are
(3.4) Φ(r) = r, Φ(r) = log r, Φ(r) = −r−1
(3.5) Φ(r) = r
k , Φ(r) = −r− 1k , k ≥ 1.
3.1. Remark. The latter choices are necessary, if the curvature function F
is not homogeneous of degree 1 but of degree k, like the symmetric polynomials
Hk. In this case we would sometimes like to define F = Hk and not H
k , since
(3.6) F ij =
is then divergence free, if the ambient space is a spaceform, cf. Lemma 5.8 on
page 24, though on the other hand we need a concave operator for technical
reasons, hence we have to take the k-th root.
The curvature flow is given by the evolution problem
(3.7)
ẋ = −σ(Φ− f̃)ν,
x(0) = x0,
where x0 is an embedding of an initial compact, spacelike hypersurfaceM0 ⊂ Ω
of class C6,α, Φ = Φ(F ), and F is evaluated at the principal curvatures of the
flow hypersurfaces M(t), or, equivalently, we may assume that F depends on
the second fundamental form (hij) and the metric (gij) of M(t); x(t) is the
embedding of M(t) and σ the signature of the normal ν = ν(t), which is
identical to the normal used in the Gaussian formula (2.3) on page 2.
The initial hypersurface should be admissible, i.e., its principal curvatures
should belong to the convex, symmetric cone Γ ⊂ Rn.
This is a parabolic problem, so short-time existence is guaranteed, cf. [18,
Chapter 2.5]
There will be a slight ambiguity in the terminology, since we shall call the
evolution parameter time, but this lapse shouldn’t cause any misunderstand-
ings, if the ambient space is Lorentzian.
At the moment we consider a sufficiently smooth solution of the initial value
problem (3.7) and want to show how the metric, the second fundamental form,
and the normal vector of the hypersurfaces M(t) evolve. All time derivatives
are total derivatives, i.e., covariant derivatives of tensor fields defined over the
curve x(t), cf. [17, Chapter 11.5]; t is the flow parameter, also referred to
as time, and (ξi) are local coordinates of the initial embedding x0 = x0(ξ)
which will also serve as coordinates for the the flow hypersurfaces M(t). The
coordinates in N will be labelled (xα), 0 ≤ α ≤ n.
3.2. Lemma (Evolution of the metric). The metric gij of M(t) satisfies the
evolution equation
(3.8) ġij = −2σ(Φ− f̃)hij .
6 CLAUS GERHARDT
Proof. Differentiating
(3.9) gij = 〈xi, xj〉
covariantly with respect to t yields
(3.10)
ġij = 〈ẋi, xj〉+ 〈xi, ẋj〉
= −2σ(Φ− f̃)〈xi, νj〉 = −2σ(Φ− f̃)hij ,
in view of the Codazzi equations. �
3.3.Lemma (Evolution of the normal). The normal vector evolves according
(3.11) ν̇ = ∇M (Φ− f̃) = gij(Φ− f̃)ixj .
Proof. Since ν is unit normal vector we have ν̇ ∈ T (M). Furthermore, differ-
entiating
(3.12) 0 = 〈ν, xi〉
with respect to t, we deduce
�(3.13) 〈ν̇, xi〉 = −〈ν, ẋi〉 = (Φ− f̃)i.
3.4. Lemma (Evolution of the second fundamental form). The second fun-
damental form evolves according to
(3.14) ḣ
i = (Φ− f̃)
i + σ(Φ− f̃)h
k + σ(Φ− f̃)R̄αβγδν
γxδkg
(3.15) ḣij = (Φ− f̃)ij − σ(Φ − f̃)hki hkj + σ(Φ − f̃)R̄αβγδναx
γxδj .
Proof. We use the Ricci identities to interchange the covariant derivatives of ν
with respect to t and ξi
(3.16)
(ναi ) = (ν̇
α)i − R̄αβγδνβx
= gkl(Φ− f̃)kixαl + gkl(Φ− f̃)kxαli − R̄αβγδνβx
For the second equality we used (3.11). On the other hand, in view of the
Weingarten equation we obtain
(3.17) D
(ναi ) =
(hki x
k ) = ḣ
k + h
Multiplying the resulting equation with ḡαβx
j we conclude
(3.18) ḣki gkj − σ(Φ − f̃)hki hkj = (Φ− f̃)ij + σ(Φ − f̃)R̄αβγδναx
or equivalently (3.14).
To derive (3.15), we differentiate
(3.19) hij = h
i gkj
with respect to t and use (3.8). �
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 7
We emphasize that equation (3.14) describes the evolution of the second
fundamental form more meaningfully than (3.15), since the mixed tensor is
independent of the metric.
3.5. Lemma (Evolution of (Φ− f̃)). The term (Φ− f̃) evolves according to
the equation
(3.20)
(Φ− f̃)
− Φ̇F ij(Φ− f̃)ij =σΦ̇F ijhikhkj (Φ− f̃) + σf̃ανα(Φ− f̃)
+ σΦ̇F ijR̄αβγδν
γxδj(Φ − f̃),
where
(3.21) (Φ− f̃)′ = d
(Φ− f̃)
(3.22) Φ̇ =
Φ(r).
Proof. When we differentiate F with respect to t we consider F to depend on
the mixed tensor h
i and conclude
(3.23) (Φ− f̃)′ = Φ̇F ij ḣ
i − f̃αẋ
The equation (3.20) then follows in view of (3.7) and (3.14). �
3.6. Remark. The preceding conclusions, except Lemma 3.5, remain valid
for flows which do not depend on the curvature, i.e., for flows
(3.24)
ẋ = −σ(−f)ν = σfν,
x(0) = x0,
where f = f(x) is defined in an open set Ω containing the initial spacelike
hypersurface M0. In the preceding equations we only have to set Φ = 0 and
f̃ = f .
The evolution equation for the mean curvature then looks like
(3.25) Ḣ = −∆f − σ{|A|2 + R̄αβνανβ}f,
where the Laplacian is the Laplace operator on the hypersurfaceM(t). This is
exactly the derivative of the mean curvature operator with respect to normal
variations as we shall see in a moment.
But first let us consider the following example.
3.7. Example. Let (xα) be a future directed Gaussian coordinate system
in N , such that the metric can be expressed in the form
(3.26) ds̄2 = e2ψ{σ(dx0)2 + σijdxidxj}.
Denote by M(t) the coordinate slices {x0 = t}, then M(t) can be looked at as
the flow hypersurfaces of the flow
(3.27) ẋ = −σ(−eψ)ν̄,
8 CLAUS GERHARDT
where we denote the geometric quantities of the slices by ḡij , ν̄, h̄ij , etc.
Here x is the embedding
(3.28) x = x(t, ξi) = (t, xi).
Notice that, if N is Riemannian, the coordinate system and the normal are
always chosen such that ν0 > 0, while, if N is Lorentzian, we always pick the
past directed normal.
Hence the mean curvature of the slices evolves according to
(3.29) ˙̄H = −∆eψ − σ{|Ā|2 + R̄αβ ν̄αν̄β}eψ.
We can now derive the linearization of the mean curvature operator of a
spacelike hypersurface, compact or non-compact.
3.8. Let M0 ⊂ N be a spacelike hypersurface of class C4. We first assume
that M0 is compact; then there exists a tubular neighbourhood U and a cor-
responding normal Gaussian coordinate system (xα) of class C3 such that ∂
is normal to M0.
Let us consider in U of M0 spacelike hypersurfaces M that can be writ-
ten as graphs over M0, M = graphu, in the corresponding normal Gaussian
coordinate system. Then the mean curvature of M can be expressed as
(3.30) H = {−∆u+ H̄ − σv−2uiujh̄ij}v,
where σ = 〈ν, ν〉, and hence, choosing u = ǫϕ, ϕ ∈ C2(M0), we deduce
(3.31)
H |ǫ=0 = −∆ϕ+ ˙̄Hϕ
= −∆ϕ− σ(|Ā|2 + R̄αβνανβ)ϕ,
in view of (3.29).
The right-hand side is the derivative of the mean curvature operator applied
to ϕ.
If M0 is non-compact, tubular neighbourhoods exist locally and the relation
(3.31) will be valid for any ϕ ∈ C2c (M0) by using a partition of unity.
The preceding linearization can be immediately generalized to a hypersurface
M0 solving the equation
(3.32) F |M0 = f,
where f = f(x) is defined in a neighbourhood of M0 and F = F (hij) is curva-
ture operator.
3.9. Lemma. Let M0 be of class C
m,α, m ≥ 2, 0 ≤ α ≤ 1, satisfy (3.32).
Let U be a (local) tubular neighbourhood of M0, then the linearization of the
operator F − f expressed in the normal Gaussian coordinate system (xα) cor-
responding to U and evaluated at M0 has the form
(3.33) − F ijuij − σ{F ijhki hkj + F ijR̄αβγδναx
γxδj + fαν
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 9
where u is a function defined in M0, and all geometric quantities are those of
M0; the derivatives are covariant derivatives with respect to the induced metric
of M0. The operator will be self-adjoint, if F
ij is divergence free.
Proof. For simplicity assume thatM0 is compact, and let u ∈ C2(M0) be fixed.
Then the hypersurfaces
(3.34) Mǫ = graph(ǫu)
stay in the tubular neighbourhood U for small ǫ, |ǫ| < ǫ0, and their second
fundamental forms (hij) can be expressed as
(3.35) v−1hij = −(ǫu)ij + h̄ij ,
where h̄ij is the second fundamental form of the coordinate slices {x0 = const}.
We are interested in
(3.36)
(F − f)|ǫ=0 .
To differentiate F with respect to ǫ it is best to consider the mixed form
i ) of the second fundamental form to derive
(3.37)
(F − f) = F ij ḣ
u = −F ijuij + F ij ˙̄h
where the equation is evaluated at ǫ = 0 and ˙̄h
i is the derivative of h̄
i with
respect to x0.
The result then follows from the evolution equation (3.14) for the flow (3.27),
i.e., we have to replace (Φ− f̃) in (3.14) by −1. �
4. Essential parabolic flow equations
From (3.14) on page 6 we deduce with the help of the Ricci identities a
parabolic equation for the second fundamental form
4.1. Lemma. The mixed tensor h
i satisfies the parabolic equation
(4.1)
i − Φ̇F
i;kl =
σΦ̇F klhrkh
i − σΦ̇Fhrih
rj + σ(Φ− f̃)hki h
− f̃αβxαi x
kj + σf̃αν
i + Φ̇F
kl,rshkl;ih
+ Φ̈FiF
j + 2Φ̇F klR̄αβγδx
− Φ̇F klR̄αβγδxαmx
rj − Φ̇F klR̄αβγδxαmx
+ σΦ̇F klR̄αβγδν
γxδl h
i − σΦ̇F R̄αβγδν
γxδmg
+ σ(Φ− f̃)R̄αβγδναxβi ν
γxδmg
+ Φ̇F klR̄αβγδ;ǫ{ναxβkx
mj + ναx
10 CLAUS GERHARDT
Proof. We start with equation (3.14) on page 6 and shall evaluate the term
(4.2) (Φ− f̃)ji ;
since we are only working with covariant spatial derivatives in the subsequent
proof, we may—and shall—consider the covariant form of the tensor
(4.3) (Φ− f̃)ij .
First we have
(4.4) Φi = Φ̇Fi = Φ̇F
klhkl;i
(4.5) Φij = Φ̇F
klhkl;ij + Φ̈F
klhkl;iF
rshrs;j + Φ̇F
kl,rshkl,;ihrs;j .
Next, we want to replace hkl;ij by hij;kl. Differentiating the Codazzi equation
(4.6) hkl;i = hik;l + R̄αβγδν
where we also used the symmetry of hik, yields
(4.7)
hkl;ij = hik;lj + R̄αβγδ;ǫν
+ R̄αβγδ{ναj x
i + ν
i + ν
i + ν
To replace hkl;ij by hij;kl we use the Ricci identities
(4.8) hik;lj = hik;jl + hakR
ilj + haiR
and differentiate once again the Codazzi equation
(4.9) hik;j = hij;k + R̄αβγδν
To replace f̃ij we use the chain rule
(4.10)
f̃i = f̃αx
f̃ij = f̃αβx
j + f̃αx
Then, because of the Gauß equation, Gaussian formula, and Weingarten
equation, the symmetry properties of the Riemann curvature tensor and the
assumed homogeneity of F , i.e.,
(4.11) F = F klhkl,
we deduce (4.1) from (3.14) on page 6 after reverting to the mixed representa-
tion. �
4.2. Remark. If we had assumed F to be homogeneous of degree d0 instead
of 1, then we would have to replace the explicit term F—occurring twice in the
preceding lemma—by d0F .
If the ambient semi-Riemannian manifold is a space of constant curvature,
then the evolution equation of the second fundamental form simplifies consid-
erably, as can be easily verified.
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 11
4.3. Lemma. Let N be a space of constant curvature KN , then the second
fundamental form of the curvature flow (3.7) on page 5 satisfies the parabolic
equation
(4.12)
i − Φ̇F
i;kl = σΦ̇F
klhrkh
i − σΦ̇Fhrih
rj + σ(Φ − f̃)hki h
− f̃αβxαi x
kj + σf̃αν
i + Φ̇F
kl,rshkl;ih
+ Φ̈FiF
+KN{(Φ− f̃)δji + Φ̇F δ
i − Φ̇F
klgklh
Let us now assume that the open set Ω ⊂ N containing the flow hyper-
surfaces can be covered by a Gaussian coordinate system (xα), i.e., Ω can be
topologically viewed as a subset of I × S0, where S0 is a compact Riemannian
manifold and I an interval. We assume furthermore, that the flow hypersur-
faces can be written as graphs over S0
(4.13) M(t) = { x0 = u(xi) : x = (xi) ∈ S0 };
we use the symbol x ambiguously by denoting points p = (xα) ∈ N as well as
points p = (xi) ∈ S0 simply by x, however, we are careful to avoid confusions.
Suppose that the flow hypersurfaces are given by an embedding x = x(t, ξ),
where ξ = (ξi) are local coordinates of a compact manifoldM0, which then has
to be homeomorphic to S0, then
(4.14)
x0 = u(t, ξ) = u(t, x(t, ξ)),
xi = xi(t, ξ).
The induced metric can be expressed as
(4.15) gij = 〈xi, xj〉 = σuiuj + σklxki xlj ,
where
(4.16) ui = ukx
i.e.,
(4.17) gij = {σukul + σkl}xki xlj ,
hence the (time dependent) Jacobian (xki ) is invertible, and the (ξ
i) can also
be viewed as coordinates for S0.
Looking at the component α = 0 of the flow equation (3.7) on page 5 we
obtain a scalar flow equation
(4.18) u̇ = −e−ψv−1(Φ− f̃),
which is the same in the Lorentzian as well as in the Riemannian case, where
(4.19) v2 = 1− σσijuiuj,
and where
(4.20) |Du|2 = σijuiuj
12 CLAUS GERHARDT
is of course a scalar, i.e., we obtain the same expression regardless, if we use
the coordinates xi or ξi.
The time derivative in (4.18) is a total time derivative, if we consider u to
depend on u = u(t, x(t, ξ)). For the partial time derivative we obtain
(4.21)
= u̇− ukẋki
= −e−ψv(Φ− f̃),
in view of (3.7) on page 5 and our choice of normal ν = (να)
(4.22) (να) = σe−ψv−1(1,−σui),
where ui = σijuj.
Controlling the C1-norm of the graphs M(t) is tantamount to controlling v,
if N is Riemannian, and ṽ = v−1, if N is Lorentzian. The evolution equations
satisfied by these quantities are also very important, since they are used for
the a priori estimates of the second fundamental form.
Let us start with the Lorentzian case.
4.4. Lemma (Evolution of ṽ). Consider the flow (3.7) in a Lorentzian space
N such that the spacelike flow hypersurfaces can be written as graphs over S0.
Then, ṽ satisfies the evolution equation
(4.23)
˙̃v − Φ̇F ij ṽij =− Φ̇F ijhikhkj ṽ + [(Φ− f̃)− Φ̇F ]ηαβνανβ
− 2Φ̇F ijhkjxαi x
kηαβ − Φ̇F
ijηαβγx
− Φ̇F ijR̄αβγδναxβi x
− f̃βxβi x
k ηαg
where η is the covariant vector field (ηα) = e
ψ(−1, 0, . . . , 0).
Proof. We have ṽ = 〈η, ν〉. Let (ξi) be local coordinates for M(t). Differenti-
ating ṽ covariantly we deduce
(4.24) ṽi = ηαβx
α + ηαν
(4.25)
ṽij = ηαβγx
α + ηαβx
+ ηαβx
j + ηαβx
i + ηαν
The time derivative of ṽ can be expressed as
(4.26)
˙̃v = ηαβ ẋ
βνα + ηαν̇
= ηαβν
ανβ(Φ − f̃) + (Φ− f̃)kxαk ηα
= ηαβν
ανβ(Φ − f̃) + Φ̇F kxαk ηα − f̃βx
ikηα,
where we have used (3.11) on page 6.
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 13
Substituting (4.25) and (4.26) in (4.23), and simplifying the resulting equa-
tion with the help of the Weingarten and Codazzi equations, we arrive at the
desired conclusion. �
In the Riemannian case we consider a normal Gaussian coordinate system
(xα), for otherwise we won’t obtain a priori estimates for v, at least not without
additional strong assumptions. We also refer to x0 = r as the radial distance
function.
4.5. Lemma (Evolution of v). Consider the flow (3.7) in a normal Gaussian
coordinate system where the M(t) can be written as graphs of a function u(t)
over some compact Riemannian manifold S0. Then the quantity
(4.27) v =
1 + |Du|2 = (rανα)−1
satisfies the evolution equation
(4.28)
v̇ − Φ̇F ijvij = −Φ̇F ijhikhkj v − 2v−1Φ̇F ijvivj
+ rαβν
ανβ [(Φ− f̃)− Φ̇F ]v2 + 2Φ̇F ijhki rαβxαkx
+ Φ̇F ijR̄αβγδν
+ Φ̇F ijrαβγν
2 + f̃αx
mkrβx
Proof. Similar to the proof of the previous lemma. �
The previous problems can be generalized to the case when the right-hand
side f is not only defined in N or in Ω̄ but in the tangent bundle T (N) resp.
T (Ω̄). Notice that the tangent bundle is a manifold of dimension 2(n+1), i.e.,
in a local trivialization of T (N) f can be expressed in the form
(4.29) f = f(x, ν)
with x ∈ N and ν ∈ Tx(N), cf. [17, Note 12.2.14]. Thus, the case f = f(x) is
included in this general set up. The symbol ν indicates that in an equation
(4.30) F |M = f(x, ν)
we want f to be evaluated at (x, ν), where x ∈M and ν is the normal of M in
The Minkowski problem or Minkowski type problems are also covered by the
present setting, though the Minkowski problem has the additional property that
the problem is transformed via the Gauß map to a different semi-Riemannian
manifold as a dual problem and solved there. Minkowski type problems have
been treated in [5], [23], [16] and [21].
4.6. Remark. The equation (4.30) will be solved by the same methods as
in the special case when f = f(x), i.e., we consider the same curvature flow,
the evolution equation (3.7) on page 5, as before.
14 CLAUS GERHARDT
The resulting evolution equations are identical with the natural exception,
that, when f or f̃ has to be differentiated, the additional argument has to be
considered, e.g.,
(4.31) f̃i = f̃αx
i + f̃νβν
i = f̃αx
i + f̃νβx
(4.32)
f = f̃αẋ
α + f̃νβ ν̇
β = −σ(Φ− f̃)f̃ανα + f̃νβgij(Φ− f̃)ix
The most important evolution equations are explicitly stated below.
Let us first state the evolution equation for (Φ − f̃).
4.7. Lemma (Evolution of (Φ− f̃)). The term (Φ− f̃) evolves according to
the equation
(4.33)
(Φ− f̃)
− Φ̇F ij(Φ− f̃)ij = σΦ̇F ijhikhkj (Φ− f̃)
+ σf̃αν
α(Φ− f̃)− f̃ναxαi (Φ− f̃)jgij
+ σΦ̇F ijR̄αβγδν
γxδj(Φ− f̃),
where
(4.34) (Φ− f̃)′ = d
(Φ− f̃)
(4.35) Φ̇ =
Φ(r).
Here is the evolution equation for the second fundamental form.
4.8. Lemma. The mixed tensor h
i satisfies the parabolic equation
(4.36)
i − Φ̇F
= σΦ̇F klhrkh
i − σΦ̇Fhrih
rj + σ(Φ − f̃)hki h
− f̃αβxαi x
kj + σf̃αν
i − f̃ανβ (x
kj + xαl x
− f̃νανβxαl x
lj − f̃νβx
i;l g
lj + σf̃ναν
αhki h
+ Φ̇F kl,rshkl;ih
rs; + 2Φ̇F
klR̄αβγδx
− Φ̇F klR̄αβγδxαmx
rj − Φ̇F klR̄αβγδxαmx
+ σΦ̇F klR̄αβγδν
γxδl h
i − σΦ̇F R̄αβγδν
γxδmg
+ σ(Φ− f̃)R̄αβγδναxβi ν
γxδmg
mj + Φ̈FiF
+ Φ̇F klR̄αβγδ;ǫ{ναxβkx
mj + ναx
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 15
The proof is identical to that of Lemma 4.1; we only have to keep in mind
that f now also depends on the normal.
If we had assumed F to be homogeneous of degree d0 instead of 1, then, we
would have to replace the explicit term F—occurring twice in the preceding
lemma—by d0F .
4.9. Lemma (Evolution of ṽ). Consider the flow (3.7) in a Lorentzian space
N such that the spacelike flow hypersurfaces can be written as graphs over S0.
Then, ṽ satisfies the evolution equation
(4.37)
˙̃v − Φ̇F ij ṽij =− Φ̇F ijhikhkj ṽ + [(Φ− f̃)− Φ̇F ]ηαβνανβ
− 2Φ̇F ijhkjxαi x
kηαβ − Φ̇F
ijηαβγx
− Φ̇F ijR̄αβγδναxβi x
− f̃βxβi x
k ηαg
ik − f̃νβx
ikxαi ηα,
where η is the covariant vector field (ηα) = e
ψ(−1, 0, . . . , 0).
The proof is identical to the proof of Lemma 4.4.
In the Riemannian case we have:
4.10. Lemma (Evolution of v). Consider the flow (3.7) in a normal Gauss-
ian coordinate system (xα), where the M(t) can be written as graphs of a func-
tion u(t) over some compact Riemannian manifold S0. Then the quantity
(4.38) v =
1 + |Du|2 = (rανα)−1
satisfies the evolution equation
(4.39)
v̇ − Φ̇F ijvij =− Φ̇F ijhikhkj v − 2v−1Φ̇F ijvivj
+ [(Φ− f)− Φ̇F ]rαβνανβv2
+ 2Φ̇F ijhkjx
krαβv
2 + Φ̇F ijrαβγx
+ Φ̇F ijR̄αβγδν
+ f̃βx
k rαg
ikv2 + f̃νβx
ikxαi rαv
where r = x0 and (rα) = (1, 0, . . . , 0).
5. Stability of the limit hypersurfaces
5.1. Definition. Let N be semi-Riemannian, F a curvature operator, and
M ⊂ N a compact, spacelike hypersurface, such that M is admissible and
F ij , evaluated at (hij , gij), the second fundamental form and metric of M , is
divergence free, then M is said to be a stable solution of the equation
(5.1) F |M = f,
16 CLAUS GERHARDT
where f = f(x) is defined in a neighbourhood of M , if the first eigenvalue λ1
of the linearization, which is the operator in (3.33) on page 8, is non-negative,
or equivalently, if the quadratic form
(5.2)
F ijuiuj − σ
{F ijhki hkj + F ijR̄αβγδναx
γxδj + fαν
is non-negative for all u ∈ C2(M).
It is well-known that the corresponding eigenspace is then onedimensional
and spanned by a strictly positive eigenfunction η
(5.3) − F ijηij − σ{F ijhki hkj + F ijR̄αβγδναx
γxδj + fαν
α}η = λ1η.
Notice that F ij is supposed to be divergence free, which will be the case, if
F = Hk, 1 ≤ k ≤ n, and the ambient space has constant curvature, as we
shall prove at the end of this section. If k = 1, then F ij = gij and N can be
arbitrary, while in case k = 2, we have
(5.4) F ij = Hgij − hij ,
hence N Einstein will suffice.
To simplify the formulation of the assumptions let us define:
5.2. Definition. A curvature function F is said to be of class (D), if for
every admissible hypersurfaceM the tensor F ij , evaluated at M , is divergence
free.
We shall prove in this section that the limit hypersurface of a converging
curvature flow will be a stable stationary solution, if the initial flow velocity
has a weak sign.
5.3. Theorem. Suppose that the curvature flow (3.7) on page 5 exists for
all time, and that the leaves M(t) converge in C4 to a hypersurface M , where
the curvature function F is supposed to be of class (D). Then M is a stable
solution of the equation
(5.5) F |M = f
provided the velocity of the flow has a weak sign
(5.6) Φ− f̃ ≥ 0 ∨ Φ− f̃ ≤ 0
at t = 0 and M(0) is not already a solution of (5.5).
Proof. Convergence of a subsequence of the M(t) would actually suffice for
the proof, however, the assumption (5.6) immediately implies that the flow
converges, if a subsequence converges and a priori estimates in C4,α are valid.
The starting point is the evolution equation (3.20) on page 7 from which we
deduce in view of the parabolic maximum principle that Φ− f̃ has a weak sign
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 17
during the evolution, cf. [18, Proposition 2.7.1], i.e., if we assume without loss
of generality that at t = 0
(5.7) Φ− f̃ ≥ 0,
then this inequality will be valid for all t. Moreover, there holds
(5.8)
(Φ− f̃) > 0 ∀ 0 ≤ t <∞
if this relation is valid for t = 0, as we shall prove in the lemma below.
On the other hand, the assumption
(5.9)
(Φ− f̃) > 0
is a natural assumption, for otherwise the initial hypersurface would already
be a stationary solution which of course may not be stable.
Notice also that apart from the factor Φ̇ the equation (3.20) looks like the
parabolic version of the linearization of (F − f). If the technical function
Φ = Φ(r) is not the trivial one Φ(r) = r, then we always assume that f > 0
and that this is also valid for the limit hypersurface M . Only in case Φ(r) = r
and F = H , we allow f to be arbitrary.
Thus, our assumptions imply that in any case
(5.10) Φ̇ > ǫ0 > 0 ∀ t ∈ R+.
Furthermore, we derive from (3.20) that not only the elliptic part converges
to 0 but also
(5.11) (Φ− f̃)′ = Φ̇Ḟ + σΦ̇(f)fανα(Φ− f̃),
i.e.,
(5.12) lim Ḟ = 0.
Suppose now that M is not stable, then the first eigenvalue λ1 is negative
and there exists a strictly positive eigenfunction η solving the equation (5.3)
evaluated atM . Let U be a tubular neighbourhood ofM with a corresponding
future directed normal Gaussian coordinate system (xα) and extend η to U by
setting
(5.13) η(x0, x) = η(x),
where, by a slight abuse of notation, we also denote (xi) by x. Thus there holds
(5.14) ηαν
α = 0
in M , and choosing U sufficiently small, we may assume
(5.15) |ηανα| < η
for all hypersurfaces M(t) ⊂ U .
Now consider the term
(5.16)
Φ̇−1(Φ− f̃)η
18 CLAUS GERHARDT
for large t, which converges to 0. Since it is positive, in view of (5.8), there
must exist a sequence of t, not explicitly labelled, tending to infinity such that
(5.17)
0 ≥ d
Φ̇−1(Φ− f̃)η
−Φ̇−2Φ̈Ḟ (Φ− f̃)η +
Φ̇−1(Φ− f̃)′η
Φ̇−1ηαν
α(Φ− f̃)2 − σ
Φ̇−1(Φ− f̃)2Hη,
where we used the relation (3.8) on page 5 to derive the last integral.
The rest of the proof is straight-forward. Multiply the equation (3.20) by
Φ̇−1η and integrate over M(t) for those values of t satisfying the preceding
inequality to deduce
(5.18)
Φ̇−1(Φ− f̃)′ =
−F ijηij(Φ− f̃)
{F ijhki hkj + F ijR̄αβγδναx
γxδj + Φ̇
−1Φ̇(f)fαν
α}η(Φ− f̃),
and conclude further that the right-hand side can be estimated from above by
(5.19)
η(Φ− f̃)
for large t, while the left-hand side can be estimated from below by
(5.20) − ǫ(t)
(Φ− f̃)η
such that
(5.21) ǫ(t) > 0 ∧ lim ǫ(t) = 0
in view of (5.17), where we used (5.12), (5.15) as well as
(5.22) lim(Φ− f̃) = 0;
a contradiction because of (5.8). �
5.4. Lemma. Let M(t) be a solution of the curvature flow (3.7) on page 5
defined on a maximal time interval [0, T ∗), 0 < T ∗ ≤ ∞, and suppose that
Φ− f̃ has a weak sign at t = 0, e.g.,
(5.23) (Φ− f̃) ≥ 0
and suppose furthermore that
(5.24)
(Φ− f̃) > 0,
(5.25)
(Φ− f̃) > 0 ∀ 0 ≤ t < T ∗.
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 19
Proof. Let M0 be an abstract compact Riemannian manifold that is being
isometrically embedded inN with imageM(0). Let (ξi) be a generic coordinate
system for M0 and abbreviate (Φ− f̃ ) by u. The evolution equation (3.20) can
then be looked at as a linear parabolic equation for u = u(t, ξ) on M0 with
time dependent coefficients and time dependent Riemannian metric gij(t, ξ).
By assumption u doesn’t vanish identically at t = 0, i.e., there exists a ball
Bρ = Bρ(ξ0) such that
(5.26) u(0, ξ) > 0 ∀ ξ ∈ B̄ρ(ξ0).
Let C be the cylinder
(5.27) C = [0, T ∗)× B̄ρ
and assume that there exists a first t0 > 0 such that
(5.28) inf
u(t0, ·) = 0 = u(t0, ξ1).
We shall show that this is not possible: If ξ1 ∈ Bρ, then this contradicts the
strong parabolic maximum principle, cf. [18, Lemma 2.7.1], and if ξ1 ∈ ∂Bρ,
then we deduce from [18, Lemma 2.7.4] (a parabolic version of the Hopf Lemma)
(5.29)
(t0, ξ1) < 0,
where ν is the exterior normal of the ball Bρ in ξ1, contradicting the fact that
the gradient of u(t0, ·) vanishes in ξ1 because it is a minimum point; notice that
we already know u ≥ 0 in [0, T ∗)×M0. �
For some curvature operators one can prove a priori estimates for the second
fundamental form only for stationary solutions and not for the leaves of a
corresponding curvature flow. In order to use a curvature flow to obtain a
stationary solution one uses
ǫ-regularization“, i.e., instead of the curvature
function F one considers
(5.30) F̃ (hij) = F (hij + ǫHgij)
for ǫ > 0, and starts a curvature flow with F̃ and fixed ǫ > 0.
A priori estimates for the regularized flow are usually fairly easily derived,
since
(5.31) F̃ ij = F ij + ǫF klgklg
but of course the estimates depend on ǫ. Having uniform estimates one can
deduce that the flow—or at least a subsequence—converges to a limit hyper-
surfaces Mǫ satisfying
(5.32) F̃ |Mǫ = f.
Then, if uniform C4,α-estimates for the Mǫ can be derived, a subsequence will
converge to a solution M of
(5.33) F |M = f,
20 CLAUS GERHARDT
cf. [13], where this method has been used to find hypersurfaces of prescribed
scalar curvature in Lorentzian manifolds, see also Theorem 6.9 on page 36.
We shall now show that the solutions M obtained by this approach are all
stable, if F is of class (D) and the initial velocities of the regularized flows have
a weak sign. Notice that the curvature functions F̃ are in general not of class
5.5. Theorem. Let F be of class (D), then any solution M of
(5.34) F |M = f
obtained by a regularized curvature flow as described above is stable, provided
the initial velocity of the regularized flow has a weak sign, i.e., it satisfies
(5.35) Φ− f̃ ≥ 0 ∨ Φ− f̃ ≤ 0
at t = 0 and the flow hypersurfaces converge to the stationary solution in C4.
Proof. Let Mǫ be the limit hypersurfaces of the regularized flow for ǫ > 0, and
assume that the Mǫ satisfy uniform C
4,α-estimates such that a subsequence,
not relabelled, converges in C4 to a compact spacelike hypersurface M solving
the equation
(5.36) F |M = f.
Assume that M is not stable so that the first eigenvalue of the linearization
is negative and there exists a strictly positive eigenfunction η satisfying (5.3).
Extend η in a small tubular neighbourhood U of M such that (5.15) is valid
for all Mǫ, if ǫ is small, ǫ < ǫ0.
For those ǫ we then deduce
(5.37)
−F̃ ijηij − F̃ ij;ijη − 2F̃
;j ηi
− σ{F̃ ijhki hkj + F̃ ijR̄αβγδναx
γxδj + fαν
α}η < λ1
where the inequality is evaluated at Mǫ and where we used the convergence in
Now, fix ǫ, ǫ < ǫ0, then the preceding inequality is also valid for the flow
hypersurfaces M(t) converging to Mǫ, if t is large, and the same arguments as
at the end of the proof of Theorem 5.3 lead to a contradiction. Hence, M has
to be a stable solution. �
Knowing that a solution is stable often allows to deduce further geometric
properties of the underlying hypersurface like that it is either strictly stable
or totally geodesic especially if the curvature function is the mean curvature,
cf. e.g., [29], where the stability property has been extensively used to deduce
geometric properties.
We want to prove that a neighbourhood of stable solutions can be foliated
by a family of hypersurfaces satisfying the equation modulo a constant.
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 21
5.6. Theorem. Let M ⊂ N be compact, spacelike, orientable and a stable
solution of
(5.38) G|M ≡ (F − f)|M = 0,
where F is of class (D) and M as well as F , f are of class Cm,α, 2 ≤ m ≤ ∞,
0 < α < 1, then a neighbourhood of M can be foliated by a family
(5.39) Λ = {Mǫ : |ǫ| < ǫ0 }
of spacelike Cm,α-hypersurfaces satisfying
(5.40) G|Mǫ = τ(ǫ),
where τ is a real function of class Cm,α. The Mǫ can be written as graphs over
M in a tubular neighbourhood of M
(5.41) Mǫ = { (u(ǫ, x), x) : x ∈M }
such that u is of class Cm,α in both variables and there holds
(5.42) u̇ > 0.
Proof. (i) Let us assume that M is strictly stable. Consider a tubular neigh-
bourhood of M with corresponding normal Gaussian coordinates (xα) such
that M = {x0 = 0}. The nonlinear operator G can then be viewed as an
elliptic operator
(5.43) G : Bρ(0) ⊂ Cm,α(M) → Cm−2,α(M)
where ρ is so small that all corresponding graphs are admissible.
In a smaller ball DG is a topological isomorphism, sinceM is strictly stable,
and hence G is a diffeomorphism in a neighbourhood of the origin, and there
exist smooth unique solutions
(5.44) Mǫ = { u(ǫ, x) : x ∈M } |ǫ| < ǫ0
of the equations
(5.45) G|Mǫ = ǫ
such that u ∈ Cm,α((−ǫ0, ǫ0)×M).
Differentiating with respect to ǫ yields
(5.46) DGu̇ = 1.
Let us consider this equation for ǫ = 0, i.e., on M , and define
(5.47) η = min(u̇, 0).
Then we deduce
(5.48) 0 ≤
〈DGη, η〉 =
η ≤ 0,
and hence there holds
(5.49) u̇ ≥ 0,
because of the strict stability of M .
22 CLAUS GERHARDT
Applying then the maximum principle to (5.46), we deduce further
(5.50) inf
u̇ > 0,
hence the hypersurfaces form a foliation if ǫ0 is chosen small enough such that
(5.51) inf
u̇(ǫ, ·) > 0 ∀ |ǫ| < ǫ0.
(ii) Assume now thatM is not strictly stable. After introducing coordinates
corresponding to a tubular neighbourhood U of M as in part (i) any function
u ∈ Cm,α(M) with |u|m,α small enough defines an admissible hypersurface
(5.52) M(u) = graphu ⊂ U
such that G|M(u) can be expressed as
(5.53) G|M(u) = G(u).
(5.54) A = DG(0),
then A is self-adjoint, monotone
(5.55) 〈Au, u〉 ≥ 0 ∀u ∈ H1,2(M)
and the smallest eigenvalue of A is equal to zero, the corresponding eigenspace
spanned by a strictly positive eigenfunction η.
Similarly as in [2, p. 621] we consider the operator
(5.56) Ψ(u, τ) = (G(u) − τ, ϕ(u))
defined in Bρ(0) × R, Bρ(0) ⊂ Cm,α(M) for small ρ > 0, where ϕ is a linear
functional
(5.57) ϕ(u) =
Ψ is of class Cm,α and maps
(5.58) Ψ : Bρ(0)× R → Cm−2,α(M)× R,
such that
(5.59) DΨ =
DG −1
evaluated at (0, 0) is bijective as one easily checks. Indeed let (u, ǫ) satisfy
(5.60) DΨ(u, ǫ) = (0, 0),
(5.61) Au = DGu = ǫ ∧
ηu = 0,
hence
(5.62) ǫ
η = 〈Au, η〉 = 〈u,Aη〉 = 0
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 23
and we conclude ǫ = 0 as well as u = 0.
To prove the surjectivity, let (w, δ) ∈ Cm−2,α(M)×R be arbitrary. Choosing
(5.63) ǫ = −
we deduce
(5.64)
(ǫ+ w)η = 0,
hence there exists ū ∈ Cm,α(M) solving
(5.65) Aū = ǫ+ w
(5.66) u = ū+ λη
(5.67) λ = δ −
then satisfies
(5.68)
ηu = ǫ,
i.e.,
(5.69) DΨ(u, ǫ) = (w, δ).
Applying the inverse function theorem we conclude that there exists ǫ0 > 0
and functions (u(ǫ, x), τ(ǫ)) of class Cm,α in both variables such that
(5.70) G(u(ǫ)) = τ(ǫ) ∧
ηu(ǫ) = ǫ ∀ |ǫ| < ǫ0;
τ(ǫ) is constant for fixed ǫ.
The hypersurfaces
(5.71) Λ = {Mǫ =M(u(ǫ)) : |ǫ| < ǫ0 }
will form a foliation, if we can show that
(5.72) u̇ 6= 0.
Differentiating the equations in (5.70) with respect to ǫ and evaluating the
result at ǫ = 0 yields
(5.73) Au̇(0) = τ̇(0) ∧
ηu̇(0) = 1
and we deduce further
(5.74) τ̇ (0)
η = 〈Au̇(0), η〉 = 〈u̇(0), Aη〉 = 0
and thus
(5.75) τ̇ (0) = 0 ∧ u̇(0) = η > 0,
24 CLAUS GERHARDT
if η is normalized such that 〈η, η〉 = 1, i.e., we have u̇(ǫ) > 0, if ǫ0 is chosen
small enough. �
5.7. Remark. Let M be a stable solution of
(5.76) G|M = 0
as in the preceding theorem, but not strictly stable and let Mǫ be a foliation
of a neighbourhood of M such that
(5.77) G|Mǫ = τ(ǫ) ∀ |ǫ| < ǫ0.
If M is the limit hypersurface of a curvature flow as in Theorem 5.3, then
(5.78) τ(ǫ) > 0 ∀ 0 < ǫ < ǫ0,
if the flow hypersurfacesM(t) converge to M from above, which is tantamount
(5.79) Φ(F )− f̃ ≥ 0,
or we have
(5.80) τ(ǫ) < 0 ∀ − ǫ0 < ǫ < 0,
(5.81) Φ(F )− f̃ ≤ 0,
in which case the flow hypersurfaces converge to M from below.
The direction
above“ is defined by the region the normal σν of M points
Proof. Let us assume that the flow hypersurfaces satisfy (5.79) and fix 0 < ǫ <
ǫ0. We may also suppose that the initial hypersurface M(0) doesn’t intersect
the tubular neighbourhood of M which is being foliated by Mǫ. Now, fix
0 < ǫ < ǫ0, then there must be a first t > 0 such that M(t) touches Mǫ from
above which yields, in view of the maximum principle,
(5.82) G|Mǫ = τ(ǫ) > 0,
since τ(ǫ) ≤ 0 would imply τ(ǫ) = 0 and Mǫ = M(t), cf. [18, Theorem 2.7.9],
i.e.,M(t) would be a stationary solution, which is impossible as we have proved
in Lemma 5.4. �
Finally, let us show that the symmetric polynomials Hk, 1 ≤ k ≤ n, are of
class (D), if the ambient space has constant curvature.
5.8. Lemma. Let N be a semi-Riemannian space of constant curvature,
then the symmetric polynomials F = Hk, 1 ≤ k ≤ n, are of class (D). In case
k = 2 it suffices to assume N Einstein.
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 25
Proof. We shall prove the result by induction on k. First we note that the
cones of definition Γk ⊂ Rn of the Hk form an ordered chain
(5.83) Γk ⊂ Γk−1 ∀ 1 < k ≤ n,
cf. [7], so that a hypersurface admissible for Hk is also admissible for Hk−1.
For k = 1 we have
(5.84) F ij = gij
and the result is obviously valid for arbitrary N .
Thus let us assume that the result is already proved for 1 ≤ k < n. Set
F = Hk+1, F̂ = Hk and let M be an admissible hypersurface for F with
principal curvatures κi.
From the definition of the Hk’s we immediately deduce
(5.85) F̂ =
for fixed i, no summation over i, or equivalently,
(5.86) F̂ gij = F ij + F̂ jmhim,
notice that the last term is a symmetric tensor, since for any symmetric cur-
vature function F F ij and hij commute, cf. [18, Lemma 2.1.9]. Thus there
holds
(5.87) F ij = F̂ gij − F̂ jmhim
and we deduce, using the induction hypothesis,
(5.88)
;j = F̂
i − F̂ jmhim;j = F̂ i − F̂ jmh imj;
= F̂ i − F̂ i = 0,
where we applied the Codazzi equations at one point.
If F = H2, then
(5.89) F ij = Hgij − hij
and the assumption N Einstein suffices to conclude that F ij is divergence
free. �
6. Existence results
From now on we shall assume that ambient manifold N is Lorentzian, or
more precisely, that it is smooth, globally hyperbolic with a compact, connected
Cauchy hypersurface. Then there exists a smooth future oriented time function
x0 such that the metric in N can be expressed in Gaussian coordinates (xα) as
(6.1) ds̄2 = e2ψ{−(dx0)2 + σijdxidxj},
where x0 is the time function and the (xi) are local coordinates for
(6.2) S0 = {x0 = 0}.
26 CLAUS GERHARDT
S0 is then also a compact, connected Cauchy hypersurface. For a proof of the
splitting result see [4, Theorem 1.1], and for the fact that all Cauchy hyper-
surfaces are diffeomorphic and hence S0 is also compact and connected, see [3,
Lemma 2.2].
One advantage of working in globally hyperbolic spacetimes with a compact
Cauchy hypersurface is that all compact, connected spacelike Cm-hypersurfaces
M can be written as graphs over S0.
6.1. Lemma. Let N be as above and M ⊂ N a connected, spacelike hyper-
surface of class Cm, 1 ≤ m, then M can be written as a graph over S0
(6.3) M = graphu|S0
with u ∈ Cm(S0).
We proved this lemma under the additional hypothesis that M is achronal,
[10, Proposition 2.5], however, this assumption is unnecessary as has been
shown in [25, Theorem 1.1].
We are looking at the curvature flow (3.7) on page 5 and want to prove that
it converges to a stationary solution hypersurface, if certain assumptions are
satisfied.
The existence proof consists of four steps:
(i) Existence on a maximal time interval [0, T ∗).
(ii) Proof that the flow stays in a compact subset.
(iii) Uniform a priori estimates in an appropriate function space, e.g., C4,α(S0)
or C∞(S0), which, together with (ii), would imply T ∗ = ∞.
(iv) Conclusion that the flow—or at least a subsequence of the flow hypersur-
faces—converges if t tends to infinity.
The existence on a maximal time interval is always guaranteed, if the data
are sufficiently regular, since the problem is parabolic. If the flow hypersurfaces
can be written as graphs in a Gaussian coordinate system, as will always be the
case in a globally hyperbolic spacetime with a compact Cauchy hypersurface
in view of Lemma 6.1, the conditions are better than in the general case:
6.2. Theorem. Let 4 ≤ m ∈ N and 0 < α < 1, and assume the semi-
Riemannian space N to be of class Cm+2,α. Let the strictly monotone curvature
function F , the functions f and Φ be of class Cm,α and let M0 ∈ Cm+2,α be
an admissible compact, spacelike, connected, orientable3 hypersurface. Then the
curvature flow (3.7) on page 5 with initial hypersurface M0 exists in a maximal
time interval [0, T ∗), 0 < T ∗ ≤ ∞, where in case that the flow hypersurfaces
cannot be expressed as graphs they are supposed to be smooth, i.e, the conditions
should be valid for arbitrary 4 ≤ m ∈ N in this case.
3Recall that oriented simply means there exists a continuous normal, which will always
be the case in a globally hyperbolic spacetime.
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 27
A proof can be found in [18, Theorem 2.5.19, Lemma 2.6.1].
The second step, that the flow stays in a compact set, can only be achieved
by barrier assumptions, cf. Definition 2.1. Thus, let Ω ⊂ N be open and
precompact such that ∂Ω has exactly two components
(6.4) ∂Ω =M1
where M1 is a lower barrier for the pair (F, f) and M2 an upper barrier. More-
over, M1 has to lie in the past of M2
(6.5) M1 ⊂ I−(M2),
cf. [18, Remark 2.7.8].
Then the flow hypersurfaces will always stay inside Ω̄, if the initial hyper-
surface M0 satisfies M0 ⊂ Ω, [18, Theorem 2.7.9]. This result is also valid if
M0 coincides with one the barriers, since then the velocity (Φ− f̃) has a weak
sign and the flow moves into Ω for small t, if it moves at all, and the arguments
of the proof are applicable.
In Lorentzian manifolds the existence of barriers is associated with the pres-
ence of past and future singularities. In globally hyperbolic spacetimes, when
N is topologically a product
(6.6) N = I × S0,
where I = (a, b), singularities can only occur, when the endpoints of the interval
are approached. A singularity, if one exists, is called a crushing singularity, if
the sectional curvatures become unbounded, i.e.,
(6.7) R̄αβγδR̄
αβγδ → ∞
and such a singularity should provide a future resp. past barrier for the mean
curvature function H .
6.3. Definition. Let N be a globally hyperbolic spacetime with compact
Cauchy hypersurface S0 so that N can be written as a topological product
N = I × S0 and its metric expressed as
(6.8) ds̄2 = e2ψ(−(dx0)2 + σij(x0, x)dxidxj).
Here, x0 is a globally defined future directed time function and (xi) are lo-
cal coordinates for S0. N is said to have a future resp. past mean curvature
barrier, if there are sequences M+k resp. M
k of closed, spacelike, admissible
hypersurfaces such that
(6.9) lim
= ∞ resp. lim
(6.10) lim sup inf
x0 > x0(p) ∀ p ∈ N
28 CLAUS GERHARDT
resp.
(6.11) lim inf sup
x0 < x0(p) ∀ p ∈ N,
If one stipulates that the principal curvatures of the M+k resp. M
k tend to
plus resp. minus infinity, then these hypersurfaces could also serve as barriers
for other curvature functions. The past barriers would most certainly be non-
admissible for any curvature function except H .
6.4.Remark. Notice that the assumptions (6.9) alone already implies (6.10)
resp. (6.11), if either
(6.12) lim sup inf
x0 > a
resp.
(6.13) lim inf sup
x0 < b
where (a, b) = x0(N), or, if
(6.14) R̄αβν
ανβ ≥ −Λ ∀ 〈ν, ν〉 = −1.
where Λ ≥ 0.
Proof. It suffices to prove that the relation (6.10) is automatically satisfied
under the assumptions (6.12) or (6.14) by switching the light cone and replacing
x0 by −x0 in case of the past barrier.
Fix k, and let
(6.15) τk = inf
then the coordinate slice
(6.16) Mτk = {x0 = τk}
touches Mk from below in a point pk ∈ Mk where τk = x0(pk) and the maxi-
mum principle yields that in that point
(6.17) H |Mτk
≥ H |Mk ,
hence, if k tends to infinity the points (pk) cannot stay in a compact subset,
i.e.,
(6.18) lim supx0(pk) → b
(6.19) lim supx0(pk) → a.
We shall show that only (6.18) can be valid. The relation (6.19) evidently
contradicts (6.12).
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 29
In case the assumption (6.14) is valid, we consider a fixed coordinate slice
M0 = {x0 = const}, then all hypersurfaces Mk satisfying
(6.20) H |M0 < infMk
nΛ < inf
have to lie in the future of M0, cf. [18, Lemma 4.7.1], hence the result. �
A future mean curvature barrier certainly represents a singularity, at least
if N satisfies the condition
(6.21) R̄αβν
ανβ ≥ −Λ ∀ 〈ν, ν〉 = −1
where Λ ≥ 0, because of the future timelike incompleteness, which is proved in
[1], and is a generalization of Hawking’s earlier result for Λ = 0, [24]. But these
singularities need not be crushing, cf. [15, Section 2] for a counterexample.
The uniform a priori estimates for the flow hypersurfaces are the hardest
part in any existence proof. When the flow hypersurfaces can be written as
graphs it suffices to prove C1 and C2 estimates, namely, the induced metric
(6.22) gij(t, ξ) = 〈xi, xj〉
where x = x(t, ξ) is a local embedding of the flow, should stay uniformly
positive definite, i.e., there should exist positive constants ci, 1 ≤ i ≤ 2, such
(6.23) c1gij(0, ξ) ≤ gij(t, ξ) ≤ c2gij(0, ξ),
or equivalently, that the quantity
(6.24) ṽ = 〈η, ν〉,
where ν is the past directed normal of M(t) and η the vector field
(6.25) η = (ηα) = e
ψ(−1, 0, . . . , 0),
is uniformly bounded, which is achieved with the help of the parabolic equation
(4.37) on page 15, if it is possible at all.
However, in some special situations C1-estimates are automatically satisfied,
cf. Theorem 6.11 at the end of this section.
For the C2-estimates the principle curvatures κi of the flow hypersurfaces
have to stay in a compact set in the cone of definition Γ of F , e.g., if F is the
Gaussian curvature, then Γ = Γ+ and one has to prove that there are positive
constants ki, i = 1, 2 such that
(6.26) k1 ≤ κi ≤ k2 ∀ 1 ≤ i ≤ n
uniformly in the cylinder [0, T ∗)×M0, whereM0 is any manifold that can serve
as a base manifold for the embedding x = x(t, ξ).
The parabolic equations that are used for these curvature estimates are
(4.36) on page 14, usually for an upper estimate, and (4.33) on page 14 for the
lower estimate. Indeed, suppose that the flow starts at the upper barrier, then
(6.27) F ≥ f
30 CLAUS GERHARDT
at t = 0 and this estimate remains valid throughout the evolution because of
the parabolic maximum principle, use (4.36). Then, if upper estimates for the
κi have been derived and if f > 0 uniformly, then we conclude from (6.27) that
the κi stay in a compact set inside the open cone Γ , since
(6.28) F |∂Γ = 0.
To obtain higher order estimates we are going to exploit the fact that the
flow hypersurfaces are graphs over S0 in an essential way, namely, we look
at the associated scalar flow equation (4.21) on page 12 satisfied by u. This
equation is a nonlinear uniformly parabolic equation, where the operator Φ(F )
is also concave in hij , or equivalently, convex in uij , i.e., the C
2,α-estimates of
Krylov and Safonov, [26, Chapter 5.5] or see [28, Chapter 10.6] for a very clear
and readable presentation, are applicable, yielding uniform estimates for the
standard parabolic Hölder semi-norm
(6.29) [D2u]β,Q̄T
for some 0 < β ≤ α in the cylinder
(6.30) Q = [0, T )× S0,
independent of 0 < T < T ∗, which in turn will lead to Hm+2+α,
m+2+α
2 (Q̄T )
estimates, cf. [18, Theorem 2.5.9, Remark 2.6.2].
Hm+2+α,
m+2+α
2 (Q̄T ) is a parabolic Hölder space, cf. [27, p. 7] for the original
definition and [18, Note 2.5.4] in the present context.
The estimate (6.29) combined with the uniform C2-norm leads to uniform
C2,β(S0)-estimates independent of T .
These estimates imply that T ∗ = ∞.
Thus, it remains to prove that u(t, ·) converges in Cm+2(S0) to a stationary
solution ũ, which is then also of class Cm+2,α(S0) in view of the Schauder
theory.
Because of the preceding a priori estimates u(t, ·) is precompact in C2(S0).
Moreover, we deduce from the scalar flow equation (4.21) on page 12 that u̇
has a sign, i.e., the u(t, ·) converge monotonely in C0(S0) to ũ and therefore
also in C2(S0).
To prove that graph ũ is a solution, we again look at (4.21) and integrate it
with respect to t to obtain for fixed x ∈ S0
(6.31) |ũ(x) − u(t, x)| =
e−ψv|Φ− f̃ |,
where we used that (Φ− f̃) has a sign, hence (Φ− f̃)(t, x) has to vanish when
t tends to infinity, at least for a subsequence, but this suffices to conclude that
graph ũ is a stationary solution and
(6.32) lim
(Φ− f̃) = 0.
Using the convergence of u to ũ in C2, we can then prove:
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 31
6.5. Theorem. The functions u(t, ·) converge in Cm+2(S0) to ũ, if the data
satisfy the assumptions in Theorem 6.2, since we have
(6.33) u ∈ Hm+2+β,
m+2+β
2 (Q̄),
where Q = Q∞.
Proof. Out of convention let us write α instead of β knowing that α is the
Hölder exponent in (6.29).
We shall reduce the Schauder estimates to the standard Schauder estimates
in Rn for the heat equation with a right-hand side by using the already estab-
lished results (6.29) and
(6.34) u(t, ·) →
C2(S0)
ũ ∈ Cm+2,α(S0).
Let (Uk) be a finite open covering of S0 such that each Uk is contained in a
coordinate chart and
(6.35) diamUk < ρ,
ρ small, ρ will be specified in the proof, and let (ηk) be a subordinate finite
partition of unity of class Cm+2,α.
Since
(6.36) u ∈ Hm+2+α,
m+2+α
2 (Q̄T )
for any finite T , cf. [18, Lemma 2.6.1], and hence
(6.37) u(t, ·) ∈ Cm+2,α(S0) ∀ 0 ≤ t <∞
we shall choose u0 = u(t0, ·) as initial value for some large t0 such that
(6.38) |aij(t, ·)− ãij |0,S0 < ǫ/2 ∀ t ≥ t0,
where
(6.39) aij = v2Φ̇F ij
and ãij is defined correspondingly for M̃ = graph ũ.
However, making a variable transformation we shall always assume that
t0 = 0 and u0 = u(0, ·).
We shall prove (6.33) successively.
(i) Let us first show that
(6.40) Dxu ∈ H2+α,
2 (Q̄).
This will be achieved, if we show that for an arbitrary ξ ∈ Cm+1,α(T 1,0(S0))
(6.41) ϕ = Dξu ∈ H2+α,
2 (Q̄),
cf. [18, Remark 2.5.11].
Differentiating the scalar flow equation (4.21) on page 12 with respect to ξ
we obtain
(6.42) ϕ̇− aijϕij + biϕi + cϕ = f,
32 CLAUS GERHARDT
where of course the symbol f has a different meaning then in (4.21).
Later we want to apply the Schauder estimates for solutions of the heat flow
equation with right-hand side. In order to use elementary potential estimates
we have to cut off ϕ near the origin t = 0 by considering
(6.43) ϕ̃ = ϕθ,
where θ = θ(t) is smooth satisfying
(6.44) θ(t) =
1, t > 1,
0, t ≤ 1
This modification doesn’t cause any problems, since we already have a priori
estimates for finite t, and we are only concerned about the range 1 ≤ t < ∞.
ϕ̃ satisfies the same equation as ϕ only the right-hand side has the additional
summand wθ̇.
Let η = ηk be one of the members of the partition of unity and set
(6.45) w = ϕ̃η,
then w satisfies a similar equation with slightly different right-hand side
(6.46) ẇ − aijwij + biwi + cw = f̃
but we shall have this in mind when applying the estimates.
The w(t, ·) have compact support in one of the Uk’s, hence we can replace
the covariant derivatives of w by ordinary partial derivatives without changing
the structure of the equation and the properties of the right-hand side, which
still only depends linearly on ϕ and Dϕ.
We want to apply the well-known estimates for the ordinary heat flow equa-
(6.47) ẇ −∆w = f̂
where w is defined in R × Rn.
To reduce the problem to this special form, we pick an arbitrary x0 ∈ Uk,
set z0 = (0, x0), z = (t, x) and consider instead of (6.46)
(6.48)
ẇ − aij(z0)wij = f̂
= [aij(z)− aij(z0)]wij − biwi − cw + f̃ ,
where we emphasize that the difference
(6.49) |aij(z)− aij(z0)|
can be made smaller than any given ǫ > 0 by choosing ρ = ρ(ǫ) in (6.35) and
t0 = t0(ǫ) in (6.38) accordingly. Notice also that this equation can be extended
into R × Rn, since all functions have support in {t ≥ 1
Let 0 < T <∞ be arbitrary, then all terms belong to the required function
spaces in Q̄T and there holds
(6.50) [w]2+α,QT ≤ c[f̂ ]α,QT ,
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 33
where c = c(n, α). The brackets indicate the standard unweighted parabolic
semi-norms, cf. [18, Definition 2.5.2], which are identical to those defined in
[27, p. 7], but there the brackets are replaced by kets.
Thus, we conclude
(6.51)
[w]2+α,QT ≤ c sup
Uk×(0,T )
|aij(z)− aij(z0)|[D2w]α,QT + c[f ]α,QT
+ c1{[D2u]α,Q+ + [Du]α,QT + [u]α,QT + |w|0,QT + |D2w|0,QT },
where c1 is independent of T , but dependent on ηk. Here we also used the fact
that the lower order coefficients and ϕ,Dϕ are uniformly bounded.
Choosing now ǫ > 0 so small that
(6.52) cǫ < 1
and ρ, t0 accordingly such the difference in (6.49) is smaller than ǫ, we deduce
(6.53)
[w]2+α,QT ≤ 2c[f ]α,QT
+ 2c1{[D2u]α,Q+ + [Du]α,QT + [u]α,QT + |w|0,QT + |D2w|0,QT }.
Summing over the partition of unity and noting that ξ is arbitrary we see
that in the preceding inequality we can replace w by Du everywhere resulting
in the estimate
(6.54)
[Du]2+α,QT ≤ c1[f ]α,QT
+ c1{[D2u]α,QT + [Du]α,QT + [u]α,QT + |Du|0,QT + |D3u|0,QT },
where c1 is a new constant still independent of T .
Now the only critical terms on the right-hand side are |D3u|0,QT , which can
be estimated by (6.57), and the Hölder semi-norms with respect to t
(6.55) [Du]α
,t,QT + [u]α2 ,t,QT .
The second one is taken care of by the boundedness of u̇, see (4.21) on page 12,
while the first one is estimated with the help of equation (6.42) revealing
(6.56) |Du̇| ≤ c{sup
[0,T ]
|u|3,S0 + |f |0,QT },
since for fixed but arbitrary t we have
(6.57) |u|3,S0 ≤ ǫ[D3u]α,S0 + cǫ|u|0,S0 ,
where cǫ is independent of t.
Hence we conclude
(6.58) |Du|2+α,QT ≤ const
uniformly in T .
(ii) Repeating these estimates successively for 2 ≤ l ≤ m we obtain uniform
estimates for
(6.59)
[Dlxu]2+α,QT ,
34 CLAUS GERHARDT
which, when combined with the uniform C2-estimates, yields
(6.60) |u(t, ·)|m+2,α,S0 ≤ const
uniformly in 0 ≤ t <∞.
Looking at the equation (4.21) we then deduce
(6.61) |u̇(t, ·)|m,α,S0 ≤ const
uniformly in t.
(iii) To obtain the estimates for Drtu up to the order
(6.62) [m+2+α
we differentiate the scalar curvature equation with respect to t as often as
necessary and also with respect to the mixed derivatives DrtD
x to estimate
(6.63)
1≤2r+s<m+2+α
using (6.60), (6.61) and the results from the prior differentiations.
Combined with the estimates for the heat equation in R×Rn these estimates
will also yield the necessary a priori estimates for the Hölder semi-norms in Q̄,
where again the smallness of (6.49) has to be used repeatedly. �
6.6. Remark. The preceding regularity result is also valid in Riemannian
manifolds, if the flow hypersurfaces can be written as graphs in a Gaussian
coordinate system. In fact the proof is unaware of the nature of the ambient
space.
With the method described above the following existence results have been
proved in globally hyperbolic spacetimes with a compact Cauchy hypersurface.
Ω ⊂ N is always a precompact domain the boundary of which is decomposed
as in (6.4) and (6.5) into an upper and lower barrier for the pair (F, f). We
also apply the stability results from Section 5 and the just proved regularity of
the convergence and formulate the theorems accordingly.
By convergence of the flow in Cm+2 we mean convergence of the leaves
M(t) = graphu(t, ·) in this norm.
6.7.Theorem. LetM1, M2 be lower resp. upper barriers for the pair (H, f),
where f ∈ Cm,α(Ω̄) and the Mi are of class Cm+2,α, 4 ≤ m, 0 < α < 1, then
the curvature flow
(6.64)
ẋ = (H − f)ν
x(0) = x0,
where x0 is an embedding of the initial hypersurface M0 = M2 exists for all
time and converges in Cm+2 to a stable solution M of class Cm+2,α of the
equation
(6.65) H |M = f,
provided the initial hypersurface is not already a solution.
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 35
The existence result was proved in [11, Theorem 2.2], see also [18, Theorem
4.2.1] and the remarks following the theorem. Notice that f isn’t supposed to
satisfy any sign condition.
For spacetimes that satisfy the timelike convergence condition and for func-
tions f with special structural conditions existence results via a mean curvature
flow were first proved in [6].
The Gaussian curvature or the curvature functions F belonging to the larger
class (K∗), see [10] for a definition, require that the admissible hypersurfaces
are strictly convex.
Moreover, proving a priori estimates for the second fundamental form of
a hypersurface M in general semi-Riemannian manifolds, when the curvature
function is not the mean curvature, or does not behave similar to it, requires
that a strictly convex function χ is defined in a neighbourhood of the hypersur-
face, see Lemma 2.2 on page 3 where sufficient assumptions are stated which
imply the existence of strictly convex functions.
Furthermore, when we consider curvature functions of class (K∗), notice
that the Gaussian curvature belongs to that class, then the right-hand side f
can be defined in T (Ω̄) instead of Ω̄, i.e., in a local trivialization of the tangent
bundle f can be expressed as
(6.66) f = f(x, ν) ∧ ν ∈ Tx(N).
We shall formulate the existence results with this more general assumption,
though of course any stability claim only makes sense for f = f(x).
6.8. Theorem. Let F ∈ Cm,α(Γ+), 4 ≤ m, 0 < α < 1, be a curvature
function of class (K∗), let 0 < f ∈ Cm,α(T (Ω̄)), and let M1, M2 be lower resp.
upper barriers for (F, f) of class Cm+2,α. Then the curvature flow
(6.67)
ẋ = (Φ− f̃)ν
x(0) = x0
where Φ(r) = log r and x0 is an embedding of M0 =M2, exists for all time and
converges in Cm+2 to a stationary solution M ∈ Cm+2,α of the equation
(6.68) F |M = f
provided the initial hypersurface M2 is not already a stationary solution and
there exists a strictly convex function χ ∈ C2(Ω̄).
When f = f(x) and F is of class (D), then M is stable.
The theorem was proved in [10] when f is only defined in Ω̄ and in the
general case in [18, Theorem 4.1.1].
When F = H2 is the scalar curvature operator, then the requirement that
f is defined in the tangent bundle and not merely in N is a necessity, if the
scalar curvature is to be prescribed. To prove existence results in this case, f
36 CLAUS GERHARDT
has to satisfy some natural structural conditions, namely,
0 < c1 ≤ f(x, ν) if 〈ν, ν〉 = −1,(6.69)
|||fβ(x, ν)||| ≤ c2(1 + |||ν|||2),(6.70)
|||fνβ (x, ν)||| ≤ c3(1 + |||ν|||),(6.71)
for all x ∈ Ω̄ and all past directed timelike vectors ν ∈ Tx(Ω), where ||| · ||| is a
Riemannian reference metric.
Applying a curvature flow to obtain stationary solutions requires to approx-
imate F and f by functions Fǫ and fk and to use these functions for the flow.
The Fǫ are the ǫ-regularizations of F , which we already discussed before, cf.
(5.30) on page 19. Let us also write F̃ instead of Fǫ as before.
The functions fk have the property that |||fkβ ||| only grows linearly in |||ν|||
and |||fkνβ (x, ν)||| is bounded. To simplify the presentation we shall therefore
assume that f satisfies
(6.72) |||fβ(x, ν)||| ≤ c2(1 + |||ν|||),
(6.73) |||fνβ (x, ν)||| ≤ c3,
and also
(6.74) 0 < c1 ≤ f(x, ν) ∀ ν ∈ Tx(N), 〈ν, ν〉 < 0,
although the last assumption is only a minor point that can easily be dealt with,
see [13, Remark 2.6], and [13, Section 7 and 8] for the other approximations of
The barriers Mi, i = 1, 2, for (F, f) satisfy the barrier condition of course
only weakly, i.e., no strict inequalities; however, because of the ǫ-regularization
we need strict inequalities, so that the Mi’s are also barriers for (F̃ , f), if ǫ is
small. In [13, Remark 2.4 and Lemma 2.5] it is shown that strict inequalities
for the barriers may be assumed without loss of generality.
Now, we can formulate the existence result for the scalar curvature operator
F = H2 under these provisions.
6.9. Theorem. Let f ∈ Cm,α(T (Ω̄)), 4 ≤ m, 0 < α < 1, satisfy the con-
ditions (6.72), (6.73) and (6.74), and let M1, M2 be strict lower resp. upper
barriers of class Cm+2,α for (F, f). Let F̃ be the ǫ-regularization of the scalar
curvature operator F , then the curvature flow for F̃
(6.75)
ẋ = (Φ− f̃)
x(0) = x0
where Φ(r) = r
2 and x0 is an embedding of M0 = M2, exists for all time and
converges in Cm+2 to a stationary solution Mǫ ∈ Cm+2,α of
(6.76) F̃ |Mǫ = f
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 37
provided there exists a strictly convex function χ ∈ C2(Ω̄) and 0 < ǫ is small.
The Mǫ then converge in C
m+2 to a solution M ∈ Cm+2,α of
(6.77) F |M = f.
If f = f(x) and N Einstein, then M is stable.
These statements, except for the stability and the convergence in Cm+2, are
proved in [13].
6.10. Remark. Let us now discuss the pure mean curvature flow
(6.78) ẋ = Hν
with initial spacelike hypersurface M0 of class C
m+2,α, m ≥ 4 and 0 < α < 1.
From the corresponding scalar curvature flow (4.21) on page 12 we immediately
infer that the flow moves into the past of M0, if
(6.79) H |M0 ≥ 0
and into its future, if
(6.80) H |M0 ≤ 0.
Let us only consider the case (6.79) and also assume that M0 is not maximal.
From the a priori estimates in [11, Section 3 and Section 4] we then deduce
that the flow remains smooth as long as it stays in a compact set of N , and if
a compact, spacelike hypersurface M1 of class C
2 satisfying
(6.81) H |M1 ≤ 0
lies in the past of M0, then the flow will exist for all time and converge in
Cm+2 to a stable maximal hypersurface M , hence a neighbourhood of M can
be foliated by CMC hypersurfaces, where those in the future ofM have positive
mean curvature, in view of Remark 5.7 on page 24.
Thus, the flow will converge if and only if such a hypersurfaceM1 lies in the
past of M0.
Conversely, if there exists a compact, spacelike hypersurface M1 in N sat-
isfying (6.81), and there is no stable maximal hypersurface in its future, then
this is a strong indication that N has no future singularity, assuming that
such a singularity would produce spacelike hypersurfaces with positive mean
curvature.
An example of such a spacetime is the (n + 1)-dimensional de Sitter space
which is geodesically complete and has exactly one maximal hypersurface M
which is also totally geodesic but not stable, and the future resp. past of M
are foliated by coordinate slices with negative resp. positive mean curvature.
To conclude this section let us show which spacelike hypersurfaces satisfy
C1-estimates automatically.
38 CLAUS GERHARDT
6.11. Theorem. Let M = graphu|S0 be a compact, spacelike hypersurface
represented in a Gaussian coordinate system with unilateral bounded principal
curvatures, e.g.,
(6.82) κi ≥ κ0 ∀ i.
Then, the quantity ṽ = 1√
1−|Du|2
can be estimated by
(6.83) ṽ ≤ c(|u|,S0, σij , ψ, κ0),
where we assumed that in the Gaussian coordinate system the ambient metric
has the form as in (6.1).
Proof. We suppose as usual that the Gaussian coordinate system is future
oriented, and that the second fundamental form is evaluated with respect to
the past directed normal. We observe that
(6.84) ‖Du‖2 = gijuiuj = e−2ψ
|Du|2
hence, it is equivalent to find an a priori estimate for ‖Du‖.
Let λ be a real parameter to be specified later, and set
(6.85) w = 1
log‖Du‖2 + λu.
We may regard w as being defined on S0; thus, there is x0 ∈ S0 such that
(6.86) w(x0) = sup
and we conclude
(6.87) 0 = wi =
‖Du‖2
j + λui
in x0, where the covariant derivatives are taken with respect to the induced
metric gij , and the indices are also raised with respect to that metric.
Expressing the second fundamental form of a graph with the help of the
Hessian of the function
(6.88) e−ψv−1hij = −uij − Γ̄ 000uiuj − Γ̄ 00iuj − Γ̄ 00jui − Γ̄ 0ij .
we deduce further
(6.89)
λ‖Du‖4 = −uijuiuj
= e−ψ ṽhiju
iuj + Γ̄ 000‖Du‖4
+ 2Γ̄ 00ju
j‖Du‖2 + Γ̄ 0ijuiuj .
Now, there holds
(6.90) ui = gijuj = e
−2ψσijujv
and by assumption,
(6.91) hiju
iuj ≥ κ0‖Du‖2,
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 39
i.e., the critical terms on the right-hand side of (6.89) are of fourth order in
‖Du‖ with bounded coefficients, and we conclude that ‖Du‖ can’t be too large
in x0 if we choose λ such that
(6.92) λ ≤ −c|||Γ̄ 0αβ||| − 1
with a suitable constant c; w, or equivalently, ‖Du‖ is therefore uniformly
bounded from above. �
Especially for convex graphs over S0 the term ṽ is uniformly bounded as
long as they stay in a compact set.
7. The inverse mean curvature flow
Let us now consider the inverse mean curvature flow (IMCF)
(7.1) ẋ = −H−1ν
with initial hypersurfaceM0 in a globally hyperbolic spacetimeN with compact
Cauchy hypersurface S0.
N is supposed to satisfy the timelike convergence condition
(7.2) R̄αβν
ανβ ≥ 0 ∀ 〈ν, ν〉 = −1.
Spacetimes with compact Cauchy hypersurface that satisfy the timelike con-
vergence condition are also called cosmological spacetimes, a terminology due
to Bartnik.
In such spacetimes the inverse mean curvature flow will be smooth as long
as it stays in a compact set, and, if H |M0 > 0 and if the flow exists for all time,
it will necessarily run into the future singularity, since the mean curvature of
the flow hypersurfaces will become unbounded and the flow will run into the
future of M0. Hence the claim follows from Remark 6.4 on page 28.
However, it might be that the flow will run into the singularity in finite
time. To exclude this behaviour we introduced in [15] the so-called strong
volume decay condition, cf. Definition 7.2. A strong volume decay condition is
both necessary and sufficient in order that the IMCF exists for all time.
7.1. Theorem. Let N be a cosmological spacetime with compact Cauchy hy-
persurface S0 and with a future mean curvature barrier. Let M0 be a closed,
connected, spacelike hypersurface with positive mean curvature and assume fur-
thermore that N satisfies a future volume decay condition. Then the IMCF
(7.1) with initial hypersurface M0 exists for all time and provides a foliation of
the future D+(M0) of M0.
The evolution parameter t can be chosen as a new time function. The flow
hypersurfaces M(t) are the slices {t = const} and their volume satisfies
(7.3) |M(t)| = |M0|e−t.
Defining a new time function τ by choosing
(7.4) τ = 1− e−
40 CLAUS GERHARDT
we obtain 0 ≤ τ < 1,
(7.5) |M(τ)| = |M0|(1− τ)n,
and the future singularity corresponds to τ = 1.
Moreover, the length L(γ) of any future directed curve γ starting from M(τ)
is bounded from above by
(7.6) L(γ) ≤ c(1− τ),
where c = c(n,M0). Thus, the expression 1 − τ can be looked at as the radius
of the slices {τ = const} as well as a measure of the remaining life span of the
spacetime.
Next we shall define the strong volume decay condition.
7.2.Definition. Suppose there exists a time function x0 such that the future
end of N is determined by {τ0 ≤ x0 < b} and the coordinate slicesMτ = {x0 =
τ} have positive mean curvature with respect to the past directed normal for
τ0 ≤ τ < b. In addition the volume |Mτ | should satisfy
(7.7) lim
|Mτ | = 0.
A decay like that is normally associated with a future singularity and we
simply call it volume decay. If (gij) is the induced metric of Mτ and g =
det(gij), then we have
(7.8) log g(τ0, x)− log g(τ, x) =
2eψH̄(s, x) ∀x ∈ S0,
where H̄(τ, x) is the mean curvature ofMτ in (τ, x). This relation can be easily
derived from the relation (3.8) on page 5 and Remark 3.6 on page 7. A detailed
proof is given in [12].
In view of (7.7) the left-hand side of this equation tends to infinity if τ
approaches b for a.e. x ∈ S0, i.e.,
(7.9) lim
eψH̄(s, x) = ∞ for a.e. x ∈ S0.
Assume now, there exists a continuous, positive function ϕ = ϕ(τ) such that
(7.10) eψH̄(τ, x) ≥ ϕ(τ) ∀ (τ, x) ∈ (τ0, b)× S0,
where
(7.11)
ϕ(τ) = ∞,
then we say that the future of N satisfies a strong volume decay condition.
7.3. Remark. (i) By approximation we may assume that the function ϕ
above is smooth.
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 41
(ii) A similar definition holds for the past of N by simply reversing the time
direction. Notice that in this case the mean curvature of the coordinate slices
has to be negative.
7.4. Lemma. Suppose that the future of N satisfies a strong volume decay
condition, then there exist a time function x̃0 = x̃0(x0), where x0 is the time
function in the strong volume decay condition, such that the mean curvature H̄
of the slices x̃0 = const satisfies the estimate
(7.12) eψ̃H̄ ≥ 1.
The factor eψ̃ is now the conformal factor in the representation
(7.13) ds̄2 = e2ψ̃(−(dx̃0)2 + σijdxidxj).
The range of x̃0 is equal to the interval [0,∞), i.e., the singularity corre-
sponds to x̃0 = ∞.
A proof is given in [15, Lemma 1.4].
7.5. Remark. Theorem 7.1 can be generalized to spacetimes satisfying
(7.14) R̄αβν
ανβ ≥ −Λ ∀ 〈ν, ν〉 = −1
with a constant Λ ≥ 0, if the mean curvature of the initial hypersurface M0 is
sufficiently large
(7.15) H |M0 >
cf. [25]. In that thesis it is also shown that the future mean curvature barrier
assumption can be dropped, i.e., the strong volume decay condition is sufficient
to prove that the IMCF exists for all time and provides a foliation of the future
of M0. Hence, the strong volume decay condition already implies the existence
of a future mean curvature barrier, since the leaves of the IMCF define such a
barrier.
8. The IMCF in ARW spaces
In the present section we consider spacetimes N satisfying some structural
conditions, which are still fairly general, and prove convergence results for the
leaves of the IMCF.
Moreover, we define a new spacetime N̂ by switching the light cone and
using reflection to define a new time function, such that the two spacetimes
N and N̂ can be pasted together to yield a smooth manifold having a metric
singularity, which, when viewed from the region N is a big crunch, and when
viewed from N̂ is a big bang.
The inverse mean curvature flows in N resp. N̂ correspond to each other via
reflection. Furthermore, the properly rescaled flow in N has a natural smooth
extension of class C3 across the singularity into N̂ . With respect to this natural
diffeomorphism we speak of a transition from big crunch to big bang.
42 CLAUS GERHARDT
8.1. Definition. A globally hyperbolic spacetime N , dimN = n+1, is said
to be asymptotically Robertson-Walker (ARW) with respect to the future, if a
future end of N , N+, can be written as a product N+ = [a, b) × S0, where S0
is a Riemannian space, and there exists a future directed time function τ = x0
such that the metric in N+ can be written as
(8.1) ds̆2 = e2ψ̃{−(dx0)2 + σij(x0, x)dxidxj},
where S0 corresponds to x0 = a, ψ̃ is of the form
(8.2) ψ̃(x0, x) = f(x0) + ψ(x0, x),
and we assume that there exists a positive constant c0 and a smooth Riemann-
ian metric σ̄ij on S0 such that
(8.3) lim
eψ = c0 ∧ lim
σij(τ, x) = σ̄ij(x),
(8.4) lim
f(τ) = −∞.
Without loss of generality we shall assume c0 = 1. Then N is ARW with
respect to the future, if the metric is close to the Robertson-Walker metric
(8.5) ds̄2 = e2f{−dx02 + σ̄ij(x)dxidxj}
near the singularity τ = b. By close we mean that the derivatives of arbitrary
order with respect to space and time of the conformal metric e−2f ğαβ in (8.1)
should converge to the corresponding derivatives of the conformal limit metric
in (8.5) when x0 tends to b. We emphasize that in our terminology Robertson-
Walker metric does not imply that (σ̄ij) is a metric of constant curvature, it is
only the spatial metric of a warped product.
We assume, furthermore, that f satisfies the following five conditions
(8.6) − f ′ > 0,
there exists ω ∈ R such that
(8.7) n+ ω − 2 > 0 ∧ lim
|f ′|2e(n+ω−2)f = m > 0.
Set γ̃ = 1
(n+ ω − 2), then there exists the limit
(8.8) lim
(f ′′ + γ̃|f ′|2)
(8.9) |Dmτ (f ′′ + γ̃|f ′|2)| ≤ cm|f ′|m ∀m ≥ 1,
as well as
(8.10) |Dmτ f | ≤ cm|f ′|m ∀m ≥ 1.
If S0 is compact, then we call N a normalized ARW spacetime, if
(8.11)
det σ̄ij = |Sn|.
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 43
8.2. Remark. (i) If these assumptions are satisfied, then the range of τ is
finite, hence, we may—and shall—assume w.l.o.g. that b = 0, i.e.,
(8.12) a < τ < 0.
(ii) Any ARW spacetime with compact S0 can be normalized as one easily
checks. For normalized ARW spaces the constantm in (8.7) is defined uniquely
and can be identified with the mass of N , cf. [20].
(iii) In view of the assumptions on f the mean curvature of the coordinate
slices Mτ = {x0 = τ} tends to ∞, if τ goes to zero.
(iv) ARW spaces with compact S0 satisfy a strong volume decay condition,
cf. Definition 7.2 on page 40.
(v) Similarly one can define N to be ARW with respect to the past. In this
case the singularity would lie in the past, correspond to τ = 0, and the mean
curvature of the coordinate slices would tend to −∞.
We assume that N satisfies the timelike convergence condition and that S0
is compact. Consider the future end N+ of N and let M0 ⊂ N+ be a spacelike
hypersurface with positive mean curvature H̆ |M0 > 0 with respect to the past
directed normal vector ν̆—it will become apparent in a moment why we use the
symbols H̆ and ν̆ and not the usual ones H and ν. Then, as we have proved
in the preceding section, the inverse mean curvature flow
(8.13) ẋ = −H̆−1ν̆
with initial hypersurface M0 exists for all time, is smooth, and runs straight
into the future singularity.
If we express the flow hypersurfaces M(t) as graphs over S0
(8.14) M(t) = graphu(t, ·),
then we have proved in [14]
8.3. Theorem. (i) Let N satisfy the above assumptions, then the range of
the time function x0 is finite, i.e., we may assume that b = 0. Set
(8.15) ũ = ueγt,
where γ = 1
γ̃, then there are positive constants c1, c2 such that
(8.16) − c2 ≤ ũ ≤ −c1 < 0,
and ũ converges in C∞(S0) to a smooth function, if t goes to infinity. We shall
also denote the limit function by ũ.
(ii) Let ğij be the induced metric of the leaves M(t), then the rescaled metric
(8.17) e
tğij
converges in C∞(S0) to
(8.18) (γ̃m)
γ̃ (−ũ)
γ̃ σ̄ij .
44 CLAUS GERHARDT
(iii) The leaves M(t) get more umbilical, if t tends to infinity, namely, there
holds
(8.19) H̆−1|h̆ji − 1nH̆δ
i | ≤ ce
−2γt.
In case n+ ω − 4 > 0, we even get a better estimate
(8.20) |h̆ji − 1nH̆δ
i | ≤ ce
(n+ω−4)t.
To prove the convergence results for the inverse mean curvature flow, we con-
sider the flow hypersurfaces to be embedded in N equipped with the conformal
metric
(8.21) ds̄2 = −(dx0)2 + σij(x0, x)dxidxj .
Though, formally, we have a different ambient space we still denote it by the
same symbol N and distinguish only the metrics ğαβ and ḡαβ
(8.22) ğαβ = e
2ψ̃ ḡαβ
and the corresponding geometric quantities of the hypersurfaces h̆ij , ğij , ν̆ resp.
hij , gij , ν, etc., i.e., the standard notations now apply to the case when N is
equipped with the metric in (8.21).
The second fundamental forms h̆
i and h
i are related by
(8.23) eψ̃h̆
i = h
i + ψ̃αν
and, if we define F by
(8.24) F = eψ̃H̆,
(8.25) F = H − nṽf ′ + nψανα,
where
(8.26) ṽ = v−1,
and the evolution equation can be written as
(8.27) ẋ = −F−1ν,
since
(8.28) ν̆ = e−ψ̃ν.
The flow exists for all time and is smooth, due to the results in the preceding
section.
Next, we want to show how the metric, the second fundamental form, and
the normal vector of the hypersurfaces M(t) evolve by adapting the general
evolution equations in Section 3 on page 4 to the present situation.
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 45
8.4. Lemma. The metric, the normal vector, and the second fundamental
form of M(t) satisfy the evolution equations
(8.29) ġij = −2F−1hij ,
(8.30) ν̇ = ∇M (−F−1) = gij(−F−1)ixj ,
(8.31) ḣ
i = (−F
i + F
−1hki h
k + F
−1R̄αβγδν
γxδkg
(8.32) ḣij = (−F−1)ij − F−1hki hkj + F−1R̄αβγδναx
γxδj .
Since the initial hypersurface is a graph over S0, we can write
(8.33) M(t) = graphu(t)|S0 ∀ t ∈ I,
where u is defined in the cylinder R+×S0. We then deduce from (8.27), looking
at the component α = 0, that u satisfies a parabolic equation of the form
(8.34) u̇ =
where we emphasize that the time derivative is a total derivative, i.e.
(8.35) u̇ =
+ uiẋ
Since the past directed normal can be expressed as
(8.36) (να) = −e−ψv−1(1, ui),
we conclude from (8.34)
(8.37)
For this new curvature flow the necessary decay estimates and convergence
results can be proved, which in turn can be immediately translated to corre-
sponding convergence results for the original IMCF.
Transition from big crunch to big bang
With the help of the convergence results in Theorem 8.3, we can rescale the
IMCF such that it can be extended past the singularity in a natural way.
We define a new spacetime N̂ by reflection and time reversal such that the
IMCF in the old spacetime transforms to an IMCF in the new one.
By switching the light cone we obtain a new spacetime N̂ . The flow equation
in N is independent of the time orientation, and we can write it as
(8.38) ẋ = −H̆−1ν̆ = −(−H̆)−1(−ν̆) ≡ −Ĥ−1ν̂,
where the normal vector ν̂ = −ν̆ is past directed in N̂ and the mean curvature
Ĥ = −H̆ negative.
46 CLAUS GERHARDT
Introducing a new time function x̂0 = −x0 and formally new coordinates
(x̂α) by setting
(8.39) x̂0 = −x0, x̂i = xi,
we define a spacetime N̂ having the same metric as N—only expressed in the
new coordinate system—such that the flow equation has the form
(8.40) ˙̂x = −Ĥ−1ν̂,
where M(t) = graph û(t), û = −u, and
(8.41) (ν̂α) = −ṽe−ψ̃(1, ûi)
in the new coordinates, since
(8.42) ν̂0 = −ν̆0 ∂x̂
(8.43) ν̂i = −ν̆i.
The singularity in x̂0 = 0 is now a past singularity, and can be referred to
as a big bang singularity.
The unionN∪N̂ is a smooth manifold, topologically a product (−a, a)×S0—
we are well aware that formally the singularity {0}×S0 is not part of the union;
equipped with the respective metrics and time orientation it is a spacetime
which has a (metric) singularity in x0 = 0. The time function
(8.44) x̂0 =
x0, in N,
−x0, in N̂ ,
is smooth across the singularity and future directed.
N ∪ N̂ can be regarded as a cyclic universe with a contracting part N =
{x̂0 < 0} and an expanding part N̂ = {x̂0 > 0} which are joined at the
singularity {x̂0 = 0}.
It turns out that the inverse mean curvature flow, properly rescaled, defines
a natural C3- diffeomorphism across the singularity and with respect to this
diffeomorphism we speak of a transition from big crunch to big bang.
Using the time function in (8.44) the inverse mean curvature flows in N and
N̂ can be uniformly expressed in the form
(8.45) ˙̂x = −Ĥ−1ν̂,
where (8.45) represents the original flow in N , if x̂0 < 0, and the flow in (8.40),
if x̂0 > 0.
Let us now introduce a new flow parameter
(8.46) s =
−γ−1e−γt, for the flow in N,
γ−1e−γt, for the flow in N̂ ,
CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 47
and define the flow y = y(s) by y(s) = x̂(t). y = y(s, ξ) is then defined in
[−γ−1, γ−1]× S0, smooth in {s 6= 0}, and satisfies the evolution equation
(8.47) y′ ≡ d
−Ĥ−1ν̂ eγt, s < 0,
Ĥ−1ν̂ eγt, s > 0.
In [14] we proved:
8.5. Theorem. The flow y = y(s, ξ) is of class C3 in (−γ−1, γ−1)×S0 and
defines a natural diffeomorphism across the singularity. The flow parameter s
can be used as a new time function.
8.6. Remark. The regularity result for the transition flow is optimal, i.e.,
given any 0 < α < 1, then there is an ARW space such that the transition flow
is not of class C3,α, cf. [19].
8.7. Remark. Since ARW spaces have a future mean curvature barrier, a
future end can be foliated by CMC hypersurfaces the mean curvature of which
can be used as a new time function., see [9] and [22]. In [8] we study this folia-
tion a bit more closely and prove that, when writing the CMC hypersurfaces as
graphs Mτ = graphϕ(τ, ·) in the special coordinate system of the ARW space,
where τ is the mean curvature, of Mτ then
(8.48) τ(−ϕ)1+γ̃
→ const > 0,
notice that ϕ < 0, and hence
(8.49) lim
ϕ(τ, x)
ϕ(τ, y)
= 1 ∀x, y ∈ S0.
Moreover, the new time function
(8.50) s = −τ−q, q = γ̃
1 + γ̃
can be extended to the mirror universe N̂ by odd reflection as a function of
class C3 across the singularity with non-vanishing gradient.
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Ruprecht-Karls-Universität, Institut für Angewandte Mathematik, Im Neuen-
heimer Feld 294, 69120 Heidelberg, Germany
E-mail address: gerhardt@math.uni-heidelberg.de
URL: http://www.math.uni-heidelberg.de/studinfo/gerhardt/
http://arXiv.org/abs/math.DG/0409457
http://arXiv.org/abs/math.DG/0409465
http://arXiv.org/pdf/math.DG/0207049
http://arXiv.org/pdf/math.DG/0207054
http://arXiv.org/pdf/math.DG/0403485
http://arXiv.org/abs/math.DG/0403097
http://arXiv.org/abs/math.DG/0509217
http://arXiv.org/pdf/gr-qc/0404112
http://arXiv.org/pdf/math.DG/0403002
http://arXiv.org/abs/math.DG/0602597
http://arxiv.org/abs/math.DG/0408197
http://page.mi.fu-berlin.de/~schnuere/skripte/pde.pdf
http://arXiv.org/abs/math.DG/0109053
	1. Introduction
	2. Notations and preliminary results
	3. Evolution equations for some geometric quantities
	4. Essential parabolic flow equations
	5. Stability of the limit hypersurfaces
	6. Existence results
	7. The inverse mean curvature flow
	8. The IMCF in ARW spaces
	Transition from big crunch to big bang
	References
