Volume preserving curvature flows in Lorentzian manifolds

Volume preserving curvature flows in Lorentzian manifolds

Matthias Makowski

Abstract Let N be a (n

+

1 )-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface So and F a curvature function, either the mean curvature H, the root of the second symmetric polynomial (T2 =

-JH2

or a curvature function of class (K*), a class of curvature functions which includes the nth root of the Gaussian curvature

I

(Tn = K ^Ii. We consider curvature flows with curvature function F and a volume preserving term and prove long time existence of the flow and exponential convergence of the corre- sponding graphs in the Coo -topology to a hypersurface of constant F -curvature, provided there are barriers. Furthermore we examine stability properties and foliations of constant

F -curvature hypersurfaces.

Mathematics Subject Classification (2000) MSC 53C44 . MSC 35K55

1 Introduction

We show the long time existence and convergence to a constant F-hypersurface of the fol- lowing curvature flow in a globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface under suitable assumptions (this setting and the other notions will be explained shortly in more detail):

i = (C/J(F) - f) v,

x(O) = xo, (1.1)

where Xo is the embedding of an initial, compact, connected, spacelike hypersurface Mo of class

c

^{m+2.a, 2 :'S mEN, 0 < IX < 1, v} is the corresponding past directed normal, F is a curvature function of class c^m.a (r) evaluated at the principal curvatures of the flow

M . Makowski (~)

Universitat Konstanz, 78457 Constance, Germany e-mail: matthias.makowski@uni-konstanz.de

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-186554

hypersurfaces M(t), which lie in an appropriate cone

r,

^x(t) denotes the embedding of M(t), q; is a smooth supplementary function satisfying <!JI > 0, q;/1 :: 0, and! is a volume preserving global term, ! = !k, see the definition below.

Fut1hermore the initial hypersurface should be admissible, meaning that its principal cur- vatures belong to the defining cone

r

of the curvature function F, which will be specified below.

Depending on which type of volume has to be preserved, we define the global term as in [22]:

fM/

Hk<!J(F) dMt

!k(t) =

f'

H d

. M/ k Mt

( 1.2)

Here Hk, k

=

^{0, ... , n,} denotes the kth elementary symmetric polynomial, where Ho = 1.

For an overview of the notation (especially concerning the curvature functions), we refer to Sect. 2.

We assume that the ambient space N is a (n

+

¹ )-dimensional smooth, connected, globally hyperbolic Lorentzian manifold with a compact, smooth, connected Cauchy hypersurface So.

This means that N is diffeomorphic to 1 x So, where 1 = (a, b) is an interval with

°

^{E /,}

and there exists a future directed Gaussian coordinate system (x^a), such that the metric

(g

^ctf!)

can be expressed in the form

0.3) Here xo denotes the time function, which is defined on I, and (xi) are local coordinates for the Cauchy hypersurface So. The coordinates can be chosen such that

(1.4) The existence of a smooth, proper function! : N ^~ IR with non-vanishing timelike gradient in a merely connected, smooth Lorentzian manifold N already assures the existence of such a special coordinate system, see [14, Theorem 1.4.2], implying that N is globally hyperbolic with a compact Cauchy hypersurface. Alternatively one can deduce the existence of the spe- cial coordinate system in smooth, globally hyperbolic Lorentzian manifolds with compact Cauchy hypersurface from [4, Theorem 1.1] and [3, Lemma 2.2]. Hence if the reader is not familiar with the notions above, he can simply assume N to be 1 x So equipped with the metric (1.3).

We need one further assumption on the ambient manifold, namely we consider curva- ture flows in cosmological spacetimes, a terminology due to Bartnik, meaning a Lorentzian manifold with the above properties, which furthermore satisfies the timelike convergence condition, an assumption which is quite natural in the setting of general relativity as it cor- responds to the strong energy condition (see for example [16]). The timelike convergence condition asserts that for all pEN there holds

( 1.5) We only mention that for the proof of Theorem 1.3 (with F = H) this condition could be relaxed to the case where the lower bound is - J\ with a constant J\ > 0, where in this case one needs to assume that there holds H >

JnA

on the initial hypersurface.

In the case of general curvature functions however we will need to assume that the timelike sectional curvatures of N are non-positive, i.e. at points pEN there holds

RafJvvVaW.BVvw8 ~

°

^{Vtimelike V E TpN.} Vspacelike W E TpN. (1.6)

The possible curvature H, F = a2 or F E (K*), see a definition of the class curvature (K*), where the most

function in this class is F = an. we have to distinguish cones of

r

and the supplementary function (f), see also 16) for a definition of the cones (i) Let F = Hand k 0, x and

r

= ]Rn.

> 0 example, one could

(f)(x) = -x-I. For need to be preserved

mean curvature fI ow,

(ii) For F = a2 =

only well defined for k E 10, I, (iii) Lastly, F ^E a

k E

to, ... ,

^n} we

x or (f)(x) = -x-I.

same reasons as above.

function of degree 1 and of class (f)(x) log(x) and

r

=

r+.

We denote by (F,

r,

(f) one choices of curvature functions cones of definitions as well as functions stated above.

not

IS

order to be able to derive CD-estimates we have to add an additional assumption, we provide the necessary definition.

Definition 1.1 Let F be a continuous curvature metric cone

r c

JRH. Then we define (here we

defined on an open, convex, the cases considering future or past by brackets).

c be a constant, then we say we a curvature for (F,

r,

c) , where k E R k ~ 2, 0 <

fJ

< L if a compact, connected, spacelike admissible hypersurface M of

> c.

CI := min F and C2 Mo

we can state the following assumption:

maxF, Mo

1.2 We have a future curvature

r,

for (F,

r,

^{Cl) of class C2.} If in the case F

=

holds ^Ci 0, then we assume the

If the curvature function is not the mean curvature, we assume of a convex function Xf2 E C²(Q), where DeN is

of a strictly convex function see Lemma 3.2.

Now we state

(l.7)

(1.8)

of class C² and a past

r

= ]Rn, for i = ] or

(F,

r,

(f) be as above, m ~ 0 < a < 1, and suppose there the flow (J .1) with f =

!k

has a solution existing for all

00, such thatforfixed time M/ E

c

^{m+2.a and the M/ 's as graphs}

in to a compact, connected, of

, which is a stable solution of the equation

F = co' ^(1.9)

where Co

If MOl F and cP are smooth, then the f'FHCllM'YU,OYlf'U of the is exponential in the Coo -topology.

for k =

°

^the volume, for k volume of the 0, then far 1 < k

and if the ambient space has constant curvature

Vn+l-k is

Finally, we want to name some of the volume preserving curvature flows in , different ambient and discuss shortly obtained in

Volume preserving curvature flows have for various curvature functions

in different speaking, if one assumes a certain or

pinching condition on initial hypersurface this is preserved during the flow, then proving a priori existence of the all times t E [0,00) and the COO-topology of the flow to a sphere or a

can

case the is IRⁿ+ I. mean curvature have

previously considered by for n

=

^{1 in [11]} and by Huisken for n > 2 in [1 McCoy considered preserving mean curvature flows in IRⁿ+¹ in [21]

on extended the results to curvature functions Recently Cabezas-Rivas MiqueJ proved similar curvature flow in the hyperbolic space under the initial hypersurfaee, see

flow by powers symmetric

mean horosphere-con vexity

considered the a pinching on the principal curvatures of the initial

only result concerning volume preserving curvature flows

in found in the paper [9) by and Huisken, where

ume preserving mean curvature flow has been considered. method in the case substantially from the case. Neither

principal curvatures is but assuming (1.5) in

case a general curvature respectively, evolution equation the curvature function one can see upper and the curvature

n,'r-',",Plr\IP't1 during the flow, valid for more bounded global terms.

is crucial part one to prove under the assumption Now the C¹_ and priori estimates can be deduced by the same methods case of a time-independent force-term and do not rely on special choice of the term. The higher order can not be deduced the results of Krylov-Safonov in view of the global term (which is merely bounded at this instead we use a method already employed [6,22]. Then evolution equation

for curvature function is the convergence to a

hypersurface constant F -curvature.

From remark about term

f

^{one can}

as long is concerned, a of global terms

can than the ones used throughout the paper. In one can look as well al curvature flows that preserve volumes with different densities and obtain the same results above, as they neither disturb bounded ness of the curvature function nor the analysis out to achieve

It is also to prove the foliation by a

similar [14] by usi the curvature since the

the mean curvature flow without a global term, we H"'.~'~~, we show in section 1

°

^{that a enclosed}

by barriers the F -curvature can be foliated by hypersurfaces of constant F -curvature.

Furthermore we show that each interior this region can be obtained as the hypersurface of a nontrivial curvature preserves the volume

area.

The notation and definitions are summarized in 2. The evolution tions of various geometric quantities are quoted in 3. In

property the flows is shown and

4 the volume-preserving are derived. Sections 5, 6

and 7 the

exponential convergence and in Sect. lOwe obtain

pn~se!mca in 8. Stability is mentioned Sect. 9 a region by constant F -curvature hypersmfaces some further results concerning foliation. short existence and uniqueness of the to the is ^{ITlPnrF'n In} 11 ^~

2 Notation and definitions

The of section is to formulate equations a hypersurface

In a 1 )-dimensional manifold N and to the definitions of the

of curvature as as some well-known of certain curvature functions which will throughout this paper. Note that of hypersurfaces in Lorentzian

Riemannian arise in

the GauE equation. For more detailed definitions about curvature functions, we refer reader to Chap. 2.1, and for an account the differential geometry to [13, 11, 12] especially Chapter 12.5 therein with respect to

coordinate systems and manifolds.

Throughout this section N will be assumed to be a (n

+

I)-dimensional Lorentzian man~

ifold and, unless stated otherwise, the summation convention is throughout the paper.

We will denote in the space N by greek with

from 0 to n and usually with a bar on top of them, the and the Riemannian curvature tensor in the ambient will denoted by (gat)) and

(R

Cifly8) respectively, and geometric quantities a spacelike hypersmface M by latin indices from 1 to n, the metric the Riemannian curvature tensor on M are by (gij) and (Rijk[) respectively. Generic coordinate systems in Nand M will be denoted by (XCi)

and (~i) respectively. Ordinary partial differentiation will denoted by a comma whereas covariant differentiation will or in case of possible ambiguity

will preceded a semicolon, u N , (u_{a )} the

(uap) the Hessian, but e.g. covariant derivative curvature tensor will by (RafJvo;f)' We also out that (with obvious to quantities)

(2.1) where x denotes embedding M in N local coordinates (x^a)

induced metric of hypersurface will be denoted by gij,

(2.2) the seCI)na form will denoted by (hi) by IJ, which is a

vector, i.c. for p E M there holds

v(p) E C_p := (2.3)

and covariant tensors and we note consists of two components,

C;,

which we call future directed respectively.

The quantltles spacelike hypersurface M are connected through the GaujJ formula, which can be considered as definition of the fundamental form,

hi) v,

where we are free to choose future or past normal, but we stipulate we always use the past directed

that the sequel a covariant is always a tensor,

x~.= ..

Ij .Ij (2.5)

,-,,",'"VI,''' the Christoffel-symbols the ambient space hypersUiface

Finally, we have

as well as

Weingarten equation:

Codazzi equation hij;k

=

equation

{hik h jf - huh jd

+

Note that the last equation sign comes into play.

(2.6)

(2.7)

Now we wanl Lo define different classes curvature functions, first we provide the

definition of and some will in the

without explicitly stating them again.

Definition 2.1

P_n is

C ]R;,n be an open, convex, symmetric cone, i.e.

set of all

(Krri) E

r

^{Vn E}

U:Ul'VU.:> of order n. Let

f

^E

f is said to a curvature function of class . For simplicity we will also (j, r) as a curvature function.

(2.9)

10) to

Now denote by S symmetric endomorphisms of lR1.^f1 and by S

r

the endo- with

mapping

where Ki denote especially the

[14, Chap. 2.1]. Since

vu,,,"u,,,,- to an subset Then we can define a

eigenvalues of A. the relation properties and relation

differentiability properties are the same

(2.11)

see our

see [14, 2.1.20], we do not notions write always F the curvature function. at a point x of a we can consider a curvature function F as a function derined on a cone

r c ,

F

=

^{F (K;) for ) E}

r

resenting the principal curvatures at point x the hypersurface, as a function depending on (hf) , F F (h{) or even as a function depending on (hi}) and (gij), F = F (hi) , g ij)'

However, we distinguish the respect to

r

or We our notation important

For a sufficiently smooth curvature function F we ^U'vHVl'-' by , a contravari-

ant tensor of and by

F/ =

contravariant with to the

j respect to i, derivatives F w.r.t. hi) hj.

the F,;

¥:

and the covariant derivative F;i Fkl hkl;i Fij is diagonal if hi} is diagonal such a coordinate system there holds . For a relation between second derivatives see [14, Lemma 2.1.14]. Finally, if (r) is concave, then F is concave as a curvature function depending on _{(h ij ).}

these definitions we can turn to special classes curvature functions.

But first we recall the definition of an admissible

Definition 2.2 spacelike, orientable hypersurface M of class C² a Lorentzian manifold N is to be admissible with to a continuous curvature (F, ,if its principal curvatures with respect to the past directed normal lie

Definition We distinguish of curvature functions:

(i) A symmetric curvature function F E (r+)

n

(f+), r+ {(Ki)E

Ki > 0, 1 ~ i ::: n}, positively homogeneous of degree do (K), if it is strictly monotone,

0, is to be class

F· .1

vanishes on the boundary of fulfills the following inequality:

i ' kl _I ( i' ) 2

F}' I]ij rlkl < F F J 17ij - 17ij 17kl V 11 E S,

F is evaluated at (hi)) E and

(liij)

is the inverse (hi)).

(ii) function F E (K) is said to that

F is evaluated at (hi}) eigenvalues of

(hij).

(iii) A differentiable curvature hypersurface

of class (K*) if there exists

°

^{< EO}

.. k

< F1Jh·kh. V(h .. ) E S

- '/ J IJ

H represents the mean to be

(2.12)

(2.13)

Eo(F) such

(2.14) I.e. sum

admissible

we mials

the most important curvature functions, the elementary symmetric polyno-

(2.1

note that nth root of the gaussian curvature (Tn

=

K

~

^{is an ofa}

curvature function of (K*).

by noticing if F E (K

degree I is concave, see [14, Lemma 2.2.14J.

We note some important properties elementary symmetric Lemma 2.4 k ::s: n be fixed.

(i) . We define convex cone

(2.16) Then Hk is strictly monotone on the connected component of

Oi) (iii)

the

1 < S < t

17) cone.

I

r are concave on

j

d - _ ( f h ) f h hid n an (Tk -

G)

^t ere 0 s

< 18)

the principal curvatures to lie in

rn ⁼⁼ r

the first, in the second

rs

^{for the} inequality.

(iv) For fixed t, no summation over

t,

there holds

19) Proof cone

n

and (i) follows from [18, Sect. 2], (1) (iii) from

hp,',,·PtYl 15. (iv) follows from the definition of

A consequence the preceding Lemma is the following

Lemma N be a semi-Riemannian space of constant curvature, the symmetric polynomials F

=

^{Hk, 1} k.::; n, are of class (D). In case k 2 it suffices to assume

N is an Einstein manifold.

Proof The of the Lemma can he found [15, Lemma proof induction on k and (iv) 2.4.

Now we state a well-known inequality general curvature functions.

Lemma 2.6 Let F E C² (r) be a concave curvature function, homogeneous with F(l, .,.,1) > 0, then

F(l, .,', 1)

F ^< H.

n See [14, 2.2.20],

of o

1

(2.20)

o

tensors on N, we will a Riemannian on N. We use a

Riemannian which we define by

to of

e2v1 { (d xO)2

+

^{uijdxi dxj } ,}

norm of a veclor field 'l on N will be by

I

1II'lili

=

(gafi 'lfi)2.

functions ifJ defined on So x [0, T), we will also use the following U V L U U , " ' "

lifJlo := lifJlo.sox[o.Tl := lifJ , t)1,

[0. T)

we often omit the dependence on So x [0, T). We will also use the notation

(

' k " )

i I/Allo

:=

IIAllo.Mx[o.n

:= sup gl gl hijhkl ~ (x, t)

(x ,OEM x [0. T)

tensors A = (hij) defined on M x [0, T), (gij

(t»

^denotes

(gij(t» of M(t), with a definition for tensors.

(2.21)

(2.22)

(2.23)

(2.24) lllverse

Finally, we want to note that we will use spaces in later sections, we refer to [14, nition

3 Evolution equations

this '"'U"IJ'"w< we state some about of hypersurfaces as graphs

the geometric quantities needed throughout the paper. a derivation to [14, Chap. Note that is a but significant difference in equations compared to Riemannian case to the in the

of aU, we have in of [19, Lemma 3.1] following

Lemma Let N a smooth, globally hyperbolic Lorentzian manifold with a compact, connected Cauchy So and MeN a compact, connected, space like hypersuiface of cm,a,O .:::: a < 1, 1 .:::: mEN, M can written as a graph over So

M=

¹⁾

with u E (So).

We the additional regularity mentioned above follows by the same proof as [1 Lemma 3.1], the implicit function theorem being the main theorem used the

From now on we assume to in local coordinates of coordinate system given by (1.3).

The !low can written as over So

M(t) =

where we use the symbol x ambiguously by points p = (x<X) E N as well as points p

=

^{(xi) E So.} Now suppose flow are given by an embedding x x(t,O,

~

=

(~i) are local x : ---7 N, Mo

of a x(M). Then xO=u(t,

xi

=

(I, induced has the form

gij

=

^e2v1

manifold M, holds

u(t, x(t, ~»,

initially we have the

(3.4) where is evaluated at , x). Its inverse (giJ)

=

^{I can be} nrp''',!I'''; as

(3.5)

Uj

where we

(3.7) and

v

^{v-I. Hence, u} is spaceUke if only if

I I

< I, in (3.4).

past-directed normal has the

-v ^{- l} (3.8)

Furthermore, looking at component ex 0 in the formula, we obtain

(3.9)

the covariant derivatives are taken with to the induced metric' of the

hypersurface. For use, we reformulate the expression as in [14, 11)], such that

(3.10)

"Af""'''''''' fundamental form of the

nate covariant of u are now taken with rpe,....",,'T

to the metric In the in view of

case controlling the C l-norm of graph u is tantamount to controlling

v

.7) and

II) Finally, as the curvature flows with curvature functions we had to assume the existence a strictly convex function X E in a given domain Q, we state a

condition guaranteeing the such a For a proof following see [14, 1.8.3].

N a smooth, globally hyperbolic Lorentzian manifold, So a sur- face, (x(l') a future directed Gaussian coordinate system associated with

compact. there a con vex function X E , i.e. afunction satisfying

with a positive constant co, provided the level hypersurfaces {x o = canst} that intersea Q are strictly convex.

We consider a curvature function F E

em.ex

^{(r), 2}

::s

^{mEN, 0 < ex <} 1, a function

f

=

f

(t) and a real function ifJ E

e

^m.a OR.) and write from now on ifJ = ifJ (F).

The curvature flow is then given by the evolution problem (1.1) with f = fk, 0

::s

^{k S n,}

as defined in (1.2) (where we remark again that not all values of k are allowed for F = H or F = (52, see the remarks after Eq. 1.2).

We will assume throughout the next sections that short time existence has already been assured and we consider a solution x E H^m+^CI• ~ (Qr') of the curvature flow on a maximal interval [0, T*), 0 < T*

::s

^{00, where QT = [0, T) x M.} Short-time existence is well known for the curvature flow without the global term as we are dealing with a parabolic problem and with a fixed point argument we can extend this result to the flow including the globaJ term. This will be supplemented in Sect. 11.

Hence we consider a sufficiently smooth solution of the initial value problem (1.1) and show how the geometric quantities 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. [13, Chap. 11.5].

First, we consider the evolution equations for the hypersurfaces represented as graphs.

Looking at the component ex = 0 of the flow (1.1) we obtain the scalar flow equation

' - -VI -I(n-. f)

u - -e v ^{'V - ,} (3.13)

where the time derivative is a total time derivative of u u(t, xU, 0). If however we consider u to depend on u = u (t,

0

we obtain the partial derivative

au

_{. ·k}

=

u - UkX

at

= -e-1/! v(ifJ - f). (3.14)

Let us now state the evolution equations, where we note that all covariant derivatives appear- ing in these equations are taken with respect to the induced metric of the flow hypersurfaces.

Lemma 3.3 We have the following evolution equations:

gij

⁼ 2(ifJ - f)hij, (3.15)

-Jg

d = (ifJ - f)H

Jg,

where g = detgij,

dt (3.16)

V=gi}ifJiXj, (3.17)

h

1 ; 1 ^j = ifJ j - (ifJ - f')

{h~

I ^hk ^j

+ R

a~yo ^R "Va

x~

^{I vy x}k ^8gkj} ' (3.18)

. {k -

^{a f3}

8}

hi} = ifJ;ij - (ifJ - f) -hJ¹kj

+

^{Rx{Jy8V Xi vy x j , (3.19)}

<P-ifJIFijifJ.··=-ifJ/(ifJ-f) _,IJ {Fijh~hk'-I-Fij _{I J}

R

_aFYO ^R 'VaX~1JYx8} _{I J '} where <pI = -ifJ(r), d

dr ^(3.20)

it - ifJl Fij Uij = -e-1/! v(ifJ - f) -I- ifJl Fe-1/! v

+

^([;1 Fij { fOOOUiU j

-+

^{2foqu j}

-+ t;n '

(3.21)

~ - ¢' Fij V;ij

=

^{- ¢ '} Fij hikh~v

+

^{[(¢ - f) - ¢'} F] l]afJva vfJ - 2¢' Fij h~xr xt I]afl - ¢' Fij l]afJyXf X; va

¢ ' Fij R- a f-$ y 0 ^E kl - exf3yoV Xi Xk Xjl]EXI g ,

where I] is the covariant vector field (rIa) = e V1 (-1,0, ' , . , 0),

h!

-¢'Fklh{kl

=

-¢'Fk1hrkhi'h{ +¢'Fhrihrj

-(¢-f)h~hj+¢'Fkl.rsh,,·h j+¢"FFj

I k kl,/ rs; /

(3.22)

+2"",1FkIR- a fJ y lihm rj m'FkIR- ex fJ y ohm "j

'P afJyr,XmXi Xk XI' I g - ^'P afJyoXmXk X,. Xl i g _m'FkIR- xaxfJx(xOhmj - m'FklR- vax/i Vy xlJh!

'P afJyr, m k / I ^'P exfJyr, k I ^I m'FR- a fJ y ^IJ mj (m f)R- ex fJ y ⁸ mj

+

^'P afJyo V Xi V xmg - ^{'P -} atiyo V Xi V xmg

m'FkIR- {a{3Y8Em j +afJyr,ElIlj} (323)

+ ^'1/ af~yr,;f V Xk XI Xi xmg ^V Xi Xk XmXI g . . Proof See [14, Lemma 2.3.1 for the first two equations, then Lemma 2.3.2, Lemma 2.3.3 for the next two equations, Lemma 2.3.4, equation (3.14) together with (3.9), Lemma 2.4.4,

Lemma 2.4.1]. 0

4 Height estimates and volume preservation of the flow

First, we recall the definition of the mixed volume, for k ^E {O, . , . , n) and a hypersurface M represented by a graph u we have:

V _ {

Iso Iou e1/J Jg ,

k = 0

n+l-k - { _(n+l)k

(n))-I J'

_MHk-l ^d _J-i, ^k ₌ ¹ _{, ... ,n,} ^(4.1)

where for (t, x) E N we denote by

g

^{(t, x)}

=

det(gij)(t, x) the volume element of the level hypersurface x^O = t at the point X E So. The choice of the reference point 0 E (a, b) for the enclosed volume is arbitrary. Now we are going to prove the claimed volume preservation property of the flow (1.1) with f = fk, 0

:s

k

:s

n.

Lemma 4.1 For k

=

⁰ the enclosed volume Vn+1 andfor k

=

¹ the volume of the hypersur- faces Vn is preserved.

If the ambient space has vanishing sectional curvatures, thenfor 1 < k

:s

n the mixed volume V,1+ I-k is preserved.

Proof First we observe that for ^X E So we have

)g(u(X), x) = vjdet(gij(u(x), x», where gij (t, .) denotes the metric of the level hypersurface x^O = t.

Taking this into account, we have for k = 0 in view of (3.14):

-V+I d

dt n ⁼

J ^{a a~ e1/J )}

g(u(x), x) dx

So

= - J

^{(¢ - fa»)} g(u(x), X) dx

=

^0,

So

in view of the definition of fa. Hence the enclosed volume is preserved by the flow.

(4.2)

(4.3)

k I we have in view of (3.16)

d d

f

(n

+

^l)n-Vn = -IMII = (IP

dt dt , ^(4.4)

M/

Finally, 1 < k .::: n we assume the ambient vanishing sectional curvatures.

Then we exploit Lemma 2.5 and Lemma We

(n

+ 1)(:)

^{d / Hk-Jd/.lt = / kHk(IP}

In order to prove ing the evolution.

which we will prove by the barriers for Lemma 4.2 Let

then there holds for

M/ Mt

= k / (IP - fk)Hk dILL O.

M/

we will show that the curvature with the monotonicity of the constant

F"r"J\.lrrl~ we then obtain that the flow

CI

0<1<

C2:= max <P(F), Mo

CI < IP (F) .::: C2.

Moreover <PSUP(t) := max M, <P is monotonically decreasing and <pinf (t) monotonically

Proof We will prove (4.7) holds until To, where 0 < To < T* is arbitrary.

the first statement and the one follows by observing that the for the interval [tl, T*), where 0 < tl is arbitrary.

(4.5)

D

is bounded dur-

(4.6)

(4.7)

min M/ r:p is

We only prove the upper bound, the lower bound follows analogously.

Let

Et - E,

where E > 0 is chosen

o

^(4.9)

and from (3.20) we

f {

₎ ^J _1ik ^hk _j

+

^{Fij R-}C'fflyfJ 1J ^a Xi V X j ^{fj y} 1J} - E. 10) Suppose there is a point (to, XO), 0 to < ,XO E So such that

&(to, o

^(4.11)

and where to is the first time for this to

holds for all x E

(to, <

<PUO,

xo) (4.12)

by the definition of

&

fk

(to)

:s

$ (to, xo)· (4.l3)

(4.10) at (to, xo) yields in view of the

o

< (4.14)

condition (1.5) in case of the mean curvature flow curvature functions as well as the non-negativity of term Fij hikh).

implies

$ < E t

+

^E

+

^{C2 E To}

+

^E

+

^{C2 (4.15)}

and the follows because ^E c:;an be arbitrarily smalL o

",-,"-"'HU'U ensures in the case or F = a2, that the

2.6 for F

=

^a2.

depends on $' > O.

which can be used to prove long

time global force terms, than ones we

Remark of the preceding Lemma that the global term k~U','."'V., CI

:s

f(t)

:s

C2

for t E [0, T*), Ci and C2 are the constants 4.2.

On the previous proof more

terms

f

⁼

f

^(t) which are bounded from above arbitrary constants C2

respectively, the curvature $ (F) is bounded by the same constants if $ we assume the existence of a future curvature barrier for (F,

r,

^{C2) of}

a past curvature (F,

r,

^CI) of class . we will see that most of the carried out in the next go through in this more setting, see remark at the end of 7.

Now we want to show the monotonicity of hypersurfaces the mean curvature a cosmological spacetime

resoec:::t to their F -curvature.

found [14, 4.7.1J, for general F a minor modification.

Lemma 4.4 Let N OSlnO.lOfi,'lClU spacetime Cauchy-hypersurface

",., •• ",,.'4' curvatures, F a monotone curvature function on an convex, cone r, F E C I (r)

n ,

such that F vanishes on the boundary of

r

FI r O.

Let Mi = graph Ui, i

=

I. 2 be two compact, connected, of class C², such that the respective F -curvatures Fi

if

pESo is a point such that I (p), p) = min M2

curvatures of M2 at (p), to zero.

admissible hypersur- (4.1 we assume that the

Then there

UI < (::::J u2· ^(4.17)

Proof In view of Eg. and the maximum principle it suffices to show

(4.18) Now suppose (4.18) is not valid, so that

E(Ul) = {x E So: Uz (4.19)

Thus there exist Pi E Mi ^that

0< d(M2, ) = d(p2, PI)

=

^sup {d(p, q) :

where dis Lorentzian distance function, which is finite and continuous in our see [13, 12.5.9].

¢ be a from to M[ this distance with

pz p I and parametrized by arc length.

Denote by

d

Lorentzian function to M2, i.e. for P E /+ (Mz)

d(p) = d(q, p). 1)

Since ¢ is maximal, A

=

{(pet) : 0

:s

^{t < do}} contains no points of , cf. [23, he()relm 34, p. hence there exists an open neighbourhood

n

(A) such that d is class C² in

n,

^[14, 1.9.15] and

n

is part of the tubular neighbourhood of

and by an associated normal coordinate (xQ')

= d

ⁱⁿ {xc > OJ, see [14, Theorem 1.9.22].

In this coordinate system M2 is the level set

{d

= O} and the level sets M(t) = {p E : d(p)

=

t}

are C²-hypersurfaces.

Next we want to a formula the evolution of the F -curvature surfaces this coordinate system. Let us the flow the level

x

^{= v, (4.23)}

x(O)

=

^Xo,

where Xo is embedding of M2. we from [14, Proposition I

x,

x

^{E C1«-EQ, do) x Bp(~o», (4.25)}

whereEo>O,xo(~o)=¢ and (-EO, Xxo C (Xi,Xj) as

3.15 with

f =

1) are continuously differentiable with

to 18) we obtain equation

(4.26) we note in view (4.25) one can verify that RrX/lyli

=

RO/JO/j is continuous this coordinate system, since t~fJ =

For the F -curvature M (t) we obtain then equation

F=

^(4.27)

where the geometric quantities like

gi)' iii)

and so on denote the geometric quantities of the level hypersurfaces and they are not to be confused with the quantities of the ambient space. This implies that the F -curvature of M (t) is monotonically increasing with respect to t in view of the strict monotonicity of the F -curvature, hence the level hypersurfaces are admissible, since F vanishes only on

a r.

Next, consider a tubular neighbourhood U of M[ with corresponding normal Gaussian coordinate system (xO'). The level sets

M(s)={xO=s}, -8<s<0, (4.28)

lie in the past of M [ =

M

(0) and are all of class C² for small 8. _

Since the geodesic ¢ is normal to M J, it is also normal to M (s) and the length of the geodesic segment of ¢ from

M

^(s) to M J is exactly -s, thus equal to the distance from

tV!

^(s)

to M I, hence we deduce

d(M2, M(s» = do

+

^{s. (4.29)}

We infer that {¢(t) : 0 :::: t :::: do

+

s) also represents a maximal geodesic from M2 to M (s) and we conclude further that, for fixed s, the hypersurface MCs)

n n

is contained in the past of M(do

+

s) and touches M(do

+

^{s) in Ps}

=

^¢(do

+

^s).

Hence by the maximum principle there holds

(4.30) Furthermore, if

FIIrt>(O)

=

^{min F2,}

M2

(4.31) then by using the additional assumption we conclude that if we choose 8 > 0 small enough, then in view of (4.27) there exists E > 0 not depending on s, -8 < s < 0, such that there holds

(4.32) On the other hand the F -curvature of

M

(s) converges to the F -curvature of M I if s tends to zero, hence we conclude

FI (PI) 2: min F2

+

^E,

M2

(4.33) where E > 0 if (4.31) is satisfied (otherwise it can be equal to zero), yielding in either case

a contradiction to (4.16). 0

The barrier condition and the preceding Lemmata imply the following

Proposition 4.5 Ifu; = graph Mi, i = 1. 2, where MI and M2 denote the lower and upper barrier respectively, then there holds

(4.34) In ^{the case F = H.}

r

⁼ IRn, this proposition follows by the proof of Lemma 4.4, since now all appearing hypersurfaces are admissible.

Lemma 4.4 also yields the uniqueness of constant F -curvature hypersurfaces.

Corollary 4.6 Let N be as in Lemma 4.4 and F be a curvature function of class (K) defined on

r+

or F

=

^Ok, 1 :::: k :::: n, defined on

n.

Then a compact, connected, space like hyper- surface of class C² with F

==

c for some constant c > 0, is uniquely determined.

5 Gradient estimates

Let cp be a runclion in , meaning estimates are only valid for this CP, which

cp' > 0 cP"

::s

O. (5.1)

F be a curvature function which is monotone, concave and of degree I.

We a solution of (1.1), where term f f(t) and the curvature F are supposed to bounded and bounds on

f

F from above below by ^C2 and ^CI respectively, the situation of Remark 4.3. Furthermore we suppose

mates have already established, flow stays a compact Then we following in [10, Sect. 5],

Proposition During the evolution of the flow (1.1) the term

v

is uniformly bounded:

(5.2)

First we some Lemmata:

Lemma composite function

(5.3)

where /-L, A are constants, satisfies equation

<P cp' FiJ

<Pi)

iI {cpl ^F cp

+ f}

^/-LAeAU ¢

+

^{cp' Fij {} Uj

+

^{2rQiuj -0} } /-L AeAU¢

- [1

^{] cp' Fij UjU j} /-LA 2eAU¢. (5.4) The follows from (3.21).

Next, we two Lemmata from [12], where we recall the definition of the Riemannian metric

III ·111

2.7.

Lemma 5.3 Let M

c

Q be a L!lIIUn<l~ hypersurface class C² in N, M

=

^graph . Then there is a constant c such that for any positive function

o

< ^E

=

^{E (x) on So} we have the following the quantities of

1Fi)

The of Lemma

which can be found

Illvlll ::s c

ii, gil

::s

^c

< Fkl gklg^ij•

lFij hjxixt ^{f k} c ..

1<-

^h.hk'V

+

^_F1J

- 2 ^{I J} 2E gij, c FiJ g" _{'} ,}

(5.5) (5.6) (5.7)

(5.9) (5.10)

II ^III ::s

cil VpEM,vtETpM:gij =1. (5.11) can be found [12, Lemma 4.5], the estimate 11 ),

Lemma

Lemma N.M

Let M c

ti

be a compact, connected, spaceUke hypersurface of in ulSo ^E

=

E(x) a function in So, 0 ^E < ~. Furthemzore we the function

tP

where 0 < /l and A < O. Then there exists c ceQ) that

I :::

^{c Fij IAI/leAu}¢

+

^{(1 - 2E)Fij}

h7

^hkjV¢

+

1 Fij u.u . Ji2A ^2e2AlI 1 2E ^! Jf""

The proof Lemma 5.4 can be found [12, Lemma Now we can prove Proposition 5.1.

We consider function

w

vtP,

where

tP

is as in and we will choose

1

(5.13)

14)

(5.15) A with

II..I

large enough. Showing that w is bounded is tantamount to show that

v

^is

bounded, we have already established CD-estimates u in Lemma 4.5.

Let us furthermore assume that u < 1, otherwise we could u in the definition

of if; by (u c), C > 1

+ lu ,.

the evolution equations

4.2, and 5.4 the parabolic w w" _{IJ -}<

where the function 0 < ^E

+

-cp' Ui /lA²e^Auv¢

c [1r]atP/l' 1

+

/lAe^AU

v

^{2] ¢,}

E(X) < ~ will specified at the end constant c depends on Q, Cl and C2.

We use the maximum principle to that w is bounded, let 0 < T <

x(to, such

sup sup ^{W 1J)} ~o).

(O.TlM(t)

We choose a coordinate point

and there holds

IC] ::: K2 < ... ::: len.

Now assume v(xo) 2 and let i

=

^io an index such that

12

2:.

IIiDule.

n

proof

and Xo =

(5.17)

18)

(5.20)

HT ) il

vve set

=

^iJ~iO assume without loss of that

°

^<

Dw(xo)

=

0, taking product with (ei) yields

= JIAe^Au

h7

^{ukei -}

where second equation follows from

v =

^{l1a va and}

eifJ(-I,O, ... ,0)).

. At Xo holds

)

equation (we recall terms and taking (5.11) as well as into account, we get large IAI

- -v'

ve .

it follows Kio is and of same order as V, which already finishes the if we have a lower bound for the principal curvatures.

Next, considering coordinate system above and fact that is negative, we conclude

Since F is concave, we at xo

hence there holds

io

F< 1 _

i=l

We conclude

iO

i=1

> ... >

io

< - " F - L... ^I

i=1

Inserting this estimate in 16) yields at xo with the

° ^::s

^{gijJl2 A2e).·uij3¢}

⁺

cFij gijJlIAle^AuiJ3¢

+

1-2 r:p' Ui UjJl2 A²eAU

v

¢ - r:p'FijuiUj/J,.A2eAUij¢

+

cJLIAle^AUiJ2¢.

(5.23)

(5.24)

(5.26)

se(;on:o row is choice of Jl. first term is the dominant one, if

we

I

A

I

enough, right hand is which the

maximum of w cannot occur at a point where

v

2.

o

6 Curvature estimates

In section we prove the boundedness of the principal curvatures during flow, which with the estimates in the next section will imply time of the flow

by well-known arguments. u follow Co_, C^l and

boundedness of the principal curvatures view of Together with the lower bound

on curvature and ^Fjil

r o

we deduce principal curvatures stay in a compact subset of

r

during the flow.

We consider a solution Aow (1.1), where the term

f f

^(t) the curvature function are supposed to be bounded and we bounds on

f

and F from

below by ^{C2 C!} we assume VAL''''.'''''"''"' of are an expedient

the proofs as for a constant the curvature from the

e ²-estimates, we will see that can be term

f

c, they rely only on bounds for

as on the eO-and e I-estimates. The and Proposition 5.1.

we in the same way global term estimates we provide the curvature estimates F H, cf. [14, Lemma 4.4.1].

Proposition 6.1 principal curvatures of the flow (1.1) with curvature function F = H and function Q'J (x) x for k

=

0 or Q'J E em,a CR+, an arbitrary, but fixed function satisfying Q'JI > 0 and Q'J" < 0 for k

=

^1, are bounded the flow,

IIAllo

~ c, (6.1)

where constant c depends on

,Iii

c] and C2.

Proof ~ defined by

~= (6.2)

Let 0 < T < and xo = xo(to) with 0 ^{< to ~} T a point in M (to) that ( < sup {sup l; : 0 ^{< t}

~

^T

I ⁼

^l;

Mo M I '

(6.3)

which allows one to substitute ~ by h~ and use evolution equation for quantity to (.

We Riemannian normal coordinates ) at Xo E M (to) such that at this point we

gij

=

^and

where we assume the principal curvatures are

i] ) be the vector

/] = (0, .. , 0, 1), set

as in (5.19).

by

(6.4)

(6.5)

We note that { is I defined in a neigbourhood of (to, xo) -

{assumes maximum at xo) as Moreover at ^(to, we have

(6.7)

and the also coincide. at ^(to, the function

E

satisfies the same differential equation as . For the sake of greater clarity, we will treat like a scalar and pretend that { is defined by

In view of maximum principle and 4.2 we that holds at (to, xo) (6.9)

where

IIAI12 =

^hij

= 2::7=1

the constant c depends on Q, the bound on ii, C1 C2 This that ~ is and we already a lower bound on H we are

done. o

Before we prove the next estimates, us state following:

Remark 6.2 Let X XQ be the strictly convex function, where we assume ^Q is the region determined by CO-estimates. Then X the evolution equation

(6.10) Hence

x,

such

exist constants c and co > 0, depending on Q, c" C2 and the strict convexity of

Xij .::: C XCI. - Co (6.11 )

we treat case F = Cf2. The proof is as in [10].

Proposition 6.3 The curvatures of the flow (1.1) with F Cf2, <P(x) x or

<P(x) -x-I, are uniformly bounded during the flow, provided there exists a co1'1- vexfunction X E (Q). Thus we have

IIAlla':::c, (6.

where c f1Pr;IPnll.'i: on

,I ca,

^{C! and C2·}

Proof l; and w be respectively defined by

l; { hij

1/

^{I]j :}

III] II =

1 } ,

w

=

logl;

+

AX, (6.1

where A > 0 is a large constant. We will show that w is bounded, if we """J"VO""' A sufficiently

O<T<

xo = xo(to) with 0 < to < T a point in M (to) such

sup w sup {sup w : 0

t.:::

T} w(xo). (6.1

Mo M/

By the same procedure as in the last and we may define w by

we introduce normal coordinates at Xa x ~o),

we assume and A to

1)) = log

+

o.,.p'''t~'r than I, we deduce

o

-c Fij hikh~

+

c h~

+ C

^AC-

.. . 2

+

^{<p' F'J(log h~)i (log h~) j}

+

<pl _ _ _

K:n K:l i=l n

following gij

(6.

at (to, ~o)

(6.16)

where we have estimated bounded terms by a constant c, which depends on

lulo, Ivlo, 1<Plo, I 10,lflo

a bound 2: C >

O.

The term is to the term with the second derivatives F in evolution equation of use formula (2.1.72) in [14] and note that two pmts are both a concave curvature function, see

r

14, Proposition 2.1

We distinguish two cases.

1 Suppose that

IKiI

E 1 Kn , where we choose some fixed E I so that 0 E I <

!.

we have VIew the concavity of F, see

ij k 2 1

F hikh i > FIKI 2:

n

Since 0,

Dlog -'ADX·

(log h~) ^{i Xi <} Fⁱⁱg .. _IJ' For Kn the term in is dominating, so we can conclude Kn is a

this case.

2 that

by using (where we omit

Codazzi equations, we can the last term in (6.16)

by

factor for a moment):

2 n

- - L ( Fn 1=1

n

(Fi - ) (h~)-2.

second sum can estimated by a constant, since FI ::; CKn.

The terms (6.1 the of h~ can be

omitting the common factor <PI) 1 2£1 n

(hn/f

^{-2 2}

n

(hnn:f (h~r2

1

+

2EI

+

Fn

1=1 1

+

2EI

I

n . 2

<

L (h

ⁿⁿ

!)

-2

^{Fnll 112.}

i=1

Hence we inequality

0::; 'A2cFn

+

^{CKn +} gij

+'Ac - 'Aco

(6.17)

(6.18)

(6.19)

(6.20)

bounded

(6.21)

(6.22)

above

(6.23)

(6.24)