Scalar Curvature 2

Home , Scalar curvature

arXiv:math/0207054v3 [math.DG] 10 Apr 2003 (0.1) M endi noe cone open an in defined once pnsbe of subset open connected aigacmatCuh hypersurface Cauchy compact a having vature manifolds. YESRAE FPECIE CLRCURVATURE SCALAR PRESCRIBED OF HYPERSURFACES osdrtepolmo nigacoe yesraeo pres of hypersurface closed a finding of problem the Consider 0 xsec fasolution a of References Existence 10. 9. approximations 8. solution Stationary stationary a to 7. Convergence solutions auxiliary 6. the for estimates order 5. Lower problem curvature auxiliary 4. An results preliminary and 3. Notations functions 2. Curvature 1. Introduction 0. 2000 Date e od n phrases. and words Key ⊂ Ω C C C etme 0 2001. 30, September : F ahmtc ujc Classification. Subject Mathematics uhthat such Abstract. uvtr ngoal yeblcLrnza aiod sp barriers. is are manifolds there Lorentzian hyperbolic globally in curvature 2 1 2 siae o h ttoayapproximations stationary the for approximations estimates stationary - the for estimates - solutions auxiliary the for estimates - nagoal yeblc(+)dmninlLrnza man Lorentzian (n+1)-dimensional hyperbolic globally a in eiae oRbr ino h caino i ihit bir eightieth his of occasion the on Finn Robert to Dedicated h xsec fcoe yesrae fpecie scalar prescribed of hypersurfaces closed of existence The NLRNZA MANIFOLDS LORENTZIAN IN Γ rsrbdsaa uvtr,goal yeblcLorentz hyperbolic globally curvature, scalar Prescribed F ,f N, ⊂ | M LU GERHARDT CLAUS 0. R = n ∈ Introduction Contents hnw okfrasaelk hypersurface space-like a for look we then , f C ( x 2 ) ,α 56,5C1 34,5C0 58J05. 53C50, 53C44, 53C21, 35J60, ( Ω ¯ S ) 0 ∀ F , ob oepeie let precise, more be To . x ∈ moh ymti function symmetric smooth, a M, oe provided roved thday rbdcur- cribed ifold Ω ea be ian 14 49 48 36 31 28 26 25 17 N 8 4 1 PRESCRIBED SCALAR CURVATURE 2

where F|M means that F is evaluated at the vector (κi(x)) the components of which are the principal curvatures of M. The prescribed function f should satisfy natural structural conditions, e. g. if Γ is the positive cone and the hypersurface M is supposed to be convex, then f should be positive, but no further, merely technical, conditions should be imposed. In [1, 2, 8, 14] the case F = H, the mean curvature, has been treated, and in [15] we solved the problem for curvature functions F of class K∗ that includes the Gaussian curvature, see [15, Section 1] for the deﬁnition, but excludes the symmetric polynomials Hk for 1

(0.2) F|M = f(x, ν) ∀x ∈ M, where ν = ν(x) is the past-directed normal of M in the point x.

To give a precise statement of the existence result we need a few deﬁnitions and assumptions. First, we assume that Ω is a precompact, connected, open subset of N, that is bounded by two achronal, connected, space-like hyper- 4,α surfaces M1 and M2 of class C , where M1 is supposed to lie in the past of M2.

Let F = H2 be the scalar curvature operator deﬁned on the open cone n 2,α Γ2 ⊂ R , and f = f(x, ν) be of class C in its arguments such that

(0.3) 0

2 (0.4) |||fβ(x, ν)||| ≤ c2(1 + |||ν||| ), and

(0.5) |||fνβ (x, ν)||| ≤ c3(1 + |||ν|||), for all x ∈ Ω¯ and all past directed time-like vectors ν ∈ Tx(Ω), where ||| · ||| is a Riemannian reference metric that will be detailed in Section 2.

We suppose that the boundary components Mi act as barriers for (F,f).

Deﬁnition 0.1. M2 is an upper barrier for (F,f), if M2 is admissible, i.e. its principal curvatures (κi) with respect to the past directed normal belong to Γ2, and if

(0.6) F |M2 ≥ f(x, ν) ∀ x ∈ M2. PRESCRIBED SCALAR CURVATURE 3

M1 is a lower barrier for (F,f), if at the points Σ ⊂ M1, where M1 is admissible, there holds

(0.7) F |Σ ≤ f(x, ν) ∀ x ∈ Σ. Σ may be empty.

Remark 0.2. This deﬁnition of upper and lower barriers for a pair (F,f) also makes sense for other curvature functions F deﬁned in an open convex cone Γ , with a corresponding meaning of the notion admissable.

Now, we can state the main theorem.

Theorem 0.3. Let M1 be a lower and M2 an upper barrier for (F,f), where F = H2. Then, the problem

(0.8) F|M = f(x, ν) has an admissible solution M ⊂ Ω¯ of class C4,α that can be written as a graph 2 over S0 provided there exists a strictly convex function χ ∈ C (Ω¯).

Remark 0.4. As we have shown in [15, Lemma 2.7] the existence of a strictly convex function χ is guaranteed by the assumption that the level hypersurfaces {x0 = const} are strictly convex in Ω¯, where (xα) is a Gaussian coordinate system associated with S0. Looking at Robertson-Walker space-times it seems that the assumption of the existence of a strictly convex function in the neighbourhood of a given compact set is not too restrictive: in Minkowski space e.g. χ = −|x0|2 + |x|2 is a globally deﬁned strictly convex function. The only obstruction we are aware of is the existence of a compact maximal slice. In the neighbourhood of such a slice a strictly convex function cannot exist.

The existence result of our main theorem would also be valid in Riemannian manifolds if one could prove C1- estimates. For the C2- estimates the nature of the ambient space is irrelevant though the proofs are slightly diﬀerent. For prescribed curvature problems it seems more natural to assume that the right-hand side f depends on (x, ν), and we shall prove in a subsequent paper existence results for curvature functions F ∈ (K∗), where the ambient space can be Riemannian or Lorentzian, cf. [16].

The paper is organized as follows: In Section 1 we take a closer look at curvature functions and deﬁne the concept of elliptic regularization for these functions, and analyze some of its properties. PRESCRIBED SCALAR CURVATURE 4

In Section 2 we introduce the notations and common definitions we rely on, and state the equations of Gauß, Codazzi, and Weingarten for space-like hypersurfaces. In Section 3 we look at the curvature flow associated with our problem, and the corresponding evolution equations for the basic geometrical quantities of the flow hypersurfaces. In Section 4 we prove lower order estimates for the evolution problem, while a priori estimates in the C2-norm are derived in Section 5. In Section 6, we demonstrate that the evolutionary solution converges to a stationary approximation of our problem, i.e. to a solution for a curvature problem, where F is replaced by its elliptic regularization Fǫ. The uniform C1- estimates for the stationary approximations are derived in Sections 7 and 8, the C2- estimates are given in Section 9, while the final existence result is contained in Section 10.

1. Curvature functions

n Let Γ ⊂ R be an open cone containing the positive cone Γ+, and F ∈ C2,α(Γ ) ∩ C0(Γ¯) a positive symmetric function satisfying the condition ∂F (1.1) F = > 0 ; i ∂κi then, F can also be viewed as a function deﬁned on the space of symmetric matrices C, the eigenvalues of which belong to Γ , namely, let (hij ) ∈ C with eigenvalues κi, 1 ≤ i ≤ n, then deﬁne F on C by

(1.2) F (hij )= F (κi). If we deﬁne ∂F (1.3) F ij = ∂hij and ∂ 2F (1.4) F ij,kl = ∂hij ∂hkl then,

ij ∂F i 2 n (1.5) F ξiξj = |ξ | ∀ ξ ∈ R , ∂κi in an appropriate coordinate system,

ij (1.6) F is diagonal if hij is diagonal, PRESCRIBED SCALAR CURVATURE 5 and 2 ij,kl ∂ F Fi − Fj 2 (1.7) F ηij ηkl = ηiiηjj + (ηij ) , ∂κi ∂κj κi − κj Xi6=j for any (ηij ) ∈ S, where S is the space of all symmetric matrices. The second term on the right-hand side of (1.7) is non-positive if F is concave, and non- negative if F is convex, and has to be interpreted as a limit if κi = κj. The preceding considerations are also applicable if the κi are the principal curvatures of a space-like hypersurface M with metric (gij ). F can then be looked at as being deﬁned on the space of all symmetric tensors (hij ) the eigenvalues of which belong to Γ . Such tensors will be called admissible; when the second fundamental form of M is admissible, then, we also call M admissible.

For an admissible tensor (hij )

∂F (1.8) F ij = ∂hij is a contravariant tensor of second order. Sometimes it will be convenient to circumvent the dependence on the metric by considering F to depend on the mixed tensor

i ik (1.9) hj = g hkj . Then,

j ∂F (1.10) Fi = i ∂hj is also a mixed tensor with contravariant index j and covariant index i. Such functions F are called curvature functions. Important examples are the symmetric polynomials of order k, Hk, 1 ≤ k ≤ n,

(1.11) Hk(κi)= κi1 · · · κik . i1<··· 0} that contains Γ+.

Since we have in mind that the κi are the principal curvatures of a hypersurface, we use the standard symbols H and |A| for

(1.12) H = κi, i X PRESCRIBED SCALAR CURVATURE 6 and

2 2 (1.13) |A| = κi . i X The scalar curvature function F = H2 can then be expressed as

1 (1.14) F = (H2 − |A|2), 2 and we deduce that for (κi) ∈ Γ2

(1.15) |A|2 ≤ H2,

(1.16) Fi = H − κi, and hence,

(1.17) HFi ≥ F, for (1.17) is equivalent to

1 1 (1.18) Hκ ≤ H2 + |A|2, i 2 2 which is obviously valid.

In important ingredient in our existence proof will be the method of elliptic regularization

n Lemma 1.1. For each ǫ> 0, consider the linear isomorphism ϕǫ in R given by

(1.19) (˜κi)= ϕǫ(κi) = (κi + ǫH). Let F ∈ C2(Γ ) ∩ C0(Γ¯) be a curvature function such that

(1.20) F |∂Γ =0.

−1 2 0 ¯ Then, Γǫ = ϕǫ (Γ ) is an open cone and Fǫ = F ◦ ϕǫ ∈ C (Γǫ) ∩ C (Γǫ) a curvature function satisfying

F . (1.21) ǫ|∂Γǫ =0 Assume furthermore, that

(1.22) H > 0 in Γ. PRESCRIBED SCALAR CURVATURE 7

Then,

(1.23) Γ ⊂ Γǫ, and

(1.24) H > 0 in Γǫ. Proof. We only prove the assertions (1.23) and (1.24) since the other assertions are obvious. Let (κi) ∈ Γ be ﬁxed. Then,

(1.25) 0 < F (κi) ≤ F (κi + ǫH), because F is monotone, and we deduce

(1.26) (κi + ǫH) ∈ Γ ∀ ǫ> 0, in view of (1.22) and the monotonicity of F , cf. (1.1). To prove (1.24), we observe that

(1.27) κ˜i =(1+ ǫn) κi. i i X X q.e.d.

Remark 1.2. (i) Let F be as in Lemma 1.1 and assume moreover, that F is homogeneous of degree 1, and concave, then,

1 (1.28) F (κ ) ≤ F (1,..., 1)H ∀ (κ ) ∈ Γ, i n i and we conclude that condition (1.22) is satisﬁed. (ii) Let F be as in Lemma 1.1, but suppose that F is homogeneous of degree 1 d d0 > 0 and F 0 concave, then, the relation (1.22) is also valid.

Proof. The inequality (1.28) follows easily from the concavity and homogeneity

F (κi) ≤ F (1,..., 1)+ Fi(1,..., 1)(κi − 1) i (1.29) X 1 = F (1,..., 1)H, n

1 since Fi(1,..., 1) = n F (1,..., 1), while the other assertions are obvious. q.e.d. PRESCRIBED SCALAR CURVATURE 8

For better reference, we use a tensor setting in the next lemma, i.e. the (κi) ∈ Γ are the eigenvalues of an admissible tensor (hij ) with respect to a Riemannian metric (gij ). In this setting the elliptic regularization of F is given by

(1.30) F˜(hij ) ≡ F (hij + ǫHgij ).

Lemma 1.3. Let F˜ be the elliptic regularization of a curvature function F of class C2, then,

ij ij rs ij (1.31) F˜ = F + ǫF grsg , and

ij,kl ij,kl ij,ab kl F˜ = F + ǫF gabg (1.32) rs,kl ij 2 rs,ab ij kl + ǫF grsg + ǫ F grsgabg g .

If F is concave, then, F˜ is also concave.

Proof. The relations (1.31) and (1.32) are straight-forward calculations.

To prove the concavity of F˜, let (ηij ) be a symmetric tensor, then,

ij,kl ij,kl ij,rs kl F˜ ηij ηkl = F ηij ηkl +2ǫF ηij grsg ηkl (1.33) 2 rs,ab ij 2 + ǫ F grsgab(g ηij ) ≤ 0. q.e.d.

2. Notations and preliminary results

The main objective of this section is to state the equations of Gauß, Co- dazzi, and Weingarten for space-like hypersurfaces M in a (n+1)-dimensional Lorentzian space N. Geometric quantities in N will be denoted by (¯gαβ), (R¯αβγδ), 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 am- biguity they will be preceded by a semicolon, i.e. for a function u in N, (uα) PRESCRIBED SCALAR CURVATURE 9 will be the gradient and (uαβ) the Hessian, but e.g., the covariant derivative of the curvature tensor will be abbreviated by R¯αβγδ;ǫ. We also point out that

¯ ¯ ǫ (2.1) Rαβγδ;i = Rαβγδ;ǫxi with obvious generalizations to other quantities.

Let M be a space-like hypersurface, i.e. the induced metric is Riemannian, with a diﬀerentiable normal ν that is time-like. In local coordinates, (xα) and (ξi), the geometric quantities of the space-like hypersurface M are connected through the following equations

α α (2.2) xij = hij ν the so-called Gauß formula. Here, and also in the sequel, a covariant derivative is always a full tensor, i.e.

α α k α ¯α β γ (2.3) xij = x,ij − Γij xk + Γβγxi xj . The comma indicates ordinary partial derivatives. In this implicit deﬁnition the second fundamental form (hij ) is taken with respect to ν. The second equation is the Weingarten equation

α k α (2.4) νi = hi xk ,

α where we remember that νi is a full tensor. Finally, we have the Codazzi equation

¯ α β γ δ (2.5) hij;k − hik;j = Rαβγδν xi xj xk and the Gauß equation

¯ α β γ δ (2.6) Rijkl = −{hikhjl − hilhjk} + Rαβγδxi xj xk xl . Now, let us assume that N is a globally hyperbolic Lorentzian manifold with a compact Cauchy surface. N is then a topological product R×S0, where S0 is a compact Riemannian manifold, and there exists a Gaussian coordinate system α 0 i (x ), such that x represents the time, the (x )1≤i≤n are local coordinates for 0 S0, where we may assume that S0 is equal to the level hypersurface {x =0}— we don’t distinguish between S0 and {0} × S0—, and such that the Lorentzian metric takes the form

2 2ψ 02 0 i j (2.7) ds¯N = e {−dx + σij (x , x)dx dx }, PRESCRIBED SCALAR CURVATURE 10 where σij is a Riemannian metric, ψ a function on N, and x an abbreviation for the space-like components (xi), see [17], [19, p. 212], [18, p. 252], and [8, Section 6]. We also assume that the coordinate system is future oriented, i.e. the time coordinate x0 increases on future directed curves. Hence, the contravariant time-like vector(ξα) = (1, 0,..., 0) is future directed as is its 2ψ covariant version (ξα)= e (−1, 0,..., 0).

Let M = graph u|S0 be a space-like hypersurface

0 0 (2.8) M = { (x , x): x = u(x), x ∈ S0 }, then the induced metric has the form

2ψ (2.9) gij = e {−uiuj + σij }

ij −1 where σij is evaluated at (u, x), and its inverse (g ) = (gij ) can be expressed as

ui uj (2.10) gij = e−2ψ{σij + }, v v

ij −1 where (σ ) = (σij ) and

i ij u = σ uj (2.11) 2 ij 2 v =1 − σ uiuj ≡ 1 − |Du| .

Hence, graph u is space-like if and only if |Du| < 1. We also note that

−2 2ψ ij 2ψ 2 (2.12) v =1+ e g uiuj ≡ 1+ e kDuk .

The covariant form of a normal vector of a graph looks like

−1 ψ (2.13) (να)= ±v e (1, −ui). and the contravariant version is

(2.14) (να)= ∓v−1e−ψ(1,ui). Thus, we have PRESCRIBED SCALAR CURVATURE 11

Remark 2.1. Let M be space-like graph in a future oriented coordinate system. Then, the contravariant future directed normal vector has the form (2.15) (να)= v−1e−ψ(1,ui) and the past directed (2.16) (να)= −v−1e−ψ(1,ui). In the Gauß formula (2.2) we are free to choose the future or past directed normal, but we stipulate that we always use the past directed normal for reasons that we have explained in [15]. Look at the component α = 0 in (2.2) and obtain in view of (2.16)

−ψ −1 ¯0 ¯0 ¯0 ¯0 (2.17) e v hij = −uij − Γ00 uiuj − Γ0j ui − Γ0i uj − Γij .

Here, the covariant derivatives are taken with respect to the induced metric of M, and

¯0 −ψ¯ (2.18) −Γij = e hij ,

0 where (h¯ij ) is the second fundamental form of the hypersurfaces {x = const}. An easy calculation shows

¯ −ψ 1 ˙ (2.19) hij e = − 2 σ˙ ij − ψσij , where the dot indicates diﬀerentiation with respect to x0. Next, let us analyze under which condition a space-like hypersurface M can be written as a graph over the Cauchy hypersurface S0. We ﬁrst need

Deﬁnition 2.2. Let M be a closed, space-like hypersurface in N. Then, M is said to be achronal, if no two points in M can be connected by a future directed time-like curve.

In [5] it is proved, see also [15, Proposition 2.5],

Proposition 2.3. Let N be connected and globally hyperbolic, S0 ⊂ N a compact Cauchy hypersurface, and M ⊂ N a compact, connected space-like m m hypersurface of class C ,m ≥ 1. Then, M = graph u|S0 with u ∈ C (S0) iﬀ M is achronal. PRESCRIBED SCALAR CURVATURE 12

Remark 2.4. The Mi are barriers for the pair (F,f). Let us point out that without loss of generality we may assume

(2.20) F|M2 >f(x, ν) ∀ x ∈ M2, and

(2.21) F|Σ

(2.22) η|M1 > 0 and η|M2 < 0 and deﬁne for δ > 0

(2.23) fδ = f + δη.

Then, if we assume f to be strictly positive with a positive lower bound, we have for small δ

1 (2.24) f ≥ f, δ 2 and the Mi are barriers for (F,fδ) satisfying the strict inequalities; since we shall derive C4,α estimates independent of δ, we shall have proved the existence of a solution for f if we can prove it for fδ.

Lemma 2.5. Let Mi be barriers for (F,f) satisfying the strict inequalities (2.20) and (2.21), where F is supposed to be monotone and concave. Then, they are also barriers for the elliptic regularizations Fǫ for small ǫ.

Proof. In view of Lemma 1.1, we know that Γ ⊂ Γǫ and H is positive in Γǫ. Hence, M2 is certainly an upper barrier for (Fǫ,f) because of the monotonicity of F . Let Σǫ resp. Σ be the points in M1 where the principal curvatures belong to Γǫ resp. Γ and assume that Σ 6= ∅. Suppose M1 were not a lower barrier for (Fǫ,f) for small ǫ, then, there exist a sequence ǫ → 0 and a corresponding convergent sequence xǫ ∈ Σǫ, xǫ → x0 ∈ Σ, such that

(2.25) Fǫ ≥ f(xǫ, ν), and hence,

(2.26) F ≥ f(x0, ν) PRESCRIBED SCALAR CURVATURE 13 contradicting (2.21). q.e.d.

Remark 2.6. The condition (0.3) is reasonable as is evident from the Einstein equation

¯ 1 ¯ (2.27) Rαβ − 2 Rg¯αβ = Tαβ, where the energy-momentum tensor Tαβ is supposed to be positive semi- deﬁnite for time-like vectors (weak energy condition, cf. [19, p. 89]), and the relation

2 ij α β (2.28) R = −[H − hij h ]+ R¯ +2R¯αβν ν for the scalar curvature of a space-like hypersurface; but it would be convenient for the approximations we have in mind, if the estimate in (0.3) would be valid for all time-like vectors. In fact, we may assume this without loss of generality: Let ϑ be a smooth real function such that

c (2.29) 1 ≤ ϑ and ϑ(t)= t ∀ t ≥ c , 2 1 then, we can replace f by ϑ ◦ f and the new function satisﬁes our requirements for all time-like vectors. We therefore assume in the following that the relation (0.3) holds for all time-like vectors ν ∈ Tx(N) and all x ∈ Ω¯.

Sometimes, we need a Riemannian reference metric, e.g. if we want to estimate tensors. Since the Lorentzian metric can be expressed as

α β 2ψ 02 i j (2.30)g ¯αβdx dx = e {−dx + σij dx dx }, we deﬁne a Riemannian reference metric (˜gαβ) by

α β 2ψ 02 i j (2.31)g ˜αβdx dx = e {dx + σij dx dx } and we abbreviate the corresponding norm of a vectorﬁeld η by

α β 1/2 (2.32) |||η||| = (˜gαβη η ) , with similar notations for higher order tensors. PRESCRIBED SCALAR CURVATURE 14

For a space-like hypersurface M = graph u the induced metrics with respect to (¯gαβ) resp. (˜gαβ) are related as follows

α β 2ψ g˜ij =g ˜αβxi xj = e [uiuj + σij ] (2.33) 2ψ = gij +2e uiuj .

i Thus, if (ξ ) ∈ Tp(M) is a unit vector for (gij ), then

i j 2ψ i 2 (2.34)g ˜ij ξ ξ =1+2e |uiξ | , and we conclude for future reference Lemma 2.7. Let M = graph u be a space-like hypersurface in N, p ∈ M, and ξ ∈ Tp(M) a unit vector, then

β i i (2.35) |||xi ξ ||| ≤ c(1 + |uiξ |) ≤ cv,˜ where v˜ = v−1.

3. An auxiliary curvature problem

Solving the problem (0.2) involves two steps: ﬁrst, proving a priori estimates, and secondly, applying a method to show the existence of a solution. In a general Lorentzian manifold the evolution method is the method of choice, but unfortunately, one cannot prove the necessary a priori estimates during the evolution when F is the scalar curvature operator. Both the C1 and C2- estimates fail for general f = f(x, ν).

Therefore, we use the elliptic regularization and consider the existence problem for the operators

(3.1) Fǫ(κi)= F (κi + ǫH), ǫ> 0, i.e. we solve

(3.2) Fǫ|M = f(x, ν).

Then, we prove uniform C2,α- estimates for the approximating solutions Mǫ, and ﬁnally, let ǫ tend to zero.

The Fǫ—or some positive power of it—belong to a class of curvature functions F that satisfy the following condition (H): F ∈ C2,α(Γ ) ∩ C0(Γ¯), n where Γ ⊂ R is an open cone containing Γ+, F is symmetric, monotone, PRESCRIBED SCALAR CURVATURE 15 i.e. Fi > 0, homogeneous of degree 1, concave, vanishes on ∂Γ , and there exists ǫ0 = ǫ0(F ) > 0 such that

(3.3) Fi ≥ ǫ0 Fk ∀ 1 ≤ i ≤ n. Xk Furthermore, the set

(3.4) Λδ,κ = { (κi) ∈ Γ : 0 <δ ≤ F (κi), κi ≤ κ ∀ 1 ≤ i ≤ n } is compact.

Remark 3.1. If the original curvature function F ∈ C2,α(Γ ) ∩ C0(Γ¯) is concave, homogeneous of degree 1, and vanishes on ∂Γ , then, the Fǫ are of class (H) in the cone Γǫ, and satisfy (3.3) with ǫ0 = ǫ. The set

(3.5) Λ˜δ,κ = { (κi) ∈ Γǫ : 0 <δ ≤ Fǫ(κi), κi ≤ κ ∀ 1 ≤ i ≤ n } is compact for ﬁxed ǫ. If the parameters κ and δ are independent of ǫ, then the Λ˜δ,κ are contained in a compact subset of Γ uniformly in ǫ, for small ǫ, 0 ≤ ǫ ≤ ǫ1(δ, κ, F ). Proof. In view of the results in Lemma 1.3 we only have to prove the com- pactness of Λ˜δ,κ. We shall also only consider the case when the estimates hold uniformly in ǫ.

Due to the concavity and homogeneity of Fǫ we conclude from (1.28) that

1 (3.6) F (κ ) ≤ F (1,..., 1)(1 + nǫ)H. ǫ i n

For (κi) ∈ Λ˜δ,κ we, therefore, infer

1+ nǫ (3.7) δ ≤ F (κ ) ≤ F (1,..., 1)H ≤ (1 + nǫ)F (1,..., 1)κ, ǫ i n and thus,

(3.8) lim ǫH =0, ǫ→0 uniformly in Λ˜δ,κ.

Suppose Λ˜δ,κ would not stay in a compact subset of Γ for small ǫ, 0 <ǫ ≤ ǫ1(δ, κ, F ). Then, there would exist a sequence ǫ → 0 and a corresponding ǫ ˜ sequence (κi ) ∈ Λδ,κ converging to a point (κi) ∈ ∂Γ , which is impossible in view of (3.7), (3.8), and the continuity of F in Γ¯. q.e.d. PRESCRIBED SCALAR CURVATURE 16

To prove the existence of hypersurfaces of prescribed curvature F for F ∈ (H) we look at the evolution problem

x˙ = (F − f)ν, (3.9) x(0) = x0, where ν is the past-directed normal of the ﬂow hypersurfaces M(t), F the curvature evaluated at M(t), x = x(t) an embedding and x0 an embedding of an initial hypersurface M0, which we choose to be the upper barrier M2. Since F is an elliptic operator, short-time existence, and hence, existence in a maximal time interval [0,T ∗) is guaranteed. If we are able to prove uniform a priori estimates in C2,α, long-time existence and convergence to a stationary solution will follow immediately.

But before we prove the a priori estimates, we 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. The proofs are identical to those of the corresponding results in a Riemannian setting, cf. [9, Section 3] and [15, Section 4], and will be omitted.

Lemma 3.2 (Evolution of the metric). The metric gij of M(t) satisﬁes the evolution equation

(3.10)g ˙ij = 2(F − f)hij .

Lemma 3.3 (Evolution of the normal). The normal vector evolves according to ij (3.11)ν ˙ = ∇M (F − f)= g (F − f)ixj .

Lemma 3.4 (Evolution of the second fundamental form). The second fundamental form evolves according to

˙ j j k j ¯ α β γ δ kj (3.12) hi = (F − f)i − (F − f)hi hk − (F − f)Rαβγδν xi ν xkg and ˙ k ¯ α β γ δ (3.13) hij = (F − f)ij + (F − f)hi hkj − (F − f)Rαβγδν xi ν xj .

Lemma 3.5 (Evolution of (F − f)). The term (F − f) evolves according to the equation

′ ij ij k α (3.14) (F − f) − F (F − f)ij = −F hikhj (F − f) − fαν (F − f) α ij ij ¯ α β γ δ − fνα xi (F − f)j g − F Rαβγδν xi ν xj (F − f), PRESCRIBED SCALAR CURVATURE 17

From (3.9) we deduce with the help of the Ricci identities a parabolic equation for the second fundamental form j Lemma 3.6. The mixed tensor hi satisﬁes the parabolic equation

˙ j kl j hi −F hi;kl kl r j k j = −F hrkhl hi + fhi hk α β kj α j α β kj α β k lj − fαβxi xk g − fαν hi − fανβ (xi xk h + xl xk hi g ) α β k lj β k lj α k j − fνανβ xl x hi h − fνβ x hi;l g − fνα ν hi h (3.15) k k k kl,rs j kl ¯ α β γ δ m rj + F hkl;ihrs; +2F Rαβγδxmxi xk xrhl g kl ¯ α β γ δ m rj kl ¯ α β γ δ mj − F Rαβγδxmxk xr xl hi g − F Rαβγδxmxk xi xl h kl ¯ α β γ δ j ¯ α β γ δ mj − F Rαβγδν xk ν xl hi + fRαβγδν xi ν xmg kl ¯ α β γ δ ǫ mj α β γ δ ǫ mj + F Rαβγδ;ǫ{ν xk xl xi xmg + ν xi xk xmxl g }.

The proof is identical to that of the corresponding result in the Riemannian case, cf. [9, Lemma 7.1 and Lemma 7.2]; the only diﬀerence is that f now also depends on ν.

Remark 3.7. In view of the maximum principle, we immediately deduce from (3.14) that the term (F −f) has a sign during the evolution if it has one at the beginning, i.e., if the starting hypersurface M0 is the upper barrier M2, then (F − f) is non-negative

(3.16) F ≥ f.

4. Lower order estimates for the auxiliary solutions

Since the two boundary components M1,M2 of ∂Ω are space-like, achronal hypersurfaces, they can be written as graphs over the Cauchy hypersurface S0, Mi = graph ui, i =1, 2, and we have

(4.1) u1 ≤ u2, for M1 should lie in the past of M2, and the enclosed domain is supposed to be connected. Let us look at the evolution equation (3.9) with initial hypersurface M0 ∗ ∗ equal to M2 deﬁned on a maximal time interval I = [0,T ),T ≤ ∞. Since the initial hypersurface is a graph over S0, we can write

(4.2) M(t) = graph u(t)|S0 ∀ t ∈ I, PRESCRIBED SCALAR CURVATURE 18 where u is deﬁned in the cylinder QT ∗ = I × S0. We then deduce from (3.9), looking at the component α = 0, that u satisﬁes a parabolic equation of the form (4.3)u ˙ = −e−ψv−1(F − f), where we use the notations in Section 2, and where we emphasize that the time derivative is a total derivative, i.e.

∂u (4.4)u ˙ = + u x˙ i. ∂t i Since the past directed normal can be expressed as

(4.5) (να)= −e−ψv−1(1,ui), we conclude from (3.9), (4.3), and (4.4)

∂u (4.6) = −e−ψv(F − f). ∂t

∂u Thus, ∂t is non-positive in view of Remark 3.7.

Next, let us state our ﬁrst a priori estimate

Lemma 4.1. Suppose that the boundary components act as barriers for (F,f), then the ﬂow hypersurfaces stay in Ω¯ during the evolution.

The proof is identical to that of the corresponding result in [15, Lemma 4.1].

For the C1- estimate the termv ˜ = v−1 is of great importance. It satisﬁes the following evolution equation

Lemma 4.2 (Evolution ofv ˜). Consider the ﬂow (3.9) in the distinguished coordinate system associated with S0. Then, v˜ satisﬁes the evolution equation

˙ ij ij k α β v˜ − F v˜ij = − F hikhj v˜ − fηαβν ν ij k α β ij β γ α − 2F hj xi xk ηαβ − F ηαβγ xi xj ν (4.7) ij ¯ α β γ δ ǫ kl − F Rαβγδν xi xk xj ηǫxl g β α ik β ik α − fβxi xk ηαg − fνβ xk h xi ηα,

ψ where η is the covariant vector ﬁeld (ηα)= e (−1, 0,..., 0). PRESCRIBED SCALAR CURVATURE 19

The proof uses the relation

α (4.8)v ˜ = ηαν and is identical to that of [15, Lemma 4.4] having in mind that presently f also depends on ν.

Lemma 4.3. Let M(t) = graph u(t) be the ﬂow hypersurfaces, then, we have

ij −ψ ¯0 ij u˙ − F uij = e vf˜ + Γ00 F uiuj (4.9) ij ¯0 ij ¯0 +2F Γ0i uj + F Γij , where all covariant derivatives a taken with respect to the induced metric of the ﬂow hypersurfaces, and the time derivative u˙ is the total time derivative, i.e. it is given by (4.4).

Proof. We use the relation (4.3) together with (2.17). q.e.d.

As an immediate consequence we obtain

Lemma 4.4. The composite function

λu (4.10) ϕ = eµe where µ, λ are constants, satisﬁes the equation

ij −ψ λu ij ¯0 λu ϕ˙ − F ϕij =fe vµλe˜ ϕ + F uiuj Γ00 µλ e ϕ ij ¯0 λu ij ¯0 λu (4.11) +2F uiΓ0j µλe ϕ + F Γij µλe ϕ

λu ij 2 λu − [1 + µe ]F uiuj µλ e ϕ.

Before we can prove the C1- estimates we need two more lemmata.

Lemma 4.5. There is a constant c = c(Ω) such that for any positive function 0 <ǫ = ǫ(x) on S0 and any hypersurface M(t) of the ﬂow we have

(4.12) |||ν||| ≤ cv,˜

(4.13) gij ≤ cv˜2 σij ,

ij kl ij (4.14) F ≤ F gkl g , PRESCRIBED SCALAR CURVATURE 20

ǫ c (4.15) |F ij hkxαxβ η |≤ F ij hkh v˜ + F ij g v˜3, j i k αβ 2 i kj 2ǫ ij

ij β γ α 3 ij (4.16) |F ηαβγxi xj ν |≤ cv˜ F gij , and ij ¯ α β γ δ ǫ kl 3 ij (4.17) |F Rαβγδν xi xk xj ηǫxl g |≤ cv˜ F gij .

Proof. (i) The ﬁrst three inequalities are obvious. (ii) (4.15) follows from the generalized Schwarz inequality combined with (4.13) and (4.14). (iii) (4.16) is a direct consequence of (4.13) and (4.14). (iv) The proof of (4.17) is a bit more complicated and uses the symmetry properties of the Riemann curvature tensor. Let

¯ α β γ δ ǫ kl (4.18) aij = Rαβγδν xi xk xj ηǫxl g .

We shall show that the symmetrization of aij satisﬁes

1 (4.19) −cv˜3g ≤ (a + a ) ≤ cv˜3g ij 2 ij ji ij with a uniform constant c = c(Ω), which in turn yields (4.17). Let p ∈ M(t) be arbitrary, (xα) be the special Gaussian coordinate of N, and (ξi) local coordinates around p such that

u , α =0, xα i (4.20) i = k (δi , α = k.

We also note that all indices are raised with respect to gij with the exception of the contravariant vector

i ij (4.21)u ˇ = σ uj. We point out that

2 ij −2ψ 2 ij (4.22) kDuk = g uiuj = e v˜ σ uiuj,

(4.23) v˜2 =1+ e2ψkDuk2,

(4.24) (να)= −v˜(1, uˇi)e−ψ, PRESCRIBED SCALAR CURVATURE 21 and

ǫ kl ψ k (4.25) ηǫxl g = −e u .

First, let us observe that in view of (4.25) and the symmetry properties of the Riemann curvature tensor we have

j (4.26) aij u =0.

Next, we shall expand the right-hand side of (4.18) explicitly.

2 k k aij = R¯0i0j v˜kDuk + R¯0ik0vu˜ ju + R¯0ikj vu˜

k l l 2 + R¯l0k0vu˜ uˇ uiuj + R¯l00j v˜uˇ uikDuk (4.27) k l l 2 + R¯l0kj vu˜ uˇ ui + R¯li0j v˜uˇ kDuk

k l k l + R¯lik0vu˜ uˇ uj + R¯likj vu˜ uˇ

To prove the estimate (4.19), we may assume that Du 6= 0. Let ei,1 ≤ i ≤ n be an orthonormal base of Tp(M(t)) such that

Du (4.28) e = , 1 kDuk then, for 2 ≤ k ≤ n, the ek are also orthonormal with respect to the metric 2ψ e σij , and it is also valid that

i j (4.29) σij uˇ ek =0 ∀ 2 ≤ k ≤ n,

i where ek = (ek). For 2 ≤ r, s ≤ n we deduce from (4.27)

i j ¯ 2 i j ¯ k i j aij eres = R0i0j v˜kDuk eres + R0ikj vu˜ eres (4.30) ¯ l 2 i j ¯ k l i j + Rli0j v˜uˇ kDuk eres + Rlikj vu˜ uˇ eres and hence,

i j 3 (4.31) |aij eres|≤ cv˜ ∀ 2 ≤ r, s ≤ n.

i j It remains to estimate aij e1er for 2 ≤ r ≤ n, because of (4.26). We deduce from (4.27)

i j ¯ 2 −2 i j ¯ −1 k i j (4.32) aij e1er = R0i0j v˜kDuk v˜ e1er + R0ikj v˜ u e1er, PRESCRIBED SCALAR CURVATURE 22 where we used the symmetry properties of the Riemann curvature tensor. Hence, we conclude

i j 2 (4.33) |aij e1er|≤ cv˜ ∀ 2 ≤ r ≤ n, and the relation (4.19) is proved. q.e.d.

Lemma 4.6. Let M ⊂ Ω¯ be a graph over S0, M = graph u, and ǫ = ǫ(x) a 1 function given in S0, 0 <ǫ< 2 . Let ϕ be deﬁned through

λu (4.34) ϕ = eµe , where 0 < µ and λ< 0. Then, there exists c = c(Ω) such that

ij ij 3 λu ij k 2|F v˜iϕj |≤ cF gij v˜ |λ|µe ϕ + (1 − 2ǫ)F hi hkj vϕ˜ (4.35) 1 + F ij u u µ2λ2e2λuvϕ.˜ 1 − 2ǫ i j

α Proof. Sincev ˜ = ηαν , we have

α β k α v˜i = ηαβ ν xi + ηαhi xk (4.36) α β ψ k = ηαβ ν xi − e hi uk.

Thus, we derive

ij ij λu 2|F v˜iϕj | =2|F v˜iuj||λ|µe ϕ (4.37) ij 3 λu ψ ij k λu ≤ cF gij v˜ |λ|µe ϕ +2e |F hi ukuj||λ|µe ϕ.

The last term of the preceding inequality can be estimated by

1 (4.38) (1 − 2ǫ)F ijhkh vϕ˜ + v˜−1e2ψkDuk2F ij u u µ2λ2e2λuϕ i kj 1 − 2ǫ i j and we obtain the desired estimate in view of (4.23). q.e.d.

Applying Lemma 4.5 to the evolution equation forv ˜ we conclude PRESCRIBED SCALAR CURVATURE 23

Lemma 4.7. There exists a constant c = c(Ω) such that for any function ǫ, 0 <ǫ = ǫ(x) < 1, deﬁned on S0 the term v˜ satisﬁes an evolution inequality of the form

˙ ij ij k α β v˜ − F v˜ij ≤−(1 − ǫ)F hi hkj v˜ − fηαβν ν (4.39) c ij 3 2 β kl ψ + F g v˜ + c|||f |||v˜ + f β x h u e . ǫ ij β ν l k

We are now ready to prove the uniform boundedness ofv ˜.

Proposition 4.8. Assume that there are positive constants ci, 1 ≤ i ≤ 3, such that for any x ∈ Ω and any past directed time-like vector ν there holds

(4.40) −c1 ≤ f(x, ν),

(4.41) |||fβ(x, ν)||| ≤ c2(1 + |||ν|||), and

(4.42) |||fνβ (x, ν)||| ≤ c3. Then, the term v˜ remains uniformly bounded during the evolution

(4.43)v ˜ ≤ c = c(Ω,c1,c2,c3,ǫ0), where ǫ0 is the constant in (3.3). Here, and in the following, the reference that a constant depends on Ω also means that it depends on the barriers and geometric quantities of the ambient space restricted to Ω.

Proof. We proceed similar as in [14, Proposition 3.7] and show that the function

(4.44) w =vϕ, ˜

ϕ as in (4.34), is uniformly bounded, if we choose

(4.45) 0 <µ< 1 and λ << −1, appropriately, and assume furthermore, without loss of generality, that u ≤−1, for otherwise replace u by (u − c), c large, in the deﬁnition of ϕ.

With the help of the lemmata 4.4, 4.6, and 4.7 we derive from the relation

ij ij ij ij (4.46)w ˙ − F wij = [v˜˙ − F v˜ij ]ϕ + [ϕ ˙ − F ϕij ]˜v − 2F v˜iϕj PRESCRIBED SCALAR CURVATURE 24 the parabolic inequality

ij ij k −1 λu ij 3 w˙ − F wij ≤−ǫF hi hkj vϕ˜ + c[ǫ + |λ|µe ]F gij v˜ ϕ 1 + [ − 1]F ij u u µ2λ2e2λuvϕ˜ 1 − 2ǫ i j (4.47) ij 2 λu − F uiuj µλ e vϕ˜

α β −ψ λu 2 + f[−ηαβν ν + e µλe v˜ ]ϕ

2 β kl ψ + c|||fβ|||v˜ ϕ + fνβ xl h uke ϕ, where we have chosen the same function ǫ = ǫ(x) in Lemma 4.6 resp. Lemma 4.7.

Setting ǫ = e−λu and using Lemma 2.7, the assumption (3.3), which can be rewritten as

ij kl ij (4.48) F ≥ ǫ0F gklg , as well as the assumptions (4.40), (4.41), and (4.42), and observing, furthermore, that in view of the concavity and homogeneity of F

ij (4.49) F gij ≥ F (1,..., 1) > 0, we conclude

1 w˙ − F ij w ≤− F ij hkh e−λuvϕ˜ + c|λ|µeλuF ij g v˜3ϕ ij 2 i kj ij 2 (4.50) + F ij u u µ2λ2eλuvϕ˜ − F ij u u µλ2eλuvϕ˜ 1 − 2ǫ i j i j λu 2 3 2 −1 λu 3 + cc1µ|λ|e v˜ ϕ + cc2v˜ ϕ + cc3ǫ0 e v˜ ϕ, where |λ| is chosen so large that

1 (4.51) e−λu ≤ . 4

Choosing, furthermore,

1 (4.52) µ = , 8 PRESCRIBED SCALAR CURVATURE 25

ij we see that the terms involving F uiuj add up to a dominating negative quantity that can be estimated from above by

1 ǫ (4.53) − F ij u u λ2eλuvϕ˜ ≤− 0 F klg kDuk2λ2eλuvϕ,˜ 16 i j 16 kl in view of (4.48). kDuk2 is of the orderv ˜2 for largev ˜, hence, the parabolic maximum principle yields a uniform estimate for w if |λ| is chosen large enough. q.e.d.

5. C2- estimates for the auxiliary solutions

We want to prove that the principal curvatures of the ﬂow hypersurfaces are uniformly bounded.

Proposition 5.1. Let M(t), 0 ≤ t

(5.1) 0

Then, the principal curvatures of the ﬂow hypersurfaces are uniformly bounded provided the M(t) are uniformly space-like, i.e. uniform C1- estimates are valid.

Proof. As we have already mentioned in Remark 3.7, we know that

(5.2) 0

i j (5.3) ϕ = sup{ hijη η : kηk =1 }. We claim that ϕ is uniformly bounded. ∗ Let 0

(5.4) sup ϕ< sup{ sup ϕ: 0

i We then introduce a Riemannian normal coordinate system (ξ ) at x0 ∈ M(t0) such that at x0 = x(t0, ξ0) we have n (5.5) gij = δij and ϕ = hn. Letη ˜ = (˜ηi) be the contravariant vector ﬁeld deﬁned by (5.6)η ˜ = (0,..., 0, 1), and set h η˜iη˜j ϕ ij . (5.7) ˜ = i j gij η˜ η˜

ϕ˜ is well defined in neighbourhood of (t0, ξ0), andϕ ˜ assumes its maximum at (t0, ξ0). Moreover, at (t0, ξ0) we have ˙ ˙ n (5.8) ϕ˜ = hn, and the spatial derivatives do also coincide; in short, at (t0, ξ0)ϕ ˜ satisfies the n same differential equation (3.15) as hn. For the sake of greater clarity, let us n n therefore treat hn like a scalar and pretend that ϕ = hn.

At (t0, ξ0) we haveϕ ˙ ≥ 0, and, in view of the maximum principle, we deduce from Lemma 3.6

ij 2 n n 2 ij n 0 ≤−ǫ0 F gij |A| hn + f|hn| + cF gij (hn + 1) (5.9) + c(1 + |A|2)(1 + f + |||Df||| + |||D2f|||), where we used the concavity of F , the Codazzi equations, (4.48), and where

2 ij k (5.10) |A| = g hi hkj .

Thus, ϕ is uniformly bounded in view of (4.49). q.e.d.

6. Convergence to a stationary solution

We shall show that the solution of the evolution problem (3.9) exists for all time, and that it converges to a stationary solution.

Proposition 6.1. The solutions M(t) = graph u(t) of the evolution problem (3.9) with F ∈ (H), and M(0) = M2 exist for all time and converge to a stationary solution provided f ∈ C2,α satisﬁes the conditions (4.41), (4.42), and (5.1). PRESCRIBED SCALAR CURVATURE 27

Proof. Let us look at the scalar version of the ﬂow as in (4.6)

∂u (6.1) = −e−ψv(F − f). ∂t This is a scalar parabolic diﬀerential equation deﬁned on the cylinder

∗ (6.2) QT ∗ = [0,T ) × S0

4,α with initial value u(0) = u2 ∈ C (S0). In view of the a priori estimates, which we have established in the preceding sections, we know that

(6.3) |u| ≤ c 2,0,S0 and (6.4) F is uniformly elliptic in u independent of t due to the deﬁnition of the class (H). Thus, we can apply the known regularity results, see e.g. [20, Chapter 5.5], where even more general operators are considered, to conclude that uniform C2,α-estimates are valid, leading further to uniform C4,α-estimates due to the regularity results for linear operators. Therefore, the maximal time interval is unbounded, i.e. T ∗ = ∞. Now, integrating (6.1) with respect to t, and observing that the right-hand side is non-positive, yields

t t (6.5) u(0, x) − u(t, x)= e−ψv(F − f) ≥ c (F − f), Z0 Z0 i.e., ∞ (6.6) |F − f| < ∞ ∀x ∈ S0. Z0

Hence, for any x ∈ S0 there is a sequence tk → ∞ such that (F − f) → 0. On the other hand, u(·, x) is monotone decreasing and therefore (6.7) lim u(t, x)=˜u(x) t→∞ 4,α exists and is of class C (S0) in view of the a priori estimates. We, ﬁnally, conclude thatu ˜ is a stationary solution, and that

(6.8) lim (F − f)=0. t→∞ q.e.d. PRESCRIBED SCALAR CURVATURE 28

An immediate consequence of the results we have proved so far—cf. especially Lemma 2.5 and Remark 3.1— is the following theorem which is of independent interest.

Theorem 6.2. Let F ∈ C2,α(Γ ) ∩ C0(Γ¯) be a concave curvature function vanishing on ∂Γ and homogeneous of degree 1. Let f = f(x, ν) of class C2,α satisfy the conditions (4.41), (4.42), and (5.1), and suppose that the boundary components Mi act as barriers for (F,f), then, there exists an admissible 4,α hypersurface M = graph u, u ∈ C (S¯0), solving

(6.9) Fǫ|M = f(x, ν) for small ǫ> 0.

7. Stationary approximations

We want to solve the equation

(7.1) H2|M = f(x, ν), where f satisﬁes the conditions of Theorem 0.3. The curvature function F = 1 2 H2 is concave and the elliptic regularization Fǫ of class (H), cf. (3.1) and Remark 3.1. Thus, we would like to apply the preceding existence result to ﬁnd hypersurfaces Mǫ ⊂ Ω¯ such that

1 F f 2 . (7.2) ǫ|Mǫ =

But, unfortunately, the derivatives fβ grow quadratically in |||ν||| contrary to the assumption (4.41) in Proposition 4.8. ∞ R Therefore, we deﬁne a smooth cut-oﬀ function θ ∈ C ( +), 0 < θ ≤ 2k, where k ≥ k0 > 1 is to be determined later, by

t, 0 ≤ t ≤ k, (7.3) θ(t)= (2k, 2k ≤ t, such that

(7.4) 0 ≤ θ˙ ≤ 4, PRESCRIBED SCALAR CURVATURE 29 and consider the problem

F f˜ x, ν , (7.5) ǫ|Mǫ = ( ˜) where for a space-like hypersurface M = graph u with past directed normal vector ν as in (2.16), we set

(7.6) ν˜ = θ(˜v)˜v−1ν and

1 (7.7) f˜(x, ν˜)= f 2 (x, ν˜).

Then,

(7.8) |||ν˜||| ≤ ck, so that the assumptions in Proposition 4.8 are certainly satisﬁed.

The constant k0 should be so large thatν ˜ = ν in case of the barriers Mi, i =1, 2. If we now start with the evolution equation

(7.9)x ˙ = (Fǫ − f˜)ν, then, the Mi are barriers for (Fǫ, f˜) for small ǫ, cf. Lemma 2.5 and we conclude

Lemma 7.1. The ﬂow hypersurfaces Mǫ(t) = graph uǫ stay in Ω¯ during the evolution if ǫ is small 0 <ǫ ≤ ¯ǫ(Ω).

Remark 7.2. When we consider the elliptic regularizations Fǫ, we would like to generalize the meaning of admissible hypersurface by calling a hypersurface admissible if the tensor hij + ǫHgij is admissible, i.e. if its eigenvalues belong to Γ2.

j Next, let us consider the evolution equations forv ˜ and hi which look slightly diﬀerent: In (4.7) the term involving fνβ has to be replaced by

˜ −1 β ˙ −1 β −2 β α ik (7.10) −fν˜β [θ(˜v)˜v νi + θv˜iv˜ ν − θv˜ v˜iν ]xk g ηα.

But in view of (4.36), the additional terms do not cause any new problems in the proof of Proposition 4.8, and hence, the uniform C1- estimates are still valid for the modiﬁed evolution problem, where the estimates depend on k. PRESCRIBED SCALAR CURVATURE 30

The C2- estimates in Section 5 remain valid, too, since the second derivatives ˜ ˜j of f, fi , that occur on the right-hand side of (3.15), can be expressed as—we only consider the covariant form f˜ii, no summation over i—

˜ ˜ α β ˜ α β −fii = −fαβxi xi − 2fαν˜β xi ν˜i (7.11) ˜ α ˜ α β ˜ α − fαν hii − fν˜αν˜β ν˜i ν˜i − fν˜α ν˜i;i, where

α −1 α ˙ −1 α −2 α (7.12)ν ˜i = θv˜ νi + θv˜iv˜ ν − θv˜ v˜iν ,

α ˙ −1 α −2 α −1 α ν˜i;i =2θv˜iv˜ νi − 2θv˜ v˜iνi + θv˜ νi;i

−1 α −2 α −1 α (7.13) + θ¨v˜iv˜iv˜ ν − 2θ˙v˜iv˜iv˜ ν + θ˙v˜iiv˜ ν

−3 α −2 α +2θv˜ v˜iv˜iν − θv˜ v˜iiν , and

β γ α α β β α α (7.14) v˜ii = ηαβγ xi xi ν + ηαβν ν hii +2ηαβxi νi + ηανi;i. Hence, the result of Proposition 5.1 is still valid since no additional bad terms occur in inequality (5.9) as one easily checks, and since, furthermore, we also have

(7.15) f˜ ≤ Fǫ during the evolution, for the modiﬁed version of (3.14) now has the form

˜ ′ ij ˜ (Fǫ − f) − Fǫ (Fǫ − f)ij ij k ˜ ˜ α ˜ = −Fǫ hikhj (Fǫ − f) − fαν (Fǫ − f)

γ −1 −2 α β − f˜ν˜γ ν [θ˙v˜ − θv˜ ]ηαβν ν (Fǫ − f˜) (7.16) ˙ −1 −2 ˜ β α ˜ ij − [θv˜ − θv˜ ]fν˜β ν ηαxi (Fǫ − f)j g −1 ˜ α ˜ ij − θv˜ fν˜α xi (Fǫ − f)j g ij ¯ α β γ δ ˜ − Fǫ Rαβγδν xi ν xj (Fǫ − f). Here, we used the relation

β α α (7.17) v˜˙ = ηαβx˙ ν + ηαν˙ , PRESCRIBED SCALAR CURVATURE 31 which follows immediately from (4.8), together with (3.9) and (3.11). The conclusions of Section 6 are therefore applicable leading to a solution of equation (7.5).

8. C1- estimates for the stationary approximations

Consider the solutions Mǫ = graph uǫ of equation (7.5), which at the mo- ment not only depend on ǫ but also on k, the parameter of the cut-oﬀ function θ, cf. (7.3). We shall prove that the hypersurfaces Mǫ are uniformly space-like independent of ǫ and k, or, equivalently, that there exists a constant m1 such that

1 2 − 2 (8.1)v ˜ = (1 − |Duǫ| ) ≤ m1 ∀ ǫ,k, where the parameter ǫ is supposed to be small and k to be large, so that the barrier condition is satisﬁed.

Lemma 8.1. Let uǫ be a solution of (7.5), then, the estimate (8.1) is valid uniformly in ǫ and k. Hence, Mǫ = graph uǫ is a solution of equation (7.2), if we choose k ≥ 2m1.

Proof. For arbitrary but ﬁxed values of ǫ and k, let us introduce the notation F˜ for Fǫ, where from now on through the rest of the article

(8.2) F = H2, and where f = f(x, ν) satisﬁes

(8.3) 0

2 (8.4) |||fβ(x, ν)||| ≤ c2(1 + |||ν||| ), and

(8.5) |||fνβ (x, ν)||| ≤ c3(1 + |||ν|||), for all x ∈ Ω¯ and all past directed time-like vectors ν ∈ Tx(Ω). Thus, F is homogeneous of degree 2, and we recall that

(8.6) F ij = Hgij − hij , and (8.7) F˜ij = F ij + ǫ(n − 1)(1 + ǫn)Hgij, PRESCRIBED SCALAR CURVATURE 32

ij ij where F˜ is evaluated at hij and F at (hij + ǫHgij ).

We also drop the index ǫ, writing u for uǫ and M for Mǫ, i.e. M solves the equation

˜ (8.8) F |M = f(x, ν˜).

The C1- estimate will follow the arguments in the proof of Proposition 4.8, where at one point we shall introduce an additional observation especially suitable for the curvature function F = H2.

Remark 8.2. The former parabolic equations and inequalities, (4.11), (4.39), and (4.47) can now be read as elliptic equations resp. inequalities by simply assuming that the terms involved are time independent. Though, to be absolutely precise, one has to observe that the present curvature function is homogeneous of degree 2, which means that, whenever the term F —not derivatives of F —occurs explicitly in the equations or inequalities just mentioned, it has to be replaced by 2F because it was obtained as a result of Euler’s formula for homogeneous functions of degree d0

ij (8.9) d0F = F hij .

We mention it as a matter of fact only, since it doesn’t aﬀect the estimates at all.

However, we have to be aware that f now depends onν ˜ instead of ν, i.e. the elliptic version of inequality (4.47) now takes the form

˜ij ˜ij k −1 λu ˜ij 3 −F wij ≤−δ F hi hkj vϕ˜ + c[δ + |λ|µe ]F gij v˜ ϕ 1 + [ − 1]F˜ij u u µ2λ2e2λuvϕ˜ 1 − 2δ i j (8.10) ij 2 λu − F˜ uiuj µλ e vϕ˜

α β −ψ λu 2 2 +2f[−ηαβν ν + e µλe v˜ ]ϕ + c|||fβ|||v˜ ϕ

−1 β ˙ −1 β −2 β i ψ + fν˜β [θv˜ νi + θv˜iv˜ ν − θv˜ v˜iν ]u e ϕ.

Here, we used the notation δ = δ(x) for the small parameter in the Schwarz inequality instead of ǫ, which has a diﬀerent meaning in the present context, w is deﬁned as in (4.44), where the parameters µ, λ should satisfy the conditions in (4.45), and u is supposed to be less than −1. PRESCRIBED SCALAR CURVATURE 33

We claim that w is uniformly bounded provided µ and λ are chosen appropriately. Following the arguments in Section 4, we shall use the maximum principle and consider a point x0 ∈ M, where

(8.11) w(x0) = sup w. M

As before, we choose δ = e−λu. But the further conclusions are no longer valid, since we have a really bad term on the right-hand side of (8.10) that is of the orderv ˜4 due to the assumption (8.4). The only possible good term which can balance it, is

˜ij k (8.12) −δ F hi hkj vϕ.˜

To exploit this term we use the fact that Dw(x0) = 0, or, equivalently

λu −v˜i = µλe vu˜ i (8.13) ψ k α β = e hi uk − ηαβν xi , where the second equation follows from (4.8) and the deﬁnition of the covariant vectorﬁeld η = eψ(−1, 0,..., 0). Next, we choose a coordinate system (ξi) such that in the critical point

k k (8.14) gij = δij and hi = κiδi , and the labelling of the principal curvatures corresponds to

(8.15) κ1 ≤ κ2 ≤···≤ κn.

Then, we deduce from (8.13)

ψ λu α β (8.16) e κiui = µλe vu˜ i + ηαβν xi .

Assume thatv ˜(x0) ≥ 2, and let i = i0 be an index such that

1 (8.17) |u |2 ≥ kDuk2. i0 n

i ∂ i Setting (e )= ∂ξi0 and assuming without loss of generality that 0 ≤ uie in x0 we infer from Lemma 2.7

ψ i λu i α β i e κi0 uie = µλe vu˜ ie + ηαβν xi e (8.18) λu i 2 ≤ µλe vu˜ ie + cv˜ , PRESCRIBED SCALAR CURVATURE 34 and we deduce further in view of (2.12) and (8.17) that

1 (8.19) κ ≤ [µλeλu + c]˜ve−ψ ≤ µλeλuve˜ −ψ, i0 2 if |λ| is suﬃciently large, i.e. κi0 is negative and of the same order asv ˜. The Weingarten equation and Lemma 2.7 yield

1 β i k i β k i l 2 (8.20) |||νi u ||| = |||hi u xk ||| ≤ cv˜[hi u hklu ] , and therefore, we infer from (8.13)

i β i λu 3 (8.21) |v˜iu | + |||νi u ||| ≤ cµ|λ|e v˜ in critical points of w, and hence, that in those points, the term involving fν˜β on the right-hand side of inequality (8.10) can be estimated from above by

−1 β ˙ −1 β −2 β i ψ λu 4 (8.22) |fν˜β [θv˜ νi + θv˜iv˜ ν − θv˜ v˜iν ]u e ϕ|≤ cµ|λ|e v˜ ϕ.

Next, let us estimate the crucial term in (8.12). Using (8.7), the particular coordinate system (8.14), as well as the inequalities (8.15), together with the fact that κi0 is negative, we conclude

i0 ˜ij k ij k i 2 −F hi hkj ≤−F hi hkj ≤− Fi κi i=1 (8.23) X i0 i 2 ≤− Fi κi0 , i=1 X i where we recall that the argument of Fi is the n- tupel with components

(8.24)κ ˜j = κj + ǫH and observe that in the present coordinate system

i ∂F (8.25) Fi = . ∂κ˜i

Let Fˆ = log F , then, Fˆ is concave, and therefore, we have in view of (8.15)

ˆ1 ˆ2 ˆn (8.26) F1 ≥ F2 ≥···≥ Fn , PRESCRIBED SCALAR CURVATURE 35 cf. [6, Lemma 2], or equivalently,

1 2 n (8.27) F1 ≥ F2 ≥···≥ Fn .

Hence, we conclude

i0 n 1 − F i ≤−F 1 ≤− F i i 1 n i i=1 i=1 (8.28) X X 1 n − 1 = − (n − 1)[H + ǫnH] ≤− H, n n where we also used (8.6) and (8.24). Combining (8.19), (8.23), (8.28), and the estimate (1.15), we deduce further

n − 1 −F˜ij hkh ≤− Hκ2 i kj n i0 n − 1 n − 1 (8.29) ≤− c |κ |3 − Hκ2 2n n i0 2n i0 3 3 3λu 3 2 2 2λu 2 ≤−a0µ |λ| e v˜ − a1Hµ λ e v˜ with some positive constants a0 = a0(n, Ω) and a1 = a1(n, Ω). Inserting this estimate, and the estimate in (8.22) in the elliptic inequality (8.10), with δ = e−λu, we ﬁnally obtain

ij 3 3 2λu 4 2 2 λu 3 −F˜ wij ≤−a0µ |λ| e v˜ ϕ − a1Hµ λ e v˜ ϕ

2 ij 2 2 λu λu 3 + F˜ uiuj µ λ e vϕ˜ + c[1 + |λ|µ]He v˜ ϕ (8.30) 1 − 2δ ij 2 λu λu 4 − F˜ uiujµλ e vϕ˜ + c[c2 + c3µ|λ|e ]˜v ϕ

+2f[c + e−ψµλeλu]˜v2ϕ.

1 Choosing, now, µ = 4 and |λ| large, the right-hand side of the preceding inequality is negative, contradicting the maximum principle, i.e. the maximum of w cannot occur at point wherev ˜ ≥ 2. Thus, the desired uniform estimate for w and, hence,v ˜ is proved. q.e.d.

Let us close this section with an interesting observation that is an immediate consequence of the preceding proof, we have especially (8.23) and the ﬁrst line of inequality (8.28) in mind, PRESCRIBED SCALAR CURVATURE 36

Lemma 8.3. Let F ∈ C2(Γ ) be a positive symmetric curvature function such that the partial derivatives Fi are positive and Fˆ = log F is concave. Suppose

F is evaluated at a point (κi), and assume that κi0 is a component that is either negative or the smallest component of that particular n- tupel, then

n n 1 (8.31) F κ2 ≥ F κ2 . i i n i i0 i=1 i=1 X X

9. C2- estimates for the stationary approximations

We want to prove uniform C2- estimates for admissible solutions M of

˜ (9.1) F |M = f(x, ν), where we use the notations and conventions of the preceding section. The starting point is an elliptic equation for the second fundamental form.

j Lemma 9.1. The tensor hi satisﬁes the elliptic equation

˜kl j −F hi;kl ˜kl r j k j = −F hrkhl hi +2fhi hk α β kj α j α β kj α β k lj − fαβxi xk g − fαν hi − fανβ (xi xk h + xl xk hi g ) α β k lj β k lj α k j − fνανβ xl x hi h − fνβ x hi;l g − fνα ν hi h (9.2) k k k ˜kl,rs j ˜kl ¯ α β γ δ m rj + F hkl;ihrs; +2F Rαβγδxmxi xk xrhl g ˜kl ¯ α β γ δ m rj ˜kl ¯ α β γ δ mj − F Rαβγδxmxk xr xl hi g − F Rαβγδxmxk xi xl h ˜kl ¯ α β γ δ j ¯ α β γ δ mj − F Rαβγδν xk ν xl hi +2fRαβγδν xi ν xmg ˜kl ¯ α β γ δ ǫ mj α β γ δ ǫ mj + F Rαβγδ;ǫ{ν xk xl xi xmg + ν xi xk xmxl g }.

Proof. The elliptic equation can be immediately derived from the corresponding parabolic equation (3.15) by dropping the time-derivative, replacing F by F˜, and observing that the present curvature function is homogeneous of degree 2, cf. Remark 8.2. q.e.d. PRESCRIBED SCALAR CURVATURE 37

Contracting over the indices (i, j) in (9.2) we obtain a diﬀerential equation for H

kl −F˜ Hkl ˜kl r 2 = −F hrkhl H +2f|A|

α β ki α α β ki − fαβxi xk g − fαν H − 2fανβ xi xk h

α β k li k β 2 β − fνανβ xl xk hi h − fνβ (H xk + |A| ν ) (9.3) ¯ α β kl γ ˜kl,rs i − Rαβν xk g xl fνγ + F hkl;ihrs; ˜kl ¯ α β γ δ m ri ˜kl ¯ α β γ δ mi +2F Rαβγδxmxi xk xrhl g − 2F Rαβγδxmxk xi xl h ˜kl ¯ α β γ δ ¯ α β − F Rαβγδν xk ν xl H +2fRαβν ν ˜kl ¯ α β γ δ ǫ mi α β γ δ ǫ mi + F Rαβγδ;ǫ{ν xk xl xi xmg + ν xi xk xmxl g }, where we also used the symmetry properties of the Riemann curvature tensor and the Codazzi equations at one point. Next, let us improve the estimate in Proposition 5.1. Lemma 9.2. Let M = graph u be an admissible solution of (9.1), then the principal curvatures of M satisfy the estimate

(9.4) ǫ|A|2 ≤ const,

2 where the constant depends on |||Df|||, |||D f|||, the constant c1 in (8.3), and on known estimates of the C0 and C1- norm of u. Proof. We argue as in the proof of Proposition 5.1 and deﬁne

i j (9.5) ϕ = sup{ hijη η : kηk =1 }.

Let x0 ∈ M be a point, where ϕ achieves its maximum, and assume without loss of generality that, after having introduced normal Riemannian coordinates n around x0, we may write ϕ = hn, cf. the corresponding arguments in the proof of Proposition 5.1.

Applying the maximum principle in x0, we deduce from (9.2) the following inequality

2 n n 2 ˜kl,rs n 0 ≤−ǫ(n − 1)H|A| hn +2f|hn| + F hkl;nhrs; (9.6) 2 2 ij + c(1 + |||Df||| + |||D f|||)(1 + |A| )+ c(F˜ gij + f), PRESCRIBED SCALAR CURVATURE 38 where we also used (8.7), the Weingarten and Codazzi equations, and the fact that the pair (x, ν) stays in a compact subset of Ω¯ × C−(Ω¯), where C−(Ω¯) stands for the set of past directed time-like vectorﬁelds in Ω¯. Furthermore, we know that

ij (9.7) F˜ gij ≤ cH, and ˜kl,rs n ˜−1 ˜ ˜ n −1 n F hkl;nhrs; ≤ F F;nF; = f fnf (9.8) −1 2 ≤ cc1 (1 + |A| ), since log F˜ is concave, cf. Lemma 1.3.

Thus, we conclude that in x0 the following inequality is valid

(9.9) 0 ≤−ǫ|A|4 + c(1 + |A|2) with a known constant c, and the lemma is proved. q.e.d.

The estimate (9.4) will play an important role in the ﬁnal a priori estimate.

n Lemma 9.3. Let F = H2, M = M a Riemannian manifold with metric gij , hij a symmetric tensor ﬁeld on M the eigenvalues of which belong to Γ2, and p ∈ M an arbitrary point. Choose local coordinates around p such that the relations (8.14) and (8.15) are satisﬁed. Then, we have for 1 ≤ j ≤ n

2 j 2 j (9.10) κi +2F = |Fj | +2Fj κj , Xi6=j 2 n (9.11) κi +2F ≤ cFn κn, Xi6=n and i 2 F;j F;j Fi 2 −1 (9.12) + (1 − ) ≤ c|F;j | F , F j H F j i6=j j j X with c = c(n), F is evaluated at hij , and where we point out that the summation convention is not used. PRESCRIBED SCALAR CURVATURE 39

Proof. Throughout the proof we shall use the ambivalent meaning of F as a function depending on κi or on hij switching freely from one viewpoint to the other. (i) From the deﬁnition of F

1 (9.13) F = (H2 − |A|2), 2 and (8.6) we conclude

j 2 2 2F = (Fj + κj ) − κi i (9.14) X j 2 j 2 = |Fj | +2Fj κj − κi , Xi6=j which proves (9.10).

(ii) If j = n, and thus κn the largest eigenvalue, then, we derive from (8.6)

n (9.15) Fn ≤ H ≤ nκn, and (9.11) follows at once from (9.10). (iii) A simple algebraic transformation yields

F F F i F ;j + ;j (1 − i )= ;j (H − κ + κ ) j H j j j i Fj Fj HFj (9.16) F;j = j ( κk + κi), HFj Xk6=j and hence,

i 2 2 F;j F;j Fi |F;j | 2 (9.17) + (1 − ) ≤ c κi . F j H F j H2|F j|2 i6=j j j j i6=j X X

We, now, treat the cases j = n and j 6= n separately. If j = n, we apply (9.11) and (1.17) and conclude

i 2 2 2 F;j F;j Fi |F;j | |F;j | (9.18) + (1 − ) ≤ c ≤ c . F j H F j HF j F i6=j j j j X

PRESCRIBED SCALAR CURVATURE 40

If j 6= n, we deduce from (9.10)

2 j 2 (9.19) κn ≤ 6|Fj | , and deduce further

2 j 2 (9.20) κi ≤ 8|Fj | , Xi6=j where apparently we only had to worry about the case 0 ≤ κj . Thus, the right-hand side of (9.17) is estimated from above by

|F |2 (9.21) c ;j , H2 which in turn is less than

|F |2 (9.22) c ;j . F q.e.d.

Corollary 9.4. Let M be an admissible solution of (9.1) and p ∈ M arbitrary. Choose local coordinates around p such that the relations (8.14) and (8.15) are valid. Then, for any 1 ≤ j ≤ n, the following inequality is valid in p

˜i 2 fj fj Fi 2 −1 (9.23) + (1 − ) ≤ c|fj | f , F˜j H F˜j i6=j j j X where we use the notation H =e (1+ ǫn)H, and do not apply the summation convention. e

Proof. Let us recall the relation (8.7), which we can also express in the form

ij ij ij rs ij (9.24) F˜ = Hg − h˜ + ǫF grsg , where e

(9.25) h˜ij = hij + ǫHgij . PRESCRIBED SCALAR CURVATURE 41

Consider each summand in (9.23) separately. We have

˜i 2 f 2 fj fj Fi j ˜j ˜i 2 + (1 − ) = |H + Fj − Fi | F˜j H F˜j H2|F˜j |2 j j j (9.26) 2 e 2 e e fj ≤ κ˜k +˜κi , H2|F j |2 j k6=j X in view of (9.24), i.e. we are exactly ine the same situation as in the proof of Lemma 9.3 after the equation (9.16) with the following modiﬁcations: we replace hij ,κi and H by h˜ij , κ˜i resp. H and observe that F (h˜ij )= f. q.e.d.

e Lemma 9.5. Let M be an admissible solution of equation (9.1), then, the estimate

˜ij,kl r −1 ˜ij −1 2 2 (9.27) F hij;rhkl; + H F HiHj ≤ cf kDfk + cǫkDHk + c is valid in every point, where the smallest principal curvature κ1 satisﬁes

1 (9.28) max(−κ , 0) ≤ H ≡ ǫ H. 1 2(n − 1) 1

Proof. The proof is a modiﬁcation of a similar result in [3, Sections 6.1.4 and 6.1.5] or [4, Section 5.1.1]. It follows immediately from the deﬁnition of F that

1 (9.29) F ij,kl = gij gkl − (gikgjl + gilgjk) 2 and ij,kl rs ij,kl rs F˜ hij;rhkl;sg = F hij;rhkl;sg (9.30) + ǫ[2(n − 1)+ ǫ(n − 1)n]kDHk2.

In a ﬁxed point p ∈ M introduce normal Riemannian coordinates such that the relations (8.14) and (8.15) are valid, and deﬁne the matrix (akl) through

1, k 6= l, (9.31) akl = (0, k = l. We also set

(9.32) hijk = hij;k. PRESCRIBED SCALAR CURVATURE 42

Then, we conclude from (9.29)

ij,kl rs 2 ijk F hijrhklsg = kDHk − hijkh

= (hkkihlli − hklihkli) i,k,lX (9.33) kl 2 = a (hkkihlli − hkli) i X kl kl 2 = a hkkihlli − a hkli. i X Xl Xk,i

In the last summand let us interchange the roles of i and l to obtain

kl 2 ki 2 (9.34) − a hkli = − a hkil. i Xl Xk,i X Xk,l

The Codazzi equations yield

(9.35) hkil = hkli + ckil, where (ckil) is a uniformly bounded tensor in Ω¯

(9.36) |||ckil||| ≤ const, and hence,

2 2 2 (9.37) hkil = hkli +2hklickil + ckil and

ki 2 ki 2 ki − a hkil ≤− a hkli − 2 a hklickil i i X Xk,l X Xk,l Xi,k,l akih2 h2 (9.38) ≤−2 kki − ijk i X Xk [i,j,kX] ki ki − 2 a hklickil − 2 a hiikciki, Xi,k,l Xi,k where [i,j,k] means that the summation is carried out over those triples (i, j, k) where all three indices are diﬀerent from each other. P PRESCRIBED SCALAR CURVATURE 43

The ﬁrst linear term in the last inequality can be estimated from above by

ki ki ki −2 a hklickil = −2 a hkkickik − 2 a hklickil i i Xi,k,l X Xk X Xk6=l (9.39) δ δ ≤ akih2 + h2 + cδ−1 2 kki 2 ijk i X Xk [i,j,kX] for any δ > 0, and the second term similarly. Thus, we deduce from (9.33) and (9.38)

δ F ij,klh h grs ≤−2(1 − ) akih2 ijr kls 2 kki i k (9.40) X X kl −1 + a hkkihlli + cδ i X for any 0 <δ ≤ 1. Next, let us consider the second term on the left-hand side of (9.27); we have

ǫn H−1F˜ij H H = H−1F˜ij H H + H−1F˜ij H H i j i j 1+ ǫn i j (9.41) −1 ii 2 2 ≤ He F˜ ( hkki) + ǫckDHk , i X Xk e where c = c(n), in view of (8.7), (9.25), and (1.15). Combining (9.40) and the preceding estimate, we conclude

ij,kl rs −1 ij F hij;rhkl;sg + H F˜ HiHj

δ ki 2 kl −1 ≤−2(1 − ) a h + a hkkihlli + cδ (9.42) 2 kki i i X Xk X −1 ii 2 2 + H F˜ ( hkki) + ǫckDHk . i X Xk e For each index i, let us estimate the corresponding summand separately, i.e. let us look at—no summation over i—

δ ki 2 1 kl 1 ˜ii 2 (9.43) −(1 − ) a hkki + a hkkihlli + F ( hkki) , 2 2 2H Xk Xk where we have divided the terms by 2. e PRESCRIBED SCALAR CURVATURE 44

Denote by ′ a sum where the index i is omitted during the summation, then, (9.43) can be expressed as P δ ′ −(1 − ) akih2 + h akih + h h 2 kki iii kki kki lli k k k

kk (9.45) fi ≡ F˜i = F˜ hkki to derive

′ 1 kk (9.46) hiii = (fi − F˜ hkki). F˜ii Xk Inserting (9.46) in (9.44) we obtain, after some simple algebraic manipula- tions, cf. [3, equ. (36) on p. 78] or [4, equ. (17)],

δ ′ ′ F˜k F˜i F˜k 2 − (1 − ) h2 − k − i 1 − k h2 kki ˜i ˜i kki 2 Fi 2H Fi Xk Xk h i ′ F˜k + F˜l F˜i F˜k F˜l − k l − 1 − i 1 − ke 1 − l h h (9.47) ˜i ˜i ˜i kki lli Fi H Fi Fi Xk

δ ′ ′ f f F˜k (9.48) I = − h2 + i + i 1 − k h , 1 kki ˜i ˜i kki 2 Fi H Fi Xk Xk h i ˜i 2 Fi fi e (9.49) I2 = , 2H F˜i i and ′ e ′ F˜k F˜i F˜k 2 I = −(1 − δ) h2 − k − i 1 − k h2 3 kki ˜i ˜i kki k k Fi 2H Fi (9.50) X X h i ′ F˜k + F˜l F˜i F˜k F˜l − k l − 1 − i 1 − k e 1 − l h h . ˜i ˜i ˜i kki lli Fi H Fi Fi Xk

In view of (9.23) we can estimate I1 from above by

−1 −1 2 (9.51) I1 ≤ cδ f |fi| .

I2 is estimated by

1 2 −1 2 (9.52) I2 = |fi| ≤ cf |fi| , ˜i 2HFi because of (1.17). e 1 Finally, we claim that I3 ≤ 0 if we choose δ = 4 . To verify this assertion, ˜i let us multiply I3 by 2HFi to obtain

′ ′ e ˜i 2 ˜k ˜i ˜k 2 2 − 2(1 − δ)HFi hkki − [2HFk − (Fi − Fk ) ]hkki k k (9.53) X X ′ e e ˜k ˜l ˜i ˜i ˜k ˜i ˜l − [2H(Fk + Fl ) − 2HFi − 2(Fi − Fk )(Fi − Fl )]hkkihlli. Xk

j j j (9.54) Fj + ǫ(n − 1)(1 + ǫn)Hgj ≡ Fj + ǫγǫH.

The expression in (9.53) is then equal to the sum of two terms I4 +I5, where

′ ′ i 2 k i k 2 2 I4 = −2(1 − δ)HFi hkki − [2HFk − (Fi − Fk ) ]hkki k k (9.55) X X ′ e e k l i i k i l − [2H(Fk + Fl ) − 2HFi − 2(Fi − Fk )(Fi − Fl )]hkkihlli, Xk

′ ′ 2 2 (9.56) I5 = −2(1 − δ)HǫγǫH hkki − 2HǫγǫH hkki Xk Xk e e ′ − 2HǫγǫH hkkihlli. Xk

′ 2 ′ ′ 2 (9.57) hkki = hkki +2 hkkihlli Xk Xk Xk

1 we infer that I5 ≤ 0, while I4 is non-positive provided we choose δ = 4 and assume

1 (9.58) max(−κ˜ , 0) ≤ H, 1 2(n − 1) cf. [3, pp. 81–85] or [4, pp. 23–29]. e But the condition (9.58) is certainly satisﬁed in view of (9.28). Combining (9.30), (9.42), (9.51), and (9.52) gives (9.27), and thus, the Lemma is proved. q.e.d.

From Lemma 9.2 and Lemma 9.5 we conclude

Corollary 9.6. Let M = graph u be an admissible solution of equation (9.1) in Ω. Then, the estimate

ij,kl rs −1 ij (9.59) F˜ hij;rhkl;sg H + F˜ (log H)i(log H)j ≤ cH−1f −1kDfk2 + cH−1kD log Hk2 + cH−1 is valid in every point p ∈ M, where (9.28) is satisﬁed. The constant c depends 2 on Ω, |||Df|||, |||D f|||, the constant c1 in (8.3), and on known estimates of the C0 and C1- norm of u.

As we already mentioned we have to assume the existence of a strictly convex function χ ∈ C2(Ω¯), i.e. χ satisﬁes

(9.60) χαβ ≥ c0 g¯αβ with a positive constant c0.

We observe that then the restriction χ = χ|M of χ to an admissible solution M ⊂ Ω¯ of (9.1) satisﬁes the elliptic inequality

˜ij ˜ α ˜ij α β −F χij = −2Fχα ν − F χαβ xi xj (9.61) ˜ α ˜ij ≤−2Fχα ν − c0F gij , where we used the homogeneity of F˜.

We can now prove uniform C2- estimates.

Theorem 9.7. Let M = graph u be an admissible solution of equation (9.1) in Ω, where f satisﬁes the estimates (8.3), (8.4) and (8.5). Then, the principal curvatures of M are uniformly bounded. PRESCRIBED SCALAR CURVATURE 47

Proof. Let χ be the strictly convex function and µ a large positive constant. We shall prove that w = log H + µχ is uniformly bounded from above.

Let x0 ∈ M be such that

(9.62) w(x0) = sup w, M and choose in x0 a local coordinate system satisfying (8.14) and (8.15). Ap- plying the maximum principle, we conclude from (9.3) and (9.61)

˜kl r ˜ij ˜ij 0 ≤−F hkrhl + cF gij + cµf − µc0F gij (9.63) + c(1 + f + |||Df||| + |||D2f|||)(1 + H + kD log Hk)

ij,kl rs −1 ij + F˜ hij;rhkl;sg H + F˜ (log H)i(log H)j , where we also assumed H to be larger than 1.

We now consider two cases.

Case 1: Suppose that

1 (9.64) |κ |≥ ǫ H ≡ H. 1 1 2(n − 1)

Then, we infer from Lemma 8.3 and (9.24)

n − 1 n − 1 (9.65) −F˜klh hr ≤− Hκ2 ≤− ǫ2H3 ≡−ǫ H3. kr l n 1 n 1 2 Moreover, the concavity of log F˜ implies

ij,kl rs −1 −1 ij kl rs −1 F˜ hij;rhkl;sg H ≤ F˜ g F˜ hkl;iF˜ hrs;j H

= f −1kDfk2H−1 (9.66) ≤ cf −1|||Df|||2|A|2H−1

≤ cf −1|||Df|||2H.

Furthermore, Dw(x0)= 0, or,

(9.67) (log H)i = −µχi . Inserting the last three relations in (9.63) we obtain

3 2 (9.68) 0 ≤−ǫ2H + c(1 + H + µ)+ cµ H, PRESCRIBED SCALAR CURVATURE 48 where, now, c depends on f and its derivatives in the ambient space. Hence, H, and therefore w, are a priori bounded in x0.

Case 2: Suppose that

(9.69) |κ1| <ǫ1H.

Then, Corollary 9.6 is applicable, and we infer from (9.63) and (9.67)

2 −1 ij (9.70) 0 ≤ c(1 + H + µ + µ H ) + (c − µc0)F˜ gij .

Choosing now µ suﬃciently large we obtain an a priori bound for H(x0), since

ij (9.71) F˜ gij ≥ (n − 1)H.

Thus, w, or equivalently H, are uniformly bounded. q.e.d.

10. Existence of a solution

We can now demonstrate the ﬁnal step in the proof of Theorem 0.3. Let Mǫ = graph uǫ be the stationary approximations. In the preceding sections we have proved uniform estimates for uǫ up to the order two. Since, by assumption, f is strictly positive, the principal curvatures of Mǫ stay in a compact subset of the cone Γ2 for small ǫ, cf. Remark 3.1, and therefore, the operator F˜ is uniformly elliptic for those ǫ. Taking the square root on both sides of equation (9.1) without changing the notation, we also know that F˜ is concave. Hence the C2,α- estimates of Evans and Krylov are applicable, cf. [7] and [20], and we deduce

(10.1) |uǫ|2,α,S0 ≤ const uniformly in ǫ. If ǫ tends to zero, a subsequence converges to a solution u ∈ 2,α C (S0) of our problem. From the Schauder estimates we further conclude 4,α u ∈ C (S0). PRESCRIBED SCALAR CURVATURE 49

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Ruprecht-Karls-Universitat,¨ Institut fur¨ Angewandte Mathematik, Im Neuen- heimer Feld 294, 69120 Heidelberg, Germany E-mail address: [email protected] URL: http://www.math.uni-heidelberg.de/studinfo/gerhardt/