Convex Hypersurfaces of Constant Curvature in Hyperbolic Space

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1 Srvey in Geometric Analysis and Relativity ALM 0, pp c Higher Edcation Press and International Press Beijing-Boston Convex Hypersrfaces of Constant Crvatre in Hyperbolic Space Bo Gan and Joel Sprck Abstract We show that for a very general and natral class of crvatre fnctions, the problem of finding a complete strictly convex hypersrface in H n+1 satisfying f(κ) = σ (0, 1) with a prescribed asymptotic bondary Γ at infinity has at least one soltion which is a vertical graph over the interior (or the exterior) of Γ. There is niqeness for a certain sbclass of these crvatre fnctions which incldes the crvatre qotients ( Hn n l, l = n, or n 1. For smooth simple Γ, as σ varies between 0 and 1, these hypersrfaces foliate the two components of the complement of the hyperbolic convex hll of Γ. H l ) Mathematics Sbject Classification: Primary 53C1, Secondary 35J60 1 Introdction In this paper we retrn to or earlier stdy [7] of complete locally strictly convex hypersrfaces of constant crvatre in hyperbolic space H n+1 with a prescribed asymptotic bondary at infinity. Given Γ H n+1 and a smooth symmetric fnction f of n variables, we seek a complete hypersrface Σ in H n+1 satisfying f(κ[σ]) = σ (1.1) Σ = Γ (1.) where κ[σ] = (κ 1,..., κ n ) denotes the positive hyperbolic principal crvatres of Σ and σ (0, 1) is a constant. Department of Mathematics, Ohio State University, Colmbs, OH 4310 Department of Mathematics, Johns Hopkins University, Baltimore, MD 118 Research of both athors was partially spported by NSF grants.

2 Bo Gan and Joel Sprck We will se the half-space model, eqipped with the hyperbolic metric H n+1 = {(x, x n+1 ) R n+1 : x n+1 > 0} ds = n+1 i=1 dx i x. (1.3) n+1 Ths H n+1 is natrally identified with R n = R n {0} R n+1 and (1.) may be nderstood in the Eclidean sense. For convenience we say Σ has compact asymptotic bondary if Σ H n+1 is compact with respect to the Eclidean metric in R n. The fnction f is assmed to satisfy the fndamental strctre conditions in and K + n := { λ R n : each component λ i > 0 } : (1.4) f i (λ) f(λ) λ i > 0 in K + n, 1 i n, (1.5) In addition, we shall assme that f is normalized f is a concave fnction in K + n, (1.6) f > 0 in K + n, f = 0 on K + n. (1.7) and satisfies the following more technical assmptions and f(1,..., 1) = 1 (1.8) f is homogeneos of degree one (1.9) lim f(λ 1,, λ n 1, λ n + R) 1 + ε 0 niformly in B δ0 (1) (1.10) R + for some fixed ε 0 > 0 and δ 0 > 0, where B δ0 (1) is the ball of radis δ 0 centered at 1 = (1,..., 1) R n. All these assmptions are satisfied by f = (H n /H l ) 1 n l, 0 l < n, where Hl is the normalized l-th elementary symmetric polynomial (H 0 = 1, H 1 = H and H n = K the mean and extrinsic Gass crvatres, respectively). See [] for proof of (1.5) and (1.6). For (1.10) one easily comptes that ( n ) 1 lim f(λ n l 1,, λ n 1, λ n + R) =. R + l Moreover, if g k, k = 1,... N satisfy (1.5)-(1.10), then so does the concave sm f = N k=1 α kg k or concave prodct f = Π N k=1 (gk ) α k where α k > 0, N k=1 α k =

3 Hypersrfaces of constant crvatre 3 1. Since f is symmetric, by (1.6), (1.8) and (1.9) we have f(λ) f(1) + f i (1)(λ i 1) = f i (1)λ i = 1 n λi in K + n (1.11) and fi (λ) = f(λ) + f i (λ)(1 λ i ) f(1) = 1 in K + n. (1.1) In this paper all hypersrfaces in H n+1 we consider are assmed to be connected and orientable. If Σ is a complete hypersrface in H n+1 with compact asymptotic bondary at infinity, then the normal vector field of Σ is chosen to be the one pointing to the niqe nbonded region in R n+1 + \ Σ, and the (both hyperbolic and Eclidean) principal crvatres of Σ are calclated with respect to this normal vector field. As in or earlier work [11, 10, 5, 7, 6], we will take Γ = Ω where Ω R n is a smooth domain and seek Σ as the graph of a fnction (x) over Ω, i.e. Σ = {(x, x n+1 ) : x Ω, x n+1 = (x)}. Then the coordinate vector fields and pper nit normal are given by X i = e i + i e n+1, n = ν = ( ie i + e n+1 ), w where w = 1 +. The first fndamental form g ij is then given by g ij = X i, X j = 1 (δ ij + i j ) = ge ij. (1.13) To compte the second fndamental form h ij we se to obtain Γ k ij = 1 x n+1 { δ jk δ in+1 δ ik δ jn+1 + δ ij δ kn+1 } (1.14) Xi X j = ( δ ij x n+1 + ij i j x n+1 )e n+1 je i + i e j x n+1. (1.15) Then h ij = Xi X j, ν = 1 w (δ ij + ij i j + i j ) = 1 (1.16) w (δ ij + i j + ij ) = he ij + νn+1 ge ij. The hyperbolic principal crvatres κ i of Σ are the roots of the characteristic eqation det(h ij κg ij ) = n det(h e ij 1 (κ 1 w )ge ij) = 0. Therefore, κ i = κ e i + ν n+1. (1.17)

4 4 Bo Gan and Joel Sprck The relations (1.16) and (1.17) are easily seen to hold for parametric hypersrfaces. One beatifl conseqence of (1.16) is the following reslt of [7]. Theorem 1.1. Let Σ be a complete locally strictly convex C hypersrface in H n+1 with compact asymptotic bondary at infinity. Then Σ is the (vertical) graph of a fnction C (Ω) C 0 (Ω), > 0 in Ω and = 0 on Ω, for some domain Ω R n : Σ = { (x, (x)) R n+1 + : x Ω } sch that That is, the fnction + x is strictly convex. {δ ij + i j + ij } > 0 in Ω. (1.18) According to Theorem 1.1, or assmption that Σ is a graph is completely general and the asymptotic bondary Γ mst be the bondary of some bonded domain Ω in R n. Problem (1.1)-(1.) then redces to the Dirichlet problem for a flly nonlinear second order eqation which we shall write in the form with the bondary condition G(D, D, ) = σ, > 0 in Ω R n (1.19) = 0 on Ω. (1.0) We seek soltions of eqation (1.19) satisfying (1.18). Following the literatre we call sch soltions admissible. By [] condition (1.5) implies that eqation (1.19) is elliptic for admissible soltions. Or goal is to show that the Dirichlet problem (1.19)-(1.0) admits smooth admissible soltions for all 0 < σ < 1, which is optimal. Or main reslt of the paper may be stated as follows. Theorem 1.. Let Γ = Ω {0} R n+1 where Ω is a bonded smooth domain in R n. Sppose σ (0, 1) and that f satisfies conditions (1.5)-(1.10) in K + n. Then there exists a complete locally strictly convex hypersrface Σ in H n+1 satisfying (1.1)-(1.) with niformly bonded principal crvatres κ[σ] C on Σ. (1.1) Moreover, Σ is the graph of an admissible soltion C (Ω) C 1 ( Ω) of the Dirichlet problem (1.19)-(1.0). Frthermore, C (Ω) C 1,1 (Ω) and D C in Ω, 1 + D = 1 σ on Ω (1.)

5 Hypersrfaces of constant crvatre 5 For Gass crvatre, f(λ) = (H n ) 1 n, Theorem 1. was proved by Rosenberg and Sprck [11]. Eqation (1.19) is singlar where = 0. It is therefore natral to approximate the bondary condition (1.0) by = ɛ > 0 on Ω. (1.3) When ɛ is sfficiently small, we showed in [7] that the Dirichlet problem (1.19),(1.3) is solvable for all σ (0, 1). Theorem 1.3. Let Ω be a bonded smooth domain in R n and σ (0, 1). Sppose f satisfies (1.5)-(1.10) in K + n. Then for any ɛ > 0 sfficiently small, there exists an admissible soltion ɛ C ( Ω) of the Dirichlet problem (1.19), (1.3). Moreover, ɛ satisfies the a priori estimates 1 + Dɛ 1 σ + Cɛ, ɛ D ɛ C on Ω, (1.4) and ɛ D ɛ C ɛ in Ω. (1.5) where C is independent of ɛ. Remark 1.4. The global gradient estimate, Lemma 3.4 of [7] is not correct as stated. This may be corrected sing the convexity argment of [11] or sing Corollary 3.3 of section 3 of this paper. Theorem 1.3 above as well as Theorem 1. of [7] remain valid. However no apriori niqeness can be asserted. In Theorem 1.6 we prove a niqeness reslt for a special class of crvatre fnctions. Or main technical difficlty in proving Theorem 1. is that the estimate (1.5) does not allow s to pass to the limit. In [7] we were able to obtain a global estimate independent of ɛ for the hyperbolic principal crvatres for σ > 1 8. In this paper we obtain sch estimates for all σ (0, 1) by proving a maximm principle for the largest hyperbolic principal crvatre. Theorem 1.5. Let Ω be a bonded smooth domain in R n and σ (0, 1). Sppose f satisfies (1.5)-(1.10) in K + n. Then for any admissible soltion ɛ C ( Ω) of the Dirichlet problem (1.19), (1.3), max κ max(x) C(1 + max κ max(x)) (1.6) x Σ ɛ x Σ ɛ where Σ ɛ = graph ɛ and C is independent of ɛ. By Theorem 1.5, the hyperbolic principal crvatres of the admissible soltion ɛ given in Theorem 1.3 are niformly bonded above independent of ɛ. Since f(κ[ ɛ ]) = σ and f = 0 on K + n, the hyperbolic principal crvatres admit a niform positive lower bond independent of ɛ and therefore (1.19) is niformly elliptic on compact sbsets of Ω for the soltion ɛ. By the interior estimates of Evans and Krylov, we obtain niform C,α estimates for any compact sbdomain

6 6 Bo Gan and Joel Sprck of Ω. The proof of Theorem 1. is now rotine. Finally we prove a niqeness reslt and as an application prove a reslt abot foliations. This latter reslt is relevant to the stdy of foliations of the complement of the convex core of qasi-fchsian manifolds (see [8], [11], [13]). Theorem 1.6. Sppose f satisfies (1.5)-(1.10) in K + n and in addition, fi > λ i f i in K + n {0 < f < 1}. (1.7) Then the soltions given in Theorem 1. and Theorem 1.3 are niqe. In particlar niqeness holds for f = ( Hn H l ) 1 n l with l 1 or l. Theorem 1.7. a. Let f satisfy the conditions of Theorem 1.6 and assme that Γ is smooth and connected. Then for each σ (0, 1) there are exactly two embedded strictly locally convex hypersrfaces satisfying (1.1), (1.). Each srface is a graph of σ C (Ω ± ) C 1 (Ω ± ) where Ω ± are the components of the complement of Γ. Moreover the soltion hypersrfaces Σ σ = graph σ have niformly bonded principal crvatres and foliate each component of H n+1 \ CH(Γ), the complement of the hyperbolic convex hll of Γ. b. Let f = Hn H n 1 and Γ = Ω where Ω is a simply connected Jordan domain. If n >, assme in addition that Γ is reglar for Laplace s eqation. Then the conclsions of part a. hold with σ (x) C (Ω ± ) C 0 (Ω ± ) and the principal crvatres are niformly bonded on compact sbsets. Remark 1.8. i. Graham Smith pointed ot to s that in the special case n = 3, f = ( K H ) 1 is his special Lagrangian crvatre with angle θ = π and interior crvatre bonds follow from the geometric ideas of his paper [14]. Moreover in Lemma 7.4 of [13] he showed that special Lagrangian crvatre with angle θ (n 1) π satisfies or niqeness condition (1.7). Ths by Theorems 1.6 and 1.7, the existence of foliations of constant special Lagrangian crvatre can be proven for θ (n 1) π for all n. This incldes the special case f = K 1 when n = and f = ( K H ) 1 3 for n = 3 mentioned above. ii. Rosenberg and Sprck [11] proved part b of Theorem 1.7 for f = K 1 in case n =. Here and also in [11], no claim is made abot the higher reglarity of the CH(Γ). In other words, the crvatre estimates obtained in the proof of Theorem 1.7 (global for Γ smooth and interior for Γ Jordan) blow p as σ 0. We have not yet derived interior crvatre estimates for the case f = Hn H n in the general case. The organization of the paper is as follows. In section we establish some basic identities on a hypersrface Σ satisfying (1.1) that will form the basis of the global gradient estimates derived in section 3 and the maximm principle for κ max, the largest principal crvatre of Σ, which is carried ot in section 4. Finally in section 5 we prove the niqeness Theorem 1.6 and the foliation Theorem 1.7.

7 Hypersrfaces of constant crvatre 7 Formlas on hypersrfaces In this section we will derive some basic identities on a hypersrface by comparing the indced hyperbolic and Eclidean metrics. Let Σ be a hypersrface in H n+1. We shall se g and to denote the indced hyperbolic metric and Levi-Civita connection on Σ, respectively. As Σ is also a sbmanifold of R n+1, we shall sally distingish a geometric qantity with respect to the Eclidean metric by adding a tilde over the corresponding hyperbolic qantity. For instance, g denotes the indced metric on Σ from R n+1, and is its Levi-Civita connection. Let x be the position vector of Σ in R n+1 and set = x e where e is the nit vector in the positive x n+1 direction in R n+1, and denotes the Eclidean inner prodct in R n+1. We refer as the height fnction of Σ. Throghot the paper we assme Σ is orientable and let n be a (global) nit normal vector field to Σ with respect to the hyperbolic metric. This also determines a nit normal ν to Σ with respect to the Eclidean metric by the relation ν = n. We denote ν n+1 = e ν. Let (z 1,..., z n ) be local coordinates and τ i = z i, i = 1,..., n. The hyperbolic and Eclidean metrics of Σ are given by g ij = τ i, τ j, g ij = τ i τ j = g ij, while the second fndamental forms are h ij = D τi τ j, n = D τi n, τ j, h ij = ν D τi τ j = τ j D τi ν, (.1) where D and D denote the Levi-Civita connection of H n+1 and R n+1, respectively. The following relations are well known (see (1.16), (1.17)): and h ij = 1 h ij + νn+1 g ij (.) κ i = κ i + ν n+1, i = 1,, n (.3) where κ 1,, κ n and κ 1,, κ n are the hyperbolic and Eclidean principal crvatres, respectively. The Christoffel symbols are related by the formla Γ k ij = Γ k ij 1 ( iδ kj + j δ ik g kl l g ij ). (.4)

8 8 Bo Gan and Joel Sprck It follows that for v C (Σ) where (and in seqel) In particlar, and Moreover, In R n+1, ij v = v ij Γ k ijv k = ij v + 1 ( iv j + j v i g kl k v l g ij ) (.5) Therefore, by (.3) and (.7), v i = v, v ij = v, etc. z i z i z j ij = ij + i j 1 gkl k l g ij (.6) ij 1 = 1 ij gkl k l g ij. (.7) v ij = v 1 ij + 1 ij v 1 gkl k v l g ij. (.8) g kl k l = = 1 (ν n+1 ) ij = h ij ν n+1. (.9) 1 νn+1 ij = h ij (1 (νn+1 ) ) g ij = 1 (g ij ν n+1 h ij ). (.10) We note that (.8) and (.10) still hold for general local frames τ 1,..., τ n. In particlar, if τ 1,..., τ n are orthonormal in the hyperbolic metric, then g ij = δ ij and g ij = δ ij. We now consider eqation (1.1) on Σ. Let A be the vector space of n n matrices and A + = {A = {a ij } A : λ(a) K n + }, where λ(a) = (λ 1,..., λ n ) denotes the eigenvales of A. Let F be the fnction defined by F (A) = f(λ(a)), A A + (.11) and denote F ij (A) = F (A), F ij,kl (A) = F (A). (.1) a ij a ij a kl Since F (A) depends only on the eigenvales of A, if A is symmetric then so is the matrix {F ij (A)}. Moreover, F ij (A) = f i δ ij when A is diagonal, and F ij (A)a ij = f i (λ(a))λ i = F (A), (.13)

9 Hypersrfaces of constant crvatre 9 F ij (A)a ik a jk = f i (λ(a))λ i. (.14) Eqation (1.1) can therefore be rewritten in a local frame τ 1,..., τ n in the form F (A[Σ]) = σ (.15) where A[Σ] = {g ik h kj }. Let F ij = F ij (A[Σ]), F ij,kl = F ij,kl (A[Σ]). Lemma.1. Let Σ be a smooth hypersrface in H n+1 satisfying eqation (1.1). Then in a local orthonormal frame, and F ij 1 σνn+1 ij = + 1 fi. (.16) F ij ij ν n+1 = σ νn+1 fi κ i. (.17) Proof. The first identity follows immediately from (.10), (.13) and assmption (1.9). To prove (.17) we recall the identities in R n+1 (ν n+1 ) i = h ij g jk k, ij ν n+1 = g kl (ν n+1 hil hkj + l k hij ). (.18) By (.), (.13), (.14), and g ik = δ jk / we see that F ij g kl hil hkj = 1 F ij hik hkj = F ij (h ik h kj ν n+1 h ij + (ν n+1 ) δ ij ) = f i κ i ν n+1 σ + (ν n+1 ) f i. (.19) As a hypersrface in R n+1, it follows from (.3) that Σ satisfies f( κ 1 + ν n+1,..., κ n + ν n+1 ) = σ, or eqivalently, F ({ g ik ( h kj + ν n+1 g kj )}) = σ. (.0) Differentiating eqation (.0) and sing g ik = δ ik, g ik = δ ik /, we obtain F ij ( k hij + k hij + (ν n+1 ) k δ ij ) = 0. (.1) That is, F ij k hij + (ν n+1 ) k F ii = k F ij hij = k F ij (h ij ν n+1 δ ij ) ( = k σ ν n+1 f i ). (.)

10 10 Bo Gan and Joel Sprck Finally, combining (.8), (.16), (.18), (.19), (.), and the first identity in (.9), we derive F ij ν n+1 ij = ν n+1 F ij 1 ij + F ij hij νn+1 3 F ij hik hkj ( = νn+1 fi ν σ) n+1 + (σ ν ) n+1 f i (f νn+1 i κ i ν n+1 σ + (ν n+1 ) ) f i = σ νn+1 fi κ i. This proves (.17). (.3) 3 The asymptotic angle maximm principle and gradient estimates In this section we show that the pward nit normal of a soltion tends to a fixed asymptotic angle on approach to the bondary. This implies a global gradient bond on soltions. Theorem 3.1. Let Σ be a smooth strictly locally convex hypersrface in H n+1 satisfying eqation (1.1). Sppose Σ is globally a graph: Σ = {(x, (x)) : x Ω} where Ω is a domain in R n H n+1. Then and so, F ij ij σ ν n+1 σ ν n+1 σ(1 σ) ( f i 1) sp Σ σ ν n+1 0 (3.1) on Σ. (3.) Moreover, if = ɛ > 0 on Ω, then there exists ɛ 0 > 0 depending only on Ω, sch that for all ɛ ɛ 0, σ ν n+1 1 σ ε(1 + σ) + r 1 r1 on Σ (3.3) where r 1 is the maximal radis of exterior tangent spheres to Ω. Proof. Set η = σ νn+1. By (.16) and (.17) we have F ij ij η = σ ( ) ( fi 1 + νn+1 fi κ i σ ).

11 On the other hand, Hypersrfaces of constant crvatre 11 κ i f i ( κ i f i ) fi = σ fi. Hence, F ij ij η σ ( ) fi 1 )(1 σνn+1 fi σ(1 σ) ( fi 1) 0. So (3.) follows from the maximm principle, while (3.3) follows from (3.) and the approximate asymptotic angle condition, η 1 σ ε(1 + σ) + r 1 r1 on Σ which is proved in Lemma 3. of [7]. Proposition 3.. Let Σ be a smooth strictly locally convex graph Σ = {(x, (x)) : x Ω} in H n+1 satisfying ε in Ω, = ε on Ω. Then 1 { ν n+1 max maxω, max Ω 1 } ν n+1. (3.4) Proof. Let h = ν = w and sppose that h assmes its maximm at an interior n+1 point x 0. Then at x 0, i h = i w + k ki w = (δ ki + k i + ki ) k w = 0 1 i n. Since Σ is strictly locally convex, this implies that = 0 at x 0 so the proposition follows immediately. Combining Theorem 3.1 and Proposition 3. gives Corollary 3.3. Let Ω be a bonded smooth domain in R n and σ (0, 1). Sppose f satisfies (1.5)-(1.10) in K + n. Then for any ɛ > 0 sfficiently small, any admissible soltion ɛ C ( Ω) of the Dirichlet problem (1.19),(1.3) satisfies the apriori estimate where C is independent of ɛ. ɛ C in Ω (3.5)

12 1 Bo Gan and Joel Sprck 4 Crvatre estimates In this section we prove a maximm principal for the largest principal crvatre of locally strictly convex graphs satisfying f(κ) = σ. Let Σ be a smooth hypersrface in H n+1 satisfying f(κ) = σ. For a fixed point x 0 Σ we choose a local orthonormal frame τ 1,..., τ n arond x 0 sch that h ij (x 0 ) = κ i δ ij. The calclations below are done at x 0. For convenience we shall write v ij = ij v, h ijk = k h ij, h ijkl = lk h ij = l k h ij, etc. Since H n+1 has constant sectional crvatre 1, by the Codazzi and Gass eqations we have h ijk = h ikj and h iijj = h jjii + (h ii h jj 1)(h ii h jj ) = h jjii + (κ i κ j 1)(κ i κ j ). (4.1) Conseqently for each fixed j, F ii h jjii = F ii h iijj + (1 + κ j) f i κ i κ j fi κ j κ i f i. (4.) Theorem 4.1. Let Σ be a smooth strictly locally convex graph in H n+1 satisfying f(κ) = σ and ν n+1 a > 0 on Σ. (4.3) For x Σ let κ max (x) be the largest principal crvatre of Σ at x. Then Proof. Let max Σ κ { max 4n ν n+1 a max a 3, max Σ M 0 = max x Σ κ } max ν n+1. (4.4) a κ max (x) ν n+1 a. (4.5) Assme M 0 > 0 is attained at an interior point x 0 Σ. Let τ 1,..., τ n be a local orthonormal frame arond x 0 sch that h ij (x 0 ) = κ i δ ij, where κ 1,..., κ n are the principal crvatres of Σ at x 0. We may assme κ 1 = κ max (x 0 ). Ths, at x 0, h 11 ν n+1 a has a local maximm. Therefore, h 11i iν n+1 h 11 ν n+1 = 0, a (4.6) h 11ii iiν n+1 h 11 ν n+1 0. a (4.7) Using (4.), we find after differentiating the eqation F (h ij ) = σ twice that Lemma 4.. At x 0, F ii h 11ii = F ij,rs h ij1 h rs1 + σ(1 + κ 1) κ 1 fi κ 1 κ i f i. (4.8)

13 Hypersrfaces of constant crvatre 13 By Lemma.1 we immediately derive Lemma 4.3. Let Σ be a smooth hypersrface in H n+1 satisfying f(κ) = σ. Then in a local orthonormal frame, F ij ij ν n+1 = F ij i j ν n+1 +σ(1+(ν n+1 ) ) ν n+1( fi + f i κ i ). (4.9) Using Lemma 4. and Lemma 4.3 we find from (4.7) 0 F ij,rs h ij1 h rs1 + σ (1 + κ (νn+1 ) ν n+1 a κ 1 + aκ 1 ν n+1 a ( fi + κ i f i ) κ 1 ν n+1 a F ij i jν n+1. ) (4.10) Next we se an ineqality de to Andrews [1] and Gerhardt [3] which states F ij,kl h ij1 h kl,1 i j Recall that (see (.18)) Then at x 0, we obtain from (4.6) f i f j h ij1 f i f 1 h κ j κ i κ 1 κ i11. (4.11) i i i ν n+1 = i (νn+1 κ i ). h 11i = Inserting this into (4.11) we derive ( F ij,kl κ 1 h ij1 h kl,1 ν n+1 a κ 1 i ν n+1 a (νn+1 κ i ). (4.1) ) i f i f 1 κ 1 κ i i (κ i ν n+1 ). (4.13) Note that we may write fi + κ i f i = (1 (ν n+1 ) ) f i + (κ i ν n+1 ) f i + σν n+1. (4.14) Combining (4.11), (4.13) and (4.14) gives ( 0 σ 1 + κ 1 (1 + (νn+1 ) ) ) ν n+1 κ 1 a + aκ 1 ((1 (ν n+1 ) ) f i + (κ i ν n+1 ) f i + σν n+1) ν n+1 a κ 1 + ν n+1 a κ 1 + (ν n+1 a) fi i (κ i ν n+1 ) i f i f 1 κ 1 κ i i (κ i ν n+1 ). (4.15)

14 14 Bo Gan and Joel Sprck Note that (assming κ 1 a ) all the terms of (4.15) are positive except possibly the ones in the sm involving (κ i ν n+1 ) and only if κ i < ν n+1. Therefore define J = {i : κ i ν n+1 < 0, f i < θ 1 f 1 }, L = {i : κ i ν n+1 < 0, f i θ 1 f 1 }, where θ (0, 1) is to be chosen later. Since i / = = 1 (ν n+1 ) 1 and κ 1 f 1 σ, we have i J (κ i ν n+1 )f i i n θ f 1 nσ θκ 1. (4.16) Finally, κ 1 ν n+1 a i L f i f 1 κ 1 κ i i (κ i ν n+1 ) (1 θ)κ 1 ν n+1 a = κ 1 i L + κ 1 ν n+1 a κ 1 i L (κ i ν n+1 ) i f i i L f i i (κ i ν n+1 ) θκ 1 ν n+1 a i L (κ i ν n+1 ) i f i i L f i i (κ i (a + ν n+1 )κ i + aν n+1 ) f i i (κ i ν n+1 ) θ a κ 1 (κ i ν n+1 ) f i 6σ a κ 1. i L (4.17) In deriving the last ineqality in (4.17) we have sed that κ i f i σ for each i and that ν n+1 a. We now fix θ = a 4. From (4.16) and (4.17) we see that the right hand side of (4.15) is strictly positive provided that κ 1 > 4n a, completing the proof of Theorem Uniqeness and foliations In this section we identify a class of crvatre fnctions for which there is niqeness. This implies that for these crvatre fnctions and smooth asymptotic bondaries Γ which are Jordan, there is a foliation of each component of H n+1 \ CH(Γ) (the complement of the hyperbolic convex hll of Γ) by soltions f(κ) = σ as σ varies between 0 and 1. Theorem 5.1. Let f(κ) satisfy (1.5)-(1.10) in the positive cone K n + and in addition satisfy f i > λ i f i in K n + {0 < f < 1}. (5.1) i

15 Hypersrfaces of constant crvatre 15 Let Σ i, i = 1, be strictly locally convex hypersrfaces (oriented p) in H n+1 satisfying f(κ) = σ i (0, 1), σ 1 σ, with the same bondary in the horosphere x n+1 = ɛ or with the same asymptotic bondary Γ = Ω. Then Σ lies below Σ 1, that is, if Σ i are represented as graphs x n+1 = i (x) over Ω R n, then 1 in Ω. Proof. We bild on an idea of Schlenker [1]. Sppose for contradiction that Σ contains points in the nbonded region of R n+1 + \ Σ 1 and let P be a point of Σ farthest from Σ 1 (necessarily P is not a bondary point) where the maximal distance, say t is achieved. Then the local parallel hypersrfaces Σ t to Σ obtained by moving a distance t (on the concave side of Σ near P ) are convex and contact Σ 1 at a point Q in Σ 1 when t = t. Moreover Σ t locally lies below Σ 1 by the maximality of the distance t. We claim that the distance fnction d(x, Σ ) is smooth in a neighborhood of Q. To show this we need only show (see [9]) that P is the niqe closest point to Q on Σ. If P was a second point of Σ at distance t from Q, then the local parallel hypersrfaces Σ t to Σ obtained by moving a distance t (on the concave side of Σ near P ) are also convex and when t = t, contact Σ 1 at Q and also locally lies below Σ 1 by the previos argment. This is clearly impossible since Σ 1 has a niqe tangent plane at Q. The principal crvatres of Σ t at points along the normal geodesic emanating from any point of Σ 1 (say near P ) are given by the ode (see [4]): κ i(t) = 1 κ i. In particlar, if κ i (0) < 1, then κ i (0) κ i (t) < 1 while if κ i (0) > 1, then 1 < κ i (t) κ i (0). Of corse if κ i (0) = 1, then κ i (t) 1. Moreover by (5.1), d dt f(κ(t)) = f i k i f i > 0 in K + n {0 < f < 1}. (5.) It follows that the Σ t satisfy f(κ) > σ and so are strict sbsoltions of the eqation f(κ) = σ 1. On the other hand at t = t we have Σ t lies below Σ 1 bt toches Σ 1 at Q violating the maximm principle. Corollary 5.. Let f(κ) satisfy (1.5)-(1.10) in the positive cone K n + and in addition satisfy (5.1). Let Σ i, i = 1, be strictly locally convex graphs (oriented p) in H n+1 over Ω R n satisfying f(κ) = σ (0, 1) with the same bondary in the horosphere x n+1 = ɛ or with the same asymptotic bondary Γ = Ω. Then Σ 1 = Σ. Example 5.3. For l = n 1 or n, let f = ( Hn H l ) 1 n l in the cone K n + R n. Then (see Lemma.14 of [15]) f i = f ( 1 ) (log H l ) i n l λ i = f ( 1 H ) l 1;i, n l λ i H l where H l 1;i = H l 1 λi=0. Hence, fi = f ( n H n 1 l H ) l 1. (5.3) n l H n H l

16 16 Bo Gan and Joel Sprck Similarly, λ i f i = f ( λ nh 1 i H ) l 1;i. n l H l Using λ i H l 1;i H l = nh 1 (n l) H l+1 H l, we find Combining (5.3) and (5.4) gives fi λ i f i = By the Newton-Maclarin ineqalities, λ i f i = f H l+1 H l. (5.4) f ( n H n 1 l H l 1 (n l) H ) l+1. (5.5) n l H n H l H l H n 1 H n H l 1 H l with eqality if and only all the λ i are eqal. Hence, fi ( λ Hn 1 i f i f H ) l+1. (5.6) H n H l Therefore if l = n 1, we find fi λ i f i 1 f > 0 in K + n {0 < f < 1} (5.7) while if l = n we similarly find fi λ i f i H n 1 H n f(1 f ) 1 f > 0 in K + n {0 < f < 1}. (5.8) We now complete the proof of Theorem 1.7. Proof. a. For Γ smooth and f(κ) satisfying the conditions of Theorem 5.1, we have by Theorem 1. and Theorem 5.1 a smooth monotone decreasing family of smooth soltions Σ σ = graph σ (x), x Ω of (1.1), (1.). That is, if σ 1 < σ, then σ1 > σ in Ω. Note also that if Ω B δ (0) then σ < v σ (x) := σδ δ + 1 σ 1 σ x in Ω, where v σ (x) corresponds to the eqidistant sphere soltion of f(κ) = σ, which is a graph over B δ (0). As σ 1, v(x) 0 niformly and so the same holds for σ (x). We claim that as σ 0, Σ σ tends to the component S of CH(Γ) that is a graph over Ω. To see this note that Σ σ lies below S bt also eventally lies above

17 Hypersrfaces of constant crvatre 17 any smooth strictly locally convex hypersrface S by Theorem 5.1. This completes the proof of Theorem 1.7 part a. In order to prove part b, it sffices by a standard approximation argment, to show that the graph soltions of f(κ) = σ have niformly bonded principal crvatres on compact sbdomains of Ω, independent of the smoothness of Γ. We carry this ot for the special crvatre qotient f = Hn H n 1 in Lemma 5.4 below, ths completing the proof of part b. Lemma 5.4. Let Σ = {graph (x) : x Ω} be the niqe strictly locally convex H soltion of n H n 1 (κ) = σ (0, 1). For any compact sbdomain Ω Ω, let Σ = {graph (x) : x Ω }. Then, max κ x Σ max C, where C depends only on σ and the (Eclidean) distance from Ω to Ω. Proof. Fix a small constant θ (0, 1) and set φ = ( θ) +. Recall from Lemma.1, L = F ij i j + σν n+1 f i. (5.9) We modify the argment of section 4 by setting M 0 = max x Σ φ κ max(x). (5.10) Then M 0 > 0 is attained at an interior point x 0 Σ. Let τ 1,..., τ n be a local orthonormal frame arond x 0 sch that h ij (x 0 ) = κ i δ ij, where κ 1,..., κ n are the principal crvatres of Σ at x 0. We may assme κ 1 = κ max (x 0 ). Ths, at x 0, log φ + log h 11 has a local maximm and so, φ i φ + h 11i = 0, h 11 (5.11) φ ii φ + h 11ii 0. h 11 (5.1) As in section 4 we obtain from (5.1) and (5.9) that at x 0, 0 κ 1 φ From the calclations of Example 5.3, fi + σ(1 + κ 1) κ 1 fi κ 1 κ i f i. (5.13) 1 f i n, κ i f i = σ. (5.14) Hence from (5.13) and (5.14) we obtain φκ 1 C. Choosing θ so small that θ on Ω completes the proof.

18 18 Bo Gan and Joel Sprck References [1] B. Andrews, Contraction of convex hypersrfaces in Eclidean space, Calc. Var. PDE (1994), [] L. Caffarelli, L. Nirenberg and J. Sprck, The Dirichlet problem for nonlinear second-order elliptic eqations III: Fnctions of eigenvales of the Hessians, Acta Math. 155 (1985), [3] C. Gerhardt, Closed Weingarten hypersrfaces in Riemannian manifolds, J. Differential Geom. 43 (1996), [4] A. Gray, Tbes, Addison Wesley Pblising Company, [5] B. Gan and J. Sprck, Hypersrfaces of constant mean crvatre in hyperbolic space with prescribed asymptotic bondary at infinity, Amer. J. Math. 1 (000), [6] B. Gan and J. Sprck, Hypersrfaces of constant crvatre in Hyperbolic space II, J. Eropean Math. Soc. 1 (010), [7] B. Gan, J. Sprck and M. Szapiel, Hypersrfaces of constant crvatre in Hyperbolic space I. J. Geom. Anal. 19 (009), [8] F. Laborie, Problème de Minkowski et srfaces à corbre constante dan les variétés hyperboliqes, Bll. Soc. Math. Fr. 119 (1991), [9] Y.-Y. Li and L. Nirenberg, Reglarity of the distance fnction to the bondary, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 9 (005), [10] B. Nelli and J. Sprck, On existence and niqeness of constant mean crvatre hypersrfaces in hyperbolic space, Geometric Analysis and the Calcls of Variations, Internat. Press, Cambridge, MA, 1996, [11] H. Rosenberg, and J. Sprck, On the existence of convex hypersrfaces of constant Gass crvatre in hyperbolic space, J. Differential Geom. 40 (1994), [1] J-M. Schlenker, Réalisations de srfaces hyperboliqes complètes dans H 3, Annales de l institt Forier, 48 (1998), [13] G. Smith, Modli of flat conformal strctres of hyperbolic type, ArXiv v6 (009). [14] G. Smith, Lagrangian crvatre, ArXiv v3 (005). [15] J. Sprck, Geometric aspects of the theory of flly nonlinear elliptic eqations, Global Theory of Minimal Srfaces, Clay Mathematics Proceedings Vol. 3 (005),