Srvey in Geometric Analysis and Relativity ALM 0, pp. 41 57 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 ) 1 000 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.
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 =
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 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.)
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 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.
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 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 + 1 3 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 3 (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)
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 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 ).
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)
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)
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 1 1 + (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 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 4.1. 5 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
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 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
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 Bo Gan and Joel Sprck References [1] B. Andrews, Contraction of convex hypersrfaces in Eclidean space, Calc. Var. PDE (1994), 151 171. [] 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), 61 301. [3] C. Gerhardt, Closed Weingarten hypersrfaces in Riemannian manifolds, J. Differential Geom. 43 (1996), 61 641. [4] A. Gray, Tbes, Addison Wesley Pblising Company, 1990. [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), 1039 1060. [6] B. Gan and J. Sprck, Hypersrfaces of constant crvatre in Hyperbolic space II, J. Eropean Math. Soc. 1 (010), 797 817. [7] B. Gan, J. Sprck and M. Szapiel, Hypersrfaces of constant crvatre in Hyperbolic space I. J. Geom. Anal. 19 (009), 77 795. [8] F. Laborie, Problème de Minkowski et srfaces à corbre constante dan les variétés hyperboliqes, Bll. Soc. Math. Fr. 119 (1991), 307 35. [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), 57 64. [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, 53 66. [11] H. Rosenberg, and J. Sprck, On the existence of convex hypersrfaces of constant Gass crvatre in hyperbolic space, J. Differential Geom. 40 (1994), 379 409. [1] J-M. Schlenker, Réalisations de srfaces hyperboliqes complètes dans H 3, Annales de l institt Forier, 48 (1998), 837 860. [13] G. Smith, Modli of flat conformal strctres of hyperbolic type, ArXiv 0804.0744v6 (009). [14] G. Smith, Lagrangian crvatre, ArXiv 050630v3 (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), 83 309.