HYPERSURFACES OF CONSTANT CURVATURE IN HYPERBOLIC SPACE I.

Transcription

1 HYPERSURFACES OF CONSTANT CURVATURE IN HYPERBOLIC SPACE I. BO GUAN, JOEL SPRUCK, AND MAREK SZAPIEL Abstract. We investigate the problem of finding complete strictly convex hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of curvature functions. 1. Introduction In this paper we study Weingarten hypersurfaces of constant curvature in hyperbolic space H n+1 with a prescribed asymptotic boundary at infinity. More precisely, given a disjoint collection Γ = {Γ 1,..., Γ m } of closed embedded (n 1) dimensional submanifolds of H n+1, the ideal boundary of H n+1 at infinity, and a smooth symmetric function f of n variables, we seek a complete hypersurface Σ in H n+1 satisfying (1.1) f(κ[σ]) = σ where κ[σ] = (κ 1,..., κ n ) denotes the hyperbolic principal curvatures of Σ and σ is a constant, with the asymptotic boundary (1.) Σ = Γ. We will use the half-space model, equipped with the hyperbolic metric (1.3) ds = H n+1 = {(x, x n+1 ) R n+1 : x n+1 > 0} n+1 i=1 dx i. x n+1 Thus H n+1 is naturally identified with R n = R n {0} R n+1 and (1.) may be understood in the Euclidean sense. For convenience we say Σ has compact asymptotic boundary if Σ H n+1 is compact with respect the Euclidean metric in R n. Research of the first and second authors were supported in part by NSF grants. 000 Mathematical Subject Classification. Primary 53C1, Secondary 35J65, 58J3. 1

2 BO GUAN, JOEL SPRUCK, AND MAREK SZAPIEL The function f is assumed to satisfy the fundamental structure conditions: (1.4) f i (λ) f(λ) λ i > 0 in K, 1 i n, (1.5) f is a concave function in K, and (1.6) f > 0 in K, f = 0 on K where K R n is an open symmetric convex cone such that (1.7) K + n := { λ R n : each component λ i > 0 } K. In addition, we shall assume that f is normalized (1.8) f(1,..., 1) = 1 and satisfies the following more technical assumptions (1.9) f is homogeneous of degree one and (1.10) lim R + f(λ 1,, λ n 1, λ n + R) 1 + ε 0 uniformly in B δ0 (1) for some fixed ε 0 > 0 and δ 0 > 0, where B δ0 (1) is the ball of radius δ 0 centered at 1 = (1,..., 1) R n. All these assumptions are satisfied by f = (σ k /σ l ) 1 k l, 0 l < k n, defined in K k where σ k is the normalized k-th elementary symmetric polynomial (σ 0 = 1) and K k = {λ R n : σ j (λ) > 0, 1 j k}. See [3] for proof of (1.4) and (1.5). For (1.10) one easily computes that ( k lim f(λ 1,, λ n 1, λ n + R) = R + l Since f is symmetric, by (1.5), (1.8) and (1.9) we have ) 1 k l (1.11) f(λ) f(1) + f i (1)(λ i 1) = f i (1)λ i = 1 n λi in K K 1 and (1.1) fi (λ) = f(λ) + f i (λ)(1 λ i ) f(1) = 1 in K. Moreover, (1.4) and f(0) = 0 imply that (1.13) f > 0 in K + n..

3 HYPERSURFACES OF CONSTANT CURVATURE 3 In this paper we shall focus on the case of finding complete hypersurfaces satisfying (1.1)-(1.) with positive hyperbolic principal curvatures everywhere; for convenience we shall call such hypersurfaces (hyperbolically) locally strictly convex. In Part II [11] we will allow f satisfying (1.4)-(1.10) and general cones K. Before we state our first result we need to explain the orientation of hypersurfaces under consideration. In this paper all hypersurfaces in H n+1 we consider are assumed to be connected and orientable. If Σ is a complete hypersurface in H n+1 with compact asymptotic boundary at infinity, then the normal vector field of Σ is chosen to be the one pointing to the unique unbounded region in R n+1 + \ Σ, and the (both hyperbolic and Euclidean) principal curvatures of Σ are calculated with respect to this normal vector field. Theorem 1.1. Let Σ be a complete locally strictly convex C hypersurface in H n+1 with compact asymptotic boundary at infinity. Then Σ is the (vertical) graph of a function u C (Ω) C 0 (Ω), u > 0 in Ω and u = 0 on Ω, for some domain Ω R n : Σ = { (x, u(x)) R n+1 + : x Ω } such that (1.14) {δ ij + u i u j + uu ij } > 0 in Ω. That is, the function u + x is strictly convex. Moreover, (1.15) e u { 1 + Du max max e u, max 1 + Du } Ω Ω in Ω. According to Theorem 1.1, Problem (1.1)-(1.) for complete locally strictly convex hypersurfaces reduces to the Dirichlet problem for a fully nonlinear second order equation which we shall write in the form (1.16) G(D u, Du, u) = σ, u > 0 in Ω Rn u with the boundary condition (1.17) u = 0 on Ω. In particular, Γ must be the boundary of some bounded domain Ω in R n. The exact formula of G will be given in Section. We seek solutions of equation (1.16) satisfying (1.14). Following the literature we call such solutions admissible. By [3] condition (1.4) implies that equation (1.16) is

4 4 BO GUAN, JOEL SPRUCK, AND MAREK SZAPIEL elliptic for admissible solutions. Our goal is to show that the Dirichlet problem (1.16)- (1.17) admits smooth admissible solutions for all 0 < σ < 1 which is also a necessary condition by the comparison principle under conditions (1.8) and (1.9), as we shall see in Section 3. Due to the special nature of the problem, there are substantial technical difficulties to overcome and we have not yet succeeded in finding solutions for all σ (0, 1). However we shall prove Theorem 1.. Let Γ = Ω {0} R n+1 where Ω is a bounded smooth domain in R n. Suppose that σ (0, 1) satisfies σ > 1 8. Under conditions (1.4)-(1.10) with K = K+ n, there exists a complete locally strictly convex hypersurface Σ in H n+1 satisfying (1.1)- (1.) with uniformly bounded principal curvatures (1.18) κ[σ] C on Σ. Moreover, Σ is the graph of an admissible solution u C (Ω) C 1 ( Ω) of the Dirichlet problem (1.16)-(1.17). Furthermore, u C (Ω) C 1,1 (Ω) and (1.19) 1 + Du 1 σ, u D u C 1 + Du = 1 σ on Ω in Ω For Gauss curvature, f(λ) = (Πλ i ) 1 n, Theorem 1. was proved by Rosenberg and Spruck [15] who in fact allowed all σ (0, 1). As we shall see in Section, equation (1.16) is singular where u = 0. It is therefore natural to approximate the boundary condition (1.17) by (1.0) u = ɛ > 0 on Ω. When ɛ is sufficiently small, the Dirichlet problem (1.16),(1.0) is solvable for all σ (0, 1). Theorem 1.3. Let Ω be a bounded smooth domain in R n and σ (0, 1). Suppose f satisfies (1.4)-(1.10) with K = K n +. Then for any ɛ > 0 sufficiently small, there exists an admissible solution u ɛ C ( Ω) of the Dirichlet problem (1.16),(1.0). Moreover, u ɛ satisfies the a priori estimates (1.1) 1 + Duɛ 1 σ + ɛc in Ω

5 HYPERSURFACES OF CONSTANT CURVATURE 5 (1.) u ɛ D u ɛ C ɛ in Ω where C is independent of ɛ. The organization of the paper is as follows. Section summarizes the basic information about vertical and radial graphs that we will need in the sequel. In section 3 we prove global gradient bounds and some sharp estimates on the vertical component of the upward normal near the boundary. These are essential for the boundary second derivative estimates which we derive in Section 4. Here we make essential use of the exact form of the linearized operator to derive the mixed normal-tangential estimates and assumption (1.10) for the pure normal second derivative estimate. In Section 5 we prove a maximum principle for the maxmum hyperbolic principle curvature. Our approach uses radial graphs and is new and rather delicate. It is here that we have had to restrict the allowable σ (0, 1) to σ > 1. Otherwise our approach is 8 completely general and we expect Theorem 1. is valid for all σ (0, 1). Because the linearized operator is not necessarily elliptic we prove Theorem 1. by an iterative procedure which is carried out in Section 6. Because of this, we have derived all our estimates for a fairly general class of hypersurfaces of prescribed curvature as a function of position.. Formulas for hyperbolic principal curvatures Let Σ be a hypersurface in H n+1 and g the induced metric on Σ from H n+1. For convenience we call g the hyperbolic metric, while the Euclidean metric on Σ means the induced metric from R n+1. We use X and u to denote the position vector and height function, defined as u = X e, of Σ in R n+1, respectively. Here and throughout this paper, e is the unit vector in the positive x n+1 direction in R n+1, and denotes the Euclidean inner product in R n+1. We assume Σ is orientable and let n be a fixed global unit normal vector field to Σ with respect to the hyperbolic metric. This also determines an Euclidean unit normal ν to Σ by the relation We denote ν n+1 = e ν. ν = n u.

6 6 BO GUAN, JOEL SPRUCK, AND MAREK SZAPIEL Let e 1,, e n be a local orthnormal frame of vector fields on (Σ, g). The second fundamental form of Σ is locally given by h ij := ei e j, n where, and denote the metric and Levi-Civita connection of H n+1 respectively. The (hyperbolic) principal curvatures of Σ, denoted as κ[σ] = (κ 1,..., κ n ), are the eigenvalues of the second fundamental form. The relation between κ[σ] and the Euclidean principal curvatures κ E [Σ] = (κ E 1,..., κ E n ) is given by (.1) κ i = uκ E i + ν n+1 1 i n. We shall derive equations for Σ based on this formula when Σ satisfies (1.1)..1. Vertical graphs. Suppose Σ is locally represented as the graph of a function u C (Ω), u > 0, in a domain Ω R n : Σ = {(x, u(x)) R n+1 : x Ω}. In this case we take ν to be the upward (Euclidean) unit normal vector field to Σ: ( Du ν = w, 1 ), w = 1 + Du w. The Euclidean metric and second fundamental form of Σ are given respectively by and g E ij = δ ij + u i u j, h E ij = u ij w. According to [4], the Euclidean principal curvatures κ E [Σ] are the eigenvalues of the symmetric matrix A E [u] = [a E ij]: (.) a E ij := 1 w γik u kl γ lj, where γ ij = δ ij u iu j w(1 + w). Note that the matrix {γ ij } is invertible with inverse γ ij = δ ij + u iu j 1 + w

7 HYPERSURFACES OF CONSTANT CURVATURE 7 which is the square root of {gij}, E i.e., γ ik γ kj = gij. E From (.1) we see that the hyperbolic principal curvatures κ[u] of Σ are the eigenvalues of the matrix A v [u] = {a v ij[u]}: (.3) a v ij[u] := 1 ) (δ ij + uγ ik u kl γ lj. w Let S be the vector space of n n symmetric matrices and S K = {A S : λ(a) K}, where λ(a) = (λ 1,..., λ n ) denotes the eigenvalues of A. Define a function F by (.4) F (A) = f(λ(a)), A S K. We recall some properties of F. Throughout the paper we denote (.5) F ij (A) = F a ij (A), F ij,kl (A) = F a ij a kl (A). The matrix {F ij (A)}, which is symmetric, has eigenvalues f 1,..., f n, and therefore is positive definite for A S K if f satisfies (1.4), while (1.5) implies that F is concave for A S K (see [3]), that is (.6) F ij,kl (A)ξ ij ξ kl 0, {ξ ij } S, A S K. We have (.7) F ij (A)a ij = f i (λ(a))λ i, (.8) F ij (A)a ik a jk = f i (λ(a))λ i. Finally, the function G in equation (1.16) is determined by (.9) G(D u, Du, u) = 1 u F (Av [u]). where A v [u] = {a v ij[u]} is given by (.3). Note that (.10) and G st [u] := G = 1 u st w F ij γ is γ tj G st [u]u st = 1 ( F ij a ij 1 F ii) u w G u = 1 F ii u w (.11) G pq,st [u] := G u pq u st = u w F ij,kl γ is γ tj γ kp γ ql

8 8 BO GUAN, JOEL SPRUCK, AND MAREK SZAPIEL where F ij = F ij (A v [u]), etc. It follows that, under condition (1.4), equation (1.16) is elliptic for u if A v [u] S K, while (1.5) implies that G(D u, Du, u) is concave with respect to D u. For later use in section 6 note that if u is a solution of (.1) G(D u, Du, u) = G(D u, Du, u) ψ(x, u) = 0 then from (.10) (.13) Gu = 1 ( Ψ uψ u u 1 ) fi w where Ψ(x, u) = uψ(x, u). Since f i 1, we obtain from (.13) (.14) Gu 1 u ( Ψ uψ u 1 w )... Radial graphs. Let denote the covariant derivative on the standard unit sphere S n in R n+1 and y = e z for z S n R n+1. Let τ 1,, τ n be an orthnormal local frame of smooth vector fields on the upper hemisphere S n so that τ i τ j = δ ij. For a function v on S n, we denote v i = i v = τi v, v ij = j i v, etc. Suppose that locally Σ is a radial graph over the upper hemisphere S n + R n+1, i.e., it is locally represented as (.15) X = e v z, z S n + R n+1. The Euclidean metric, outward unit normal vector and second fundamental from of Σ are and g E ij = e v (δ ij + v i v j ) ν = z v w, w = (1 + v ) 1/ h E ij = 1 w ev (v ij v i v j δ ij ) respectively. Therefore the Euclidean principal curvatures are the eigenvalues of the matrix (.16) a E ij = 1 w e v γ ik (v kl v k v l δ kl )γ lj = 1 w e v (γ ik v kl γ lj δ ij ) where γ ij = δ ij v iv j w(1 + w).

9 HYPERSURFACES OF CONSTANT CURVATURE 9 Note that the height function u = ye v. We see that the hyperbolic principal curvatures are the eigenvalues of the matrix A s [v] = {a s ij[v]}: (.17) a s ij[v] := 1 w (yγik v kl γ lj e vδ ij ). In this case equation (1.1) takes the form (.18) F (A s [v]) = σ. 3. Locally convex hypersurfaces. C 1 estimates 3.1. Equidistance spheres. There are two important facts which will be used repeatedly. One is the invariance of equation (1.1) under scaling X λx in R n+1, as it is an isometry of H n+1. The other is that the Euclidean spheres, known as equidistance spheres, have constant hyperbolic principal curvatures. Let B R (a) be a ball of radius R centered at a = (a, σr) R n+1 where σ (0, 1) and S = B R (a) H n+1. Then κ i [S] = σ for all 1 i n with respect to its outward normal. These spheres may serve as barriers in many situations. Especially, we have the following estimates which were first derived in [8] for hypersurfaces of constant mean curvature. Lemma 3.1. Suppose f satisfies (1.4), (1.8) and (1.9). Let Σ be a hypersurface in H n+1 with κ[σ] K and σ 1 f(κ[σ]) σ where 0 σ 1 σ 1 are constants, and Σ P (ɛ) {x n+1 = ɛ}, ɛ 0. Let Ω be the region in R n bounded by the projection of Σ to R n = {x n+1 = 0} (such that R n \ Ω contains an unbounded component), and u denote the height function of Σ. (i) For any point (x, u) Σ, (3.1) ɛσ 1 + σ + d(x) 1 σ u L 1 + σ 1 σ1 1 + σ 1 + ɛ, where d(x) and L denote the distance from x R n to Ω and the (Euclidean) diameter of Ω, respectively. (ii) Assume that Σ C. For ɛ > 0 sufficiently small, (3.) σ 1 ɛ 1 σ 1 r 1 ɛ (1 + σ 1 ) r 1 < ν n+1 < σ + ɛ 1 σ r + ɛ (1 σ ) r on Σ

10 10 BO GUAN, JOEL SPRUCK, AND MAREK SZAPIEL where r 1 and r are the maximal radii of exterior and interior spheres to Ω, respectively. In particular, if σ 1 = σ = σ then ν n+1 σ on Σ as ɛ 0. While Lemma 3.1 was proved in [8] only for the mean curvature case, the proof remains valid for more general symmetric functions of principal curvatures with minor modifications. So we omit the proof here. Another important class of hypersurfaces of constant principal curvatures are the horospheres P (ɛ) {x n+1 = ɛ}, ɛ > 0. Indeed, from (.1) we see that κ[p (ɛ)] = 1. By the comparison principle we immediately obtain the following necessary condition for the solvability of problem (1.1)-(1.). Lemma 3.. Suppose that f satisfies (1.4) and (1.8), and that there is a hypersurface Σ in H n+1 which satisfies (1.1) and (1.) with κ[σ] K. Then σ < Locally strictly convex hypersurfaces. We now consider hypersurfaces of positive principal curvatures in H n+1 ; we call such hypersurfaces locally strictly convex. Lemma 3.3. Let Σ {x n+1 c} be a locally strictly convex hypersurface of class C in H n+1 with compact (asymptotic) boundary Σ {x n+1 = c} for some constant c 0. Then Σ is a vertical graph. In particular, Σ is is boundary of a bounded domain in {x n+1 = c}. Proof. Let T be the set of t c such that Σ t := Σ {x n+1 t} is a vertical graph and let t 0 be the minimum of T which is clearly nonempty. Suppose t 0 > c. Then there must be a point p Σ t0 with ν n+1 (p) = 0, that is, the normal vector to Σ at p is horizontal. It follows from (.1) that κ E i = κ i /t 0 > 0 for all 1 i n at p. On the other hand, the curve Σ P (near p) clearly has nonpositive curvature at p (with respect to the normal ν(p)), where P is the plane through p spanned by e and ν(p). This is a contradiction, proving that t 0 = c. By the formula (.3) the graph of a function u is locally strictly convex if and only if the function U = x + u is (locally) strictly convex, i.e., its Hessian D U is positive definite. We define the class of admissible functions in a domain Ω R n as (3.3) A(Ω) = { u C (Ω) : u > 0, x + u is locally strictly convex in Ω }. By the convexity of x + u we immediately have (3.4) Du 1 ( ) L + max u u Du Ω

11 HYPERSURFACES OF CONSTANT CURVATURE 11 where L is the diameter of Ω. The following gradient estimate, which improves (3.4) in the sense that it is independent of the (positive) lower bound of u, will be crucial to our results. Lemma 3.4. Let u A(Ω). Then (3.5) e u { } 1 + Du max sup e u, max eu 1 + Du. Ω Ω Proof. If e u 1 + Du attains its maximum at an interior point x 0 Ω then at x 0, u j (δ ij + u i u j + u ij ) = e u (e u 1 + Du x ) = 0, 1 i n. i j If follows that Du(x 0 ) = 0 as the matrix {δ ij + u i u j + u ij } is positive definite. Lemma 3.5. Let u A(Ω) satisfy σ 1 f(κ[u]) σ, (3.6) u ɛ, u = ɛ, in Ω in Ω on Ω where 0 < σ 1 σ < 1, ɛ 0 and Ω C. Suppose f satisfies (1.4), (1.8) and (1.9). Then, for ɛ sufficiently small, 1 (3.7) σ 1 Cɛ(ɛ + 1 σ 1 + Du 1) in Ω. Proof. By Lemma 3.1 we have, for ɛ sufficiently small, (3.8) 1 + Du on Ω. 1 + σ Fix λ > 0 (sufficiently large) such that L 1 σ1 (3.9) ln λ 1 + σ σ where L is the diameter of Ω, and let u λ (x) = 1 u(λx), x Ωλ λ where Ω λ = Ω. Then λ κ[uλ ](x) = κ[u](λx) in Ω λ. It follow that u λ A(Ω λ ) and σ 1 f(κ[u λ ]) σ, in Ω λ (3.10) u λ ɛ, in Ωλ λ u λ = ɛ, on Ωλ λ

12 1 BO GUAN, JOEL SPRUCK, AND MAREK SZAPIEL Applying Lemma 3.1 to u λ, we have by (3.8) and (3.9), or u λ ɛ λ L λ (3.11) sup e uλ Ω λ 1 σ1 1 + σ 1 max Ω λ ln 1 + Du λ in Ω λ max e ε λ 1 + Duλ. Ω λ By (3.11), Lemma 3.4 and Lemma 3.1 (part ii, formula (3.)), (3.1) This proves (3.7) Duλ e(uλ ɛ λ ) min Ω λ Duλ 1 min Ω λ 1 + Duλ σ 1 Cɛ(ɛ + 1 σ 1). 4. Boundary estimates for second derivatives Let Ω be a bounded smooth domain in R n. In this section we establish boundary estimates for second derivatives of admissible solutions to the Dirichlet problem { G(D u, Du, u) = ψ(x, u), in Ω (4.1) u = ɛ, on Ω where G is defined in (.9) and ψ C (Ω R + ). We assume that ψ satisfies the following conditions: (4.) 0 < ψ(x, z) 1 ɛ 1, D x ψ(x, z) + ψ z (x, z) C z z, x Ω δ, z (0, ɛ 1 ) (4.3) ψ(x, z) = σ z, x Ω, z (0, ɛ 1). where C is a large fixed constant, ɛ 1 > 0 is a small fixed constant, δ = ɛ C Ω δ = {x Ω : d(x, Ω) < δ}. Remark 4.1. We have in mind Ψ := uψ(x, u) = σ + M(u v(x)) where M [0, 1] ɛ and v = ɛ on Ω, v C in Ω. We will need this generality because (see (.14)) G u = G u ψ u may be positive in Ω causing us some trouble when we try to prove Theorem 1.3. Note also that conditions (4.), (4.3) imply osc Ωδ Ψ C δ 1 which is ɛ C used at the end of the proof when we need to appeal to Lemma 3.5. and

13 HYPERSURFACES OF CONSTANT CURVATURE 13 Theorem 4.. Let Ω be a bounded domain in R n, Ω C 3, and u C 3 ( Ω) A(Ω) a solution of problem (4.1). Suppose that f satisfies (1.4)-(1.10) and ψ satisfies (4.) and (4.3). Then, if ɛ is sufficiently small, (4.4) u D u C on Ω where C is independent of ɛ. The notation of this section follows that of subsection.1. We first consider the partial linearized operator of G at u: where G st is defined in (.10) and L = G st s t + G s s (4.5) G s := G = u s u s w u F ij a ij ( wu F ij wuk γ sj + u j γ ks ) a ik w w u F ij u i γ sj by the formula (.1) in [10], where F ij = F ij (A v [u]) and a ij = a v ij[u]. Since {F ij } and {a ij } are both positive definite and can be diagonalized simultaneously, we see that (4.6) F ij a ik ξ k ξ j 0, ξ R n. Moreover, by the concavity of f we have the following inequality similar to Lemma.3 in [10] (4.7) G s C (1 + ) F ii. u Since γ sj u s = u j /w, (4.8) G s u s = 1 u It follows from (.10) and (4.8) that {( 1 w 1 )F ij a ij w F ij a ik u k u j + w 3 F ij u i u j }. (4.9) Lu = 1 w u F ij a ij 1 F ii wu w u F ij a ik u k u j + w 3 u F ij u i u j Lemma 4.3. Suppose that f satisfies (1.4), (1.5), (1.8) and (1.9). Then ( (4.10) L 1 ɛ ) ɛ(1 uψ ) ( w 1 + ) F ii u wu 3 in Ω.

14 14 BO GUAN, JOEL SPRUCK, AND MAREK SZAPIEL Proof. By (4.9), (4.6) and (1.9), L 1 u = 1 u Lu + u 3 Gst u s u t (4.11) = 1 u Lu + w 3 u F ij u 3 i u j 1 ( F ii uψ ). wu 3 w Thus (4.10) follows from (1.1) and (4.). Lemma 4.4. For 1 i, j n, (4.1) L(x i u j x j u i ) = (ψ u G u )(x i u j x j u i ) + x i ψ xj x j ψ xi where (4.13) G u := G u = 1 F ii. wu Proof. For θ R let Differentiate the equation y i = x i cos θ x j sin θ y j = x i sin θ + x j cos θ y k = x k, k i, j. G(D u(y), Du(y), u(y)) = ψ(y, u(y)) i with respect to θ and set θ = 0 afterwards. We obtain L(x i u j x j u i ) + G u (x i u j x j u i ) = (L + G u ) u = θ θ=0 θ ψ(y, u(y)) θ=0 which yields (4.1). Proof of Theorem 4.. Consider an arbitrary point on Ω, which we may assume to be the origin of R n and choose the coordinates so that the positive x n axis is the interior normal to Ω at the origin. There exists a uniform constant r > 0 such that Ω B r (0) can be represented as a graph (4.14) x n = ρ(x ) = 1 B αβ x α x β + O( x 3 ), x = (x 1,..., x n 1 ). α,β<n

15 HYPERSURFACES OF CONSTANT CURVATURE 15 Since u = ɛ on Ω, we see that u(x, ρ(x )) = ɛ and (4.15) u αβ (0) = u n ρ αβ α, β < n. Consequently, (4.16) u αβ (0) C Du(0), α, β < n where C depends only on the maximal (Euclidean principal) curvature of Ω. Next, following [] we consider for fixed α < n the operator (4.17) T = α + β<n B αβ (x β n x n β ). We have (4.18) T u C, in Ω B δ (0) T u C x, on Ω B δ since u = ɛ on Ω. By Lemma 4.4 and (4.), (4.7), (4.19) Let L(T u) = T G(D u, Du, u) G u T u = T ψ(x, u) G u T u C ( 1 + F ). ii u ( φ = A 1 ɛ ) + B x ± T u. u By (4.), (4.7), (4.19) and Lemma 4.3, ( (4.0) Lφ ɛ ) 1ɛA (1 + F ii w + ) Cu B(u + δ) + Cu in Ω B u 3 δ. We first choose B = C 1 δ with C 1 = C the constant in (4.18) so that φ 0 on (Ω B δ ). Then choosing A >> C 1 /ɛ 1 makes Lφ 0 in Ω B δ. By the maximum principle φ 0 in Ω B δ. Since φ(0) = 0, we have φ n (0) 0 which gives (4.1) u αn (0) Au n(0) u(0).

16 16 BO GUAN, JOEL SPRUCK, AND MAREK SZAPIEL Finally to estimate u nn (0) we use our hypothesis (1.10). We may assume [u αβ (0)], 1 α, β < n, to be diagonal. Note that u α (0) = 0 for α < n. We have at x = 0 uu 1 + uu n A v [u] = 1 w uu uu... n w w uu n1 uu n uunn w w w By Lemma 1. in [3], if ɛu nn (0) is very large, the eigenvalues λ 1,..., λ n of A v [u] are asymptotically given by (4.) λ α = 1 w (1 + ɛu αα(0)) + o(1), α < n λ n = ɛu ( ( nn(0) 1 )) 1 + O. w 3 ɛu nn (0) By (4.16) and assumptions (1.9)-(1.10), for all ɛ > 0 sufficiently small, ɛψ(0, ɛ) = 1 w F (wav [u](0)) 1 ( 1 + ε ) 0 w if ɛu nn (0) R where R is a uniform constant. By the hypothesis (4.3) and Lemma 3.5, however, σ 1 ( 1 + ε ) ( 0 (σ Cɛ) 1 + ε ) 0 > σ w which is a contradiction. Therefore u nn (0) R ɛ and the proof is complete. 5. Global estimates for second derivatives In this section we prove a maximum principle for the largest hyperbolic principal curvature κ max (x) of solutions of general curvature equations. For later applications we keep track of how the estimates depend on the right hand side of (4.1). We consider (5.1) M(x) = κ max (x) u (x)(ν n+1 (x) a).

17 HYPERSURFACES OF CONSTANT CURVATURE 17 Theorem 5.1. Let u C 4 ( Ω) be a positive solution of f(κ[u]) = Ψ(x, u) where f satisfies (1.4)-(1.9), A v [u] S K, ν n+1 a and Ψ σ 0 > 0. Suppose M achieves its maximum at an interior point x 0 Ω. Then either κ max (x 0 ) 16(a + 1 a ) or (5.) M (x 0 ) C {( Ψ x + Ψ u ) + ( Ψ xx + Ψ ux + Ψ uu )}(x 0 ) u (x 0 ) where C is a controlled constant. If we assume, for example, that (5.3) Ψ σ 0, Ψ x + Ψ u L 1 ɛ, Ψ xx + Ψ ux + Ψ uu L ɛ with L 1, L independent of ɛ, we obtain using Theorem 4. in Ω Theorem 5.. Let u C 4 ( Ω) be a solution of problem (4.1) with A v [u] S K. Suppose that f satisfies (1.4)-(1.9) and ψ satisfies (4.), (5.3). Then if ɛ is sufficiently small, (5.4) u D u C(1 + max Ω u D u ) u where C is independent of ɛ. ɛ in Ω We begin the proof of Theorem 5.1 which is long and computational. Let Σ be the graph of u. For x Ω let κ max (x) be the largest principal curvature of Σ at the point X = (x, u(x)) Σ. We consider (5.5) M 0 = max x Ω κ max (x) φ(η, u) where η = e ν, ν is the upward (Euclidean) unit normal to Σ, and φ a smooth positive function to be chosen later. Suppose that M 0 is attained at an interior point x 0 Ω and let X 0 = (x 0, u(x 0 )). After a horizontal translation of the origin in R n+1, we may write Σ locally near X 0 as a radial graph (5.6) X = e v z, z S n + R n+1 with X 0 = e v(z 0) z 0, z 0 S n +, such that ν(x 0 ) = z 0. In the rest of this section we shall follow the notation in subsection. and rewrite the equation in (4.1) in the form (5.7) F (A s [v]) = Ψ uψ

18 18 BO GUAN, JOEL SPRUCK, AND MAREK SZAPIEL where A s [v] is given in (.17); henceforth we write A[v] = A s [v] and a ij = a s ij[v]. We choose an orthnormal local frame τ 1,..., τ n around z 0 on S n + such that v ij (z 0 ) is diagonal. Since ν(x 0 ) = z 0, v(z 0 ) = 0 and, by (.17), at z 0, (5.8) a ij = yv ij = κ i δ ij where κ 1,..., κ n are the principal curvatures of Σ at X 0. We may assume (5.9) κ 1 = κ max (X 0 ). The function a 11, which is defined locally near z φ 0, then achieves its maximum at z 0 where, therefore ( a11 ) (5.10) = 0, 1 i n φ i and (5.11) F ii ( a11 φ Proposition 5.3. At z 0, ) y φf ii a 11,ii y κ 1 F ii φ ii ii = 1 φ F ii a 11,ii κ 1 φ F ii φ ii 0. = y φf ii a ii,11 + y(φ η κ 1 φ)f ii y j a ii,j yφf ii y 1 a ii,1 + (yφ η φ)κ 1 fi κ i φ ηη κ 1 fi (y κ i ) y i (5.1) + (φ(κ 1 + y 1 + 1) y e v φ u κ 1 (1 + y )φ η κ 1 ) f i κ i + (yφ η + ye v φ u (1 + y1)φ)κ 1 fi + (yφ η + ye v φ u y e v φ uu y e v φ uη φ)κ 1 fi yi + (φ φ η κ 1 + ye v φ uη κ 1 ) f i κ i yi. Proof. In what follows all calculations are evaluated at z 0. Since v = 0, we have (5.13) w = 1, w i = 0, w ij = v ki v kj (recall w = 1 + v ). Straightforward calculations show that (5.14) a ij,k = yv ijk + y k v ij (e v) k δ ij (5.15) a kk,ii = yv kkii + y i v kki + y ii v kk (e v) ii yv kk v ii yv 3 kkδ ki.

19 HYPERSURFACES OF CONSTANT CURVATURE 19 Therefore, a 11,ii a ii,11 = y(v 11ii v ii11 v 11 vii + v ii v11) (5.16) + y ii v 11 y 11 v ii + (y i v 11i y 1 v ii1 ) + (e v) 11 (e v) ii. We recall the following formulas (5.17) v ijk = v ikj = v kij (5.18) v kkii = v iikk + (v kk v ii ) (where we have used the fact that v = 0) and, from [8], (5.19) y j = 1 y (5.0) y ij = yδ ij (5.1) (e v) i = y i v ii (5.) By (5.14) and (5.8), (e v) ij = e τ k v kij yv ij e τ j v i = y k v ijk yv ij y j v i. (5.3) v iij = a ii,j y (5.4) (e v) ii = y ja ii,j y y jκ i y + y jκ j y (1 + 1 )κ y i + 1 κ y j yj. Plug these formulas into (5.16) and note that F ij = f i δ ij. We obtain y F ii a 11,ii = y F ii a ii,11 yf ii (y j a ii,j + y 1 a ii,1 ) j (5.5) + yf ii (y j a 11,j + y i a 11,i ) κ 1 fi κ i + (κ 1 + y 1 + 1) f i κ i (1 + y 1)κ 1 fi + f i κ i y i κ 1 fi y i. Next, recall that u = ye v and η := e ν = y e v w. At z 0 we have η = y, (5.6) η i = y i (e v) i = y i (1 v ii )

20 0 BO GUAN, JOEL SPRUCK, AND MAREK SZAPIEL (5.7) η ii = y ii (e v) ii yvii ( = y κ i y )κ y i 1 (yy y j a ii,j + κ j yj ) j and (5.8) u i = e v y i, u ii = e v (κ i y). We have y F ii φ ii = y F ii (φ η η ii + φ ηη η i + φ uη u i η i + φ uu u i + φ u u ii ) = yφ η fi κ i + (y e v φ u + (1 + y )φ η ) f i κ i (5.9) (y 3 φ η + y 3 e v φ u + κ j y j φ η ) f i + φ ηη fi y i (y κ i ) + (y e v φ uu + y e v φ uη ) f i y i ye v φ uη fi κ i y i yφ η yyj F ii a ii,j. By (5.10), ( (5.30) a 11,i φ = κ 1 φ i = κ 1 (φ η η i + e v φ u y i ) = κ 1 φ η 1 κ ) i y i + e v φ u κ 1 y i. y Using (5.30) we have yφf ii (y j a 11,j + y i a 11,i ) = (5.31) (yφ η + ye v φ u )κ 1 fi yi φ η κ 1 fi κ i yi ( ) + y(1 y )φ η + y(1 y )e v φ u φ η κj yj κ 1 fi. Combining (5.11), (5.5), (5.9) and (5.31), we obtain (5.1). Lemma 5.4. ( y F ii a ii,11 yf ii φη ) y 1 a ii,1 + y (5.3) φ κ 1 1 F ii y j a ii,j y F ij,kl a ij,1 a kl,1 C{uκ 1 ( Ψ x + Ψ u ) + u ( Ψ xx + Ψ ux + Ψ uu )} where C depends on an upper bound for φη φ.

21 HYPERSURFACES OF CONSTANT CURVATURE 1 Proof. Since X = e v z, x = e v (z ye) and u = e v y. Hence, τj x = e v (τ j y j e) + v j x (5.33) τj Ψ(x, u) = e v (Ψ x (τ j y j e) + Ψ u (yv j + y j )) + (x Ψ x )v j τ1 τ 1 Ψ(x, u) = e v (D xψ(τ 1 y 1 e)(τ 1 y 1 e) y 1 Ψ ux (τ 1 y 1 e) + Ψ uu y 1) + e v (Ψ x ( τ1 τ 1 y 11 e) + Ψ u (κ 1 + y 11 )) + (x Ψ x )v 11. Using (5.33) and differentiating equation (5.7) twice gives (5.34) yf ii y j a ii,j = u(ψ x (y j τ j (1 y )e) + Ψ u (1 y )) yf ii y 1 a ii,1 = u(ψ x (y 1 τ 1 y1e) + Ψ u y1) y F ii a ii,11 = u (DxΨ(τ 1 y 1 e) (τ 1 y 1 e) + y 1 Ψ ux (τ 1 y 1 e) + Ψ uu y1) + yu(ψ x ( τ1 τ 1 + ye) + Ψ u (κ 1 y)) + (x Ψ x )yκ 1 y F ij,kl a ij,1 a kl,1. Formula (5.3) follows immediately from (5.34). We now make the choice φ(η, u) = (η a)u where 0 < a η/. We have (y a)φ η = φ, φ ηη = 0, uφ u = u φ uu = φ, u(y a)φ uη = φ. By Proposition 5.3 and Lemma 5.4, for κ 1 16(a + 1 a ) (5.35) y (y a)f ij,kl a ij,1 a kl,1 + aκ 1 fi κ i + a σ 0κ 1 +(a + y (y a))κ 1 fi + (κ 1 + y a) f i (κ i y)yi + y(y a) f i yi C{uκ 1 ( Ψ x + Ψ u ) + u ( Ψ xx + Ψ ux + Ψ uu )}. Let 0 < θ < 1 (to be chosen in a moment) and set I = {i : κ i θκ 1 }, J = {i : θκ 1 < κ i y, f i < θ 1 f 1 }, K = {i : θκ 1 < κ i y, f i θ 1 f 1 }, L = {i : κ i > y}. Note that for i L, all the terms on the left hand side of (5.35)are nonnegative.

22 BO GUAN, JOEL SPRUCK, AND MAREK SZAPIEL We have (provided that θκ 1 1), f i κ i 1 f i (κ i + θ κ 1) i I i I (5.36) θκ 1 f i ( κ i + θκ 1 ) i I θκ 1 f i ( κ i + y)yi, and (5.37) i I f i (κ i y)yi κ 1 f 1. i J According to Andrews [1] and Gerhardt [6] (see also [18], Lemma 3.1 and [16]), (5.38) F ij,kl a ij,1 a kl,1 f i f j n a f i f 1 ij,1 a κ j κ i κ i j i= 1 κ i1,1. i By (5.14) and (5.30), Therefore, Note that (5.39) It follows that ya i1,1 = ya 11,i + κ i y i κ 1 y i = (5.40) y (y a)f ij,kl a ij,1 a kl,1 ( κ i + κ 1 + y κ ) i y a κ 1 y i. y a i1,1 (1 θ)κ 1 (y κ i )yi, i K. y a f i f 1 κ 1 κ i f i θf i κ 1 + θκ 1 = (1 θ)f i (1 + θ)κ 1, i K. 4(1 θ) 1 + θ κ 1 f i (κ i y)yi. We now fix θ such that 4(1 θ) + θ. 1 + θ For example, we can choose θ = 1. From (5.36), (5.37) and (5.40) we obtain 6 (5.41) y (y a)f ij,kl a ij,1 a kl,1 + aκ 1 fi κ i + (κ 1 + y a) f i (κ i y)y i 0 provided that κ 1 16(a + 1 ). Consequently, a aσ 0 (5.4) κ 1 C{( Ψ x + Ψ u )uκ 1 + u ( Ψ xx + Ψ ux + Ψ uu )} i K

23 HYPERSURFACES OF CONSTANT CURVATURE 3 Formula (5.) follows easily from (5.4) completing the proof of Theorem Existence: Proof of Theorems 1.3 and 1. In order to prove Theorem 1.3 we will construct a monotone sequence {u k } of admissible functions satisfying (1.) in Ω starting from u 0 ɛ. Having found u 0 u 1... u k, u = u k+1 is a solution of the Dirichlet problem G(D u, Du, u) = 1 ( σ + 1 ) (6.1) u ɛ (u u k) ψ(x, u) in Ω. u = ɛ on Ω In order to solve (6.1) we use a continuity method for u = u t, 0 t 1: G(D u, Du, u) = 1 ( σ + 1 ) u ɛ (u (tu k + (1 t)u k 1 )) in Ω, k 1 (6.) G(D u, Du, u) = 1 (t(σ 1) + 1 ) u ɛ u in Ω, k = 0 u = ɛ on Ω where u A k = {u u k and u admissible} and u 0 = u k. Since u is admissible we have from section 3 that u C 1 Ω C for a uniform constant C. Now according to (.14), G u ψ u 1 (σ 1 (6.3) u w 1 ) ɛ (tu k + (1 t)u k 1 ) 1 σ + 1 C, k 1 u G u ψ u 1 ( t(σ 1) 1 ) 1 u w Cu, k = 0. Hence for Ω C +α, the linearized operator for u t is invertible and the set of t for which (6.) is solvable is open. In particular, (6.) is solvable for 0 t t 0. Using standard regularity theory for concave fully nonlinear operators, to show the closedness of this set, it suffice to show u C (Ω) C for a uniform constant C for t 0 t 1. Observe that Ψ(x, u) = uψ(x, u) = σ + 1(u (tu ɛ k + (1 t)u k 1 )) satisfies the conditions (4.), (5.3) of Theorem 5.. Hence we obtain an estimate sup D u C k Ω ɛ 3 where C k depends on k but is independent of t. Therefore (6.) is solvable for all 0 t 1 and so we have found a monotone increasing sequence of solutions to (6.1).

24 4 BO GUAN, JOEL SPRUCK, AND MAREK SZAPIEL It remains to show that the sequence {u k } converges to a solution of (1.16). For this we need second derivative estimates independent of k. Define M k (x) = κ max (x) u k (x)(νn+1 k (x) a). If M k (x) achieves its maximum on Ω, then according to Theorem 4., M k (x) C ɛ where C is independent of k and ɛ (see Remark 4.1). Otherwise applying Theorem 5.1 with Ψ(x, u) = σ + 1 ɛ (u u k) we obtain (6.4) M k+1 C ɛ 4 + C ɛ M k C ɛ M k where C is independent of k and ɛ. Iterating (6.4) gives M k C ɛ M 1 C ɛ 4. It follows that the sequence u k converges uniformly in C +α (Ω) and the proof of Theorem 1.3 is complete. To finish the proof of Theorem 1. we need to show that for σ > 1, we can obtain 8 an estimate for sup Ω κ max which is independent of ɛ as ɛ tends to zero. As in Section 5 we define M 0 = max x Ω κ max (x) φ(η, u). We now choose φ = η a, where inf η > a. If M 0 is achieved on Ω, then we obtain a uniform bound by Theorem 5.1. Otherwise at an interior maximum, Proposition 5.3 and Lemma 5.4 gives (6.5) σ(y a)κ 1 + (a (1 y )(y a))κ 1 fi 4σκ 1 where we have dropped some positive terms from the left hand side of (6.5) and used fi κ i y i σ. From (6.5) we see that we must find the minimum of the function (6.6) γ(y) = a (1 y )(y a) = y 3 ay y + 3a on [a, 1]. We have (6.7) γ (y) = (3y ay 1) γ (y) = 4(3y a)

25 HYPERSURFACES OF CONSTANT CURVATURE 5 The unique critical point of γ(y) in (a, 1) is y = a+ a +3 3 and some computation shows that γ(y ) = 7 3 a 4 7 a3 4 7 (a + 3) 3. It is also not difficult to see that γ(y ) < a = γ(a) = γ(1). We claim γ(y ) > 0 if a > 1. This is equivalent to showing 8 4(a + 3) 3 < a(63 4a ) which after squaring both sides is in turn equivalent to a a + 3 = (a 1 8 )(a 16 3 ) < 0. Thus our claim follows. Now suppose ε 0 = σ 1 > 0 and set 8 a = 1 + ε 8 0. Then σ a = ε 0 σ + a > ε 0 σ. According to Lemma 3.5 (see formula (3.7)), η σ Cɛ for a uniform constant C if ɛ is sufficiently small. Hence if Cɛ ε 0, 4σ η a (σ a) Cε ε 0 σ Cε > ε 0 4σ. Returning to formula (6.5) we find ε 0 4 κ 1 4σκ 1 or κ 1 16σ = 16σ 1 ε 0. 8 σ The proof of Theorem 1. is complete. References [1] B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. PDE (1994), [] L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations I. Monge-Ampère equations, Comm. Pure Applied Math. 37 (1984), [3] L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations III: Functions of eigenvalues of the Hessians, Acta Math. 155 (1985), [4] L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations IV. Starshaped compact Weingarten hypersurfaces, Current Topics in P.D.E. Kinokunize Co., Tokyo 1986, 1-6 (Y. Ohya, et al. editors).

26 6 BO GUAN, JOEL SPRUCK, AND MAREK SZAPIEL [5] L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations V. The Dirichlet problem for Weingarten hypersurfaces, Comm. Pure Applied Math. 41 (1988), [6] C. Gerhardt, Closed Weingarten hypersurfaces in Riemannian manifolds, J. Differential Geom. 43 (1996), [7] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Springer- Verlag [8] B. Guan and J. Spruck, Hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity, Amer. J. Math. 1 (000), [9] B. Guan and J. Spruck, The existence of hypersurfaces of constant Gauss curvature with prescribed boundary, J. Differential Geom. 6 (00), [10] B. Guan and J. Spruck, Locally convex hypersurfaces of constant curvature with boundary, Comm. Pure Appl. Math. 57 (004), [11] B. Guan and J. Spruck, Hypersurfaces of constant curvature in Hyperbolic space II, preprint. [1] M. A. Krasnosel skii and P. P. Zabreiko, Geometrical methods of nonlinear analysis, Springer- Verlag, 1984, ISBN [13] N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk. SSSR Ser. Mat. 47 (1983), (Russian); English translation in Math. USSR Izv. (1984), pp [14] B. Nelli and J. Spruck, On existence and uniqueness of constant mean curvature hypersurfaces in hyperbolic space, Geometric Analysis and the Calculus of Variations, Internat. Press, Cambridge, MA, 1996, [15] H. Rosenberg, and J. Spruck, On the existence of convex hypersurfaces of constant Gauss curvature in hyperbolic space, J. Differential Geom. 40 (1994), [16] W.-M. Sheng, J. Urbas and X.-J. Wang, Interior curvature bounds for a class of curvature equations, Duke Math. J. 13 (004), [17] J. Spruck, Geometric aspects of the theory of fully nonlinear elliptic equations, Global Theory of Minimal Surfaces, Clay Mathematics Proceedings Vol. 3 (005), [18] J. Urbas, Hessian equations on compact Riemannian manifolds, Nonlinear Problems in Mathematical Physics and Related Topics, II, , Kluwer/Plenum, New York, 00. Department of Mathematics, Ohio State University, Columbus, OH address: guan@math.osu.edu Department of Mathematics, Johns Hopkins University, Baltimore, MD 118 address: js@math.jhu.edu