j. differential geometry 62 (2002) 259-287

THE EXISTENCE OF HYPERSURFACES OF

CONSTANT GAUSS CURVATURE WITH PRESCRIBED BOUNDARY

BO GUAN & JOEL SPRUCK

Abstract

We are concerned with the problem of finding hypersurfaces of constant Gauss curvature (K-hypersurfaces) with prescribed boundary Γ in R^{n+1}, using the theory of Monge-Ampère equations. We prove that if Γ bounds a suitable locally convex hypersurface Σ, then Γ bounds a locally convex K-hypersurface. The major difficulty lies in the lack of a global coordinate system to reduce the problem to solving a fixed Dirichlet problem of Monge-Ampère type. In order to overcome this difficulty we introduced a Perron method to deform (lift) Σ to a solution. The success of this method is due to some important properties of locally convex hypersurfaces, which are of independent interest. The regularity of the resulting hypersurfaces is also studied and some interesting applications are given.

1. Introduction

In this paper we are concerned with the problem of finding hypersurfaces of constant Gauss-Kronecker curvature (K-hypersurfaces) in ℝ^{n+1} (n ≥ 2) with prescribed boundary: given a disjoint collection Γ = {Γ_{1}, …, Γ_{m}} of closed smooth embedded (n − 1) dimensional submanifolds of ℝ^{n+1}, decide whether there exist (immersed) K-hypersurfaces M in ℝ^{n+1} with ∂M = Γ. Locally this problem reduces to questions concerning Monge-Ampère type equations and we seek solutions for which the resulting equation is elliptic. This means that we must confine ourselves to the class of locally strictly convex hypersurfaces, i.e., those whose principal curvatures are all positive. Such

Both authors were partially supported by NSF grants. Received 10/29/2002.

hypersurfaces locally lie on one side of their tangent planes at any point but need not do so globally as they have nonempty boundary.

Finding hypersurfaces with prescribed curvature and boundary has been a major challenge in geometric analysis because of the highly nonlinear nature of the problem and the lack of variational methods. Beginning around 1980, some success was achieved due to breakthroughs in the theory of Monge-Ampère equations and general fully nonlinear equations, but only for hypersurfaces which are globally graphs of functions over domains with geometric restrictions (e.g., strictly convex domains). This was the case even for K-hypersurfaces where the only general existence results were consequences of the existence theory for Monge-Ampère equations (see [6], [20], [23]), and was restricted to strictly convex domains. This means that the resulting surfaces must be simply connected graphs, a very strong restriction geometrically.

The first idea that a more general result was possible came in the paper of Hoffman-Rosenberg-Spruck [18] and subsequently such a general result was developed in [12] and [11]. In these papers, the authors proved an essentially optimal existence theorem for Monge-Ampère equations in domains of arbitrary geometry and thus the limit of our understanding of K-hypersurfaces with boundary was reached, as far as global graphs (including multi-sheeted radial graphs) are concerned. This theory already led to striking geometric applications [12], [28], [8].

To solve the problem in its full parametric generality seemed to require substantial new techniques. A necessary condition for Γ to bound a locally strictly convex hypersurface is that its second fundamental form (as a submanifold of ℝ^{n+1}) is nondegenerate everywhere. This however, is not a sufficient condition; Rosenberg [27] (see also [10]) shows there are topological obstructions. It is natural to seek geometric conditions that guarantee the existence of locally strictly convex K-hypersurfaces spanning a given Γ. Based on the results in [12], the second author [29] made the following conjecture: Γ must bound an immersed K-hypersurface if it bounds a locally strictly convex immersed hypersurface. The first main result of the present paper settles this conjecture affirmatively. More precisely, we will prove:

Theorem 1.1. Assume that there exists a locally convex immersed hypersurface Σ in ℝ^{n+1} with ∂Σ = Γ and K_{Σ} ≥ K everywhere, where K is a positive constant. Suppose, in addition, that Σ is C^{2} and locally strictly convex along its boundary. Then there exists a smooth (up to the boundary) locally strictly convex immersed hypersurface M with ∂M = Γ such that K_{M} ≡ K. Moreover, M is homeomorphic to Σ.

We note that this is a huge jump in generality from our previous results in [12] as it deals with general immersed K-hypersurfaces and not just graphs (or radial graphs). Because of the presence of boundary, locally convex surfaces can be very complicated. In particular, in Theorem 1.1 M need not be embedded even if Σ is embedded.

It is also important to understand hypersurfaces of vanishing Gauss curvature. These hypersurfaces are clearly related to convex hulls of codimension 2 submanifolds in space. Our second main result in this article is the following

Theorem 1.2. Suppose Γ bounds a locally convex hypersurface which is C^{2} and locally strictly convex along its boundary. Then there exists a locally convex hypersurface M of Gauss curvature K_{M} ≡ 0 with ∂M = Γ, and M is of class C^{1,1} up to the boundary. Moreover, for any interior point p ∈ M, all the extreme points of the (intrinsic) component of M ∩ T_{p}M containing p lie on ∂M, where T_{p}M denotes the tangent plane of M at p. In particular, if Γ is extreme, i.e., Γ lies on the boundary of its convex hull, then M coincides with part of the boundary of the convex hull of Γ and, therefore, is globally convex.

The C^{1,1} regularity in Theorem 1.2 is optimal for hypersurfaces of vanishing Gauss curvature, as shown by counterexamples (see [7]). We also remark that Theorem 1.2 does not hold without the assumption that Γ bounds a locally convex hypersurface which is locally strictly convex near its boundary. Ghomi [8] has constructed a smooth extreme Jordan curve γ in ℝ^{3} with the properties that:

(a) γ bounds a convex surface of vanishing Gauss curvature which is not C^{1,1},

(b) γ does not bound any locally strictly convex surface, and

(c) γ does not bound any locally convex surface of class C^{1,1} with vanishing Gauss curvature.

As a consequence of Theorem 1.2 we have

Corollary 1.3. Suppose Γ is extreme and let Σ be a locally convex hypersurface with ∂Σ = Γ. If Σ is C^{2} up to the boundary and locally strictly convex in a neighborhood of its boundary, then the interior of Σ lies strictly outside the convex hull of Γ.

We remark that such a hypersurface need not be globally convex, nor embedded. A somewhat stronger version of Corollary 1.3 has been proved by Alexander-Ghomi [1]. In [8], Ghomi made the following conjecture: every compact connected hypersurface of positive curvature with connected extreme boundary is embedded and its interior lies outside the convex hull of its boundary. We see that Corollary 1.3 settles affirmatively part of this conjecture. On the other hand, we will construct an example which shows such a hypersurface may fail to be embedded. Furthermore, using the bridge principle of Hauswirth [16], we will show there exist smooth K-surfaces in ℝ^{3}, with connected extreme boundary, which are not embedded.

Suppose Γ is extreme and let H_{Γ} be the boundary of its convex hull. Theorem 1.2 indicates that if Γ bounds a locally convex hypersurface which is C^{2} (up to the boundary) and locally strictly convex in a neighborhood of its boundary, then one of the components of H_{Γ} \ Γ must be C^{1,1} up to the boundary. However, as we will show by an example, the other components may have interior singularities. A result of Ghomi [8] states that every component of H_{Γ} \ Γ is C^{1,1} up to the boundary if Γ is strictly convex, i.e., through every point of Γ there passes a (global) supporting hyperplane with first order contact.

Hypersurfaces of vanishing Gauss curvature are closely related to the homogeneous degenerate Monge-Ampère equation

det D^{2}u = 0. (1.1)

In general, the Dirichlet problem for (1.1), even with smooth boundary data, does not have C^{2} solutions, as shown by an example of Urbas (see [7]). Under suitable regularity assumptions on the boundary data, the interior and global C^{1,1} regularity was established by Trudinger-Urbas [30] and Caffarelli-Nirenberg-Spruck [7], respectively, for strictly convex domains. Later the first author [11] extended the global regularity result of [7] to non-convex domains. These regularity results will play important role in our proof of Theorem 1.2. For more general (nonhomogeneous) degenerate Monge-Ampère equations, the C^{1,1} regularity has been studied by Caffarelli-Kohn-Nirenberg-Spruck [5], Hong [19], Krylov [24], P.-F. Guan [13] and Guan-Trudinger-Wang [14], etc.

A major difficulty in proving Theorems 1.1 and 1.2 lies in the lack of global coordinate systems to reduce the problem to solving certain boundary value problem for Monge-Ampère type equations. To overcome this difficulty, we adopt a Perron method to deform (lift) Σ into a K-hypersurface by solving the Dirichlet problem for the Gauss curvature equation (2.1) locally. This approach, while classical for PDE’s, requires substantial technical work as we are dealing with general locally convex hypersurfaces in space. A key ingredient, among others, is an a priori estimate for the local Lipschitz constants (C^{0,1} norms) of locally convex hypersurfaces spanning Γ. This is established in Section 3 where we also derive a priori estimates for the lower and upper bounds of principal curvatures of locally strictly convex K-hypersurfaces spanning Γ. The Perron method is carried out in Section 4 where we define the deformation space L of liftings of Σ and construct M as the limit of a suitable sequence of hypersurfaces in L. In Section 5, we study the regularity of the resulting hypersurface constructed in Section 4, to complete the proofs of Theorems 1.1 and 1.2. Finally, in Section 6 we prove Corollary 1.3 and construct an extreme curve in ℝ^{3} which bounds a locally strictly convex K-surface with self-intersection and for which the boundary of its convex hull has interior singularities.

For general Monge-Ampère equations, there is a vast literature, with fundamental contributions from Pogorelov, Cheng-Yau, Lions, Ivochkina, Krylov, Caffarelli-Nirenberg-Spruck, Trudinger, Urbas and others in the 1970-1980’s, and that of Caffarelli [2], [3] on the regularity theory. For further references the reader is referred to [9], [15] and the expository article [25].

An earlier version of this article was circulated as a preprint starting March 2001. At about the same time Trudinger-Wang [31] independently proved Theorem 1.1.

2. Notation and preliminaries

Let Φ : Σ_{0}^{n} → ℝ^{n+1} be an immersion where Σ_{0} is a manifold of dimension n ≥ 2 with boundary ∂Σ_{0} which may be empty. We will often identify Φ with its image Σ := Φ(Σ_{0}) and call Σ a hypersurface of ℝ^{n+1}. Similarly, the boundary of Σ, ∂Σ, means the immersion Φ : ∂Σ_{0} → ℝ^{n+1}. When we consider a point p ∈ Σ, it should be understood as one of its preimages in Σ_{0}. For a subset U of ℝ^{n+1}, Σ∩_{p}U will denote the component of Σ ∩ U that contains p, that is, Σ ∩_{p} U = Φ(U_{0}) where U_{0} is the component of Φ^{−1}(Σ ∩ U) that contains the point identified to p in Φ^{−1}(p) ⊂ Σ_{0}. In this paper, all hypersurfaces in ℝ^{n+1} we consider are assumed to be connected, orientable and compact with or without boundary. Unless otherwise indicated, if two hypersurfaces have the same boundary, they are assumed to be oriented in such a way that they induce the same orientation on the boundary.

Let Σ be a C^{2} hypersurface in ℝ^{n+1}. We will use K_{Σ}, ν_{Σ} and d_{Σ} to denote the Gauss curvature, the unit normal vector field, and the extrinsic diameter of Σ, respectively. The orientation of Σ is assumed to be consistent with ν_{Σ} which is continuously defined on entire Σ. At a point on Σ the Gauss curvature K_{Σ} is the product of the principal curvatures which are the eigenvalues of the second fundamental form of Σ computed with respect to ν_{Σ}. We denote by κ_{min}[Σ] and κ_{max}[Σ] the minimum and maximum, respectively, of all principal curvatures of Σ. We say Σ is locally convex (locally strictly convex) if κ_{min}[Σ] ≥ 0 (κ_{min}[Σ] > 0, respectively).

We will also need to consider hypersurfaces with less regularity. In general, a hypersurface Σ in ℝ^{n+1} is said to be locally convex if at every point p ∈ Σ there exists a neighborhood which is the graph of a convex function x_{n+1} = u(x), x ∈ ℝ^{n}, for a suitable coordinate system in ℝ^{n+1}, such that locally the region x_{n+1} ≥ u(x) always lies on a fixed side of Σ. (Note that Σ is assumed to be orientable so it has two sides; for convenience we will refer to the inner side as the one facing x_{n+1} ≥ u(x).) The latter requirement that the region x_{n+1} ≥ u(x) lie on one fixed side of Σ is to ensure that the local convexity at each point is consistent with a fixed orientation; see [1] for a detailed discussion. Note that a locally convex hypersurface is necessarily of class C^{0,1} in the interior.

For a locally convex hypersurface Σ which is not necessarily C^{1}, ν_{Σ} is understood as the Gauss map from Σ to the subsets of 𝕊^{n}: for a point p ∈ Σ, ν_{Σ}(p) is the set of all unit normal vectors of local supporting hyperplanes of Σ at p. For convenience, we will say ν_{Σ} has a certain property of a vector if every element of ν_{Σ} has that property. For the definition in weak sense of Gauss curvature we refer to [26]. According to Caffarelli [2], if Σ is the graph of a locally convex function x_{n+1} = u(x) over a domain Ω in ℝ^{n} then K_{Σ} = K if and only if u is a viscosity solution of the Gauss curvature equation

det(u_{ij}) = K(1 + |∇u|^{2})^{\frac{n+2}{2}} in Ω. (2.1)

One can similarly interpret the meanings of K_{Σ} ≤ K and K_{Σ} ≥ K. We will need the following existence result which follows from, for example, Theorem 1.1 of [11] by approximation.

Theorem 2.1. Let Ω be a bounded domain in ℝ^{n} with ∂Ω ∈ C^{0,1}. Suppose there exists a locally convex viscosity subsolution \underline{u} ∈ C^{0,1}(\overline{Ω}) of (2.1), i.e.,

det(\underline{u}_{ij}) ≥ K(1 + |∇\underline{u}|^{2})^{\frac{n+2}{2}} in Ω,

where K ≥ 0 is a constant. Then there exists a unique locally convex viscosity solution u ∈ C^{0,1}(\overline{Ω}) of (2.1) satisfying u = \underline{u} on ∂Ω.

3. A priori estimates and compactness

In this section we prove some important local properties of locally convex hypersurfaces with boundary. Throughout the section, let Σ and M be locally convex hypersurfaces in ℝ^{n+1} with ∂Σ = ∂M and assume that there exists a fixed constant δ > 0 such that the hypersurface

Σ_{δ} := {x ∈ Σ : dist_{Σ}(x, ∂Σ) < δ}

is C^{2} up to the boundary and locally strictly convex, where dist_{Σ} denotes the intrinsic distance on Σ. We furthermore assume that M locally lies on the inner side of Σ along the boundary and any neighborhood of ∂M in M does not intersect Σ_{δ} in the interior. By this we mean that ν_{Σ}(p) · (q − p) > 0 for all p ∈ Σ_{δ} and q ∈ M near ∂M. In particular, both Σ and M locally lie on the same side of the tangent plane to Σ at any point of ∂Σ. Let Π denote the second fundamental form of ∂Σ (as a submanifold of ℝ^{n+1}). The main result of this section, which plays a key role in our proof of Theorems 1.1 and 1.2 and is of independent interest, may be stated as follows.

Theorem 3.1. At every point on M, locally M can be represented as the graph of a convex function u defined in a domain Ω ⊂ ℝ^{n} of a fixed lower bound in size (depending only on δ, κ_{min}[Σ_{δ}], κ_{max}[Σ_{δ}], max_{∂Σ} |Π| and d_{M}) such that

∥u∥_{C^{0,1}(\overline{Ω})} ≤ C_{1} (3.1)

where C_{1} depends on δ, κ_{min}[Σ_{δ}], κ_{max}[Σ_{δ}], max_{∂Σ} |Π| and d_{M}.

More precisely, that Ω is of a lower bound in size means that there exists some constant δ_{0} > 0 such that Ω contains a ball of radius δ_{0} or a portion of a ball of radius δ_{0} separated by a smooth hypersurface (in ℝ^{n}) with controlled geometric quantities, with the center of the ball in \overline{Ω}.

Proof.

Step 1. We first note the following simple fact. Let p be an arbitrary point on ∂Σ and X a unit tangent vector to ∂Σ at p . Since Σ is locally strictly convex near ∂Σ, we have

ν_{Σ}(p) · Π(X, X) ≥ κ_{min}[Σ_{δ}] > 0. (3.2)

Throughout this proof let β = \frac{1}{2} sin^{−1}(κ_{min}[Σ_{δ}]/ max_{∂Σ} |Π|). Then

(3.3) \frac{ν_{Σ}(p) · Π(X, X)}{|Π(X, X)|} ≥ \frac{κ_{min}[Σ_{δ}]}{max_{∂Σ} |Π|} = sin 2β > 0.

Thus the angle between ν_{Σ}(p) and Π(X, X) does not exceed \frac{π}{2} − 2β.

Now, for a fixed point p ∈ ∂Σ, we take p to be the origin and choose a coordinate system of ℝ^{n+1} such that e_{n} and e_{n+1} are normal to ∂Σ at p and

ν_{Σ}(p) = e_{n} cos β + e_{n+1} sin β. (3.4)

Here e_{k} is the unit vector in the positive x_{k}-axis direction (1 ≤ k ≤ n + 1). For later reference we will call this the special coordinate system at p. It follows that Σ (locally at p) can be represented as the graph of a strictly convex function x_{n+1} = \underline{u}(x) over a domain Ω′ with a lower bound in size which depends on δ, β and κ_{max}[Σ_{δ}]. In particular, ∂Σ is locally a graph over a portion, which we denote as Γ′, of ∂Ω′. By (3.3) and (3.4), the angle between e_{n} and Π(X, X) does not excess \frac{π}{2} − β, that is

(3.5) e_{n} · Π(X, X) ≥ |Π(X, X)| sin β ≥ κ_{min}[Σ_{δ}] sin β

for any unit tangent vector X to ∂Σ at p. Consequently, (possibly after a rotation of the (x_{1}, …, x_{n−1}) coordinates) we may represent Γ′ as a graph

x_{n} = φ(x′) ≡ ∑_{i=1}^{n−1} a_{i}x_{i}^{2} + o(|x′|^{2}), x′ = (x_{1}, …, x_{n−1}) ∈ ℝ^{n−1} (3.6)

for some constants a_{i}, 1 ≤ i ≤ n − 1, satisfying

(3.7) 0 < κ_{min}[Σ_{δ}] sin β ≤ a_{i} ≤ max_{∂Σ} |Π|, 1 ≤ i ≤ n − 1.

By shrinking the size of Ω′ as necessary, we may assume Ω′ = {φ < x_{n} < 2r} for some uniform constant r > 0.

Let v be the convex function defined on \overline{Ω′} by

(3.8) v(x) = sup\{L(x) : L is an affine function, L ≤ \underline{u} on Γ′\}.

We have

\underline{u} ≤ v ≤ max_{Γ′} \underline{u} in \overline{Ω′}, v = \underline{u} on Γ′

and

Lip_{Ω′}(v) ≤ max_{Γ′} |∇\underline{u}| + C

where Lip_{Ω′}(v) denotes the Lipschitz coefficient of v on Ω′.

By the local convexity of M we have ν_{M} · Π(X, X) ≥ 0 for any tangent vector X to ∂Σ. From (3.3) we see that the angle between ν_{M} and ν_{Σ} at any point on ∂Σ does not exceed π − 2β. Therefore,

ν_{M}(p) · e_{n+1} ≥ sin β.

That is, the angle between ν_{M}(p) and e_{n+1} does not exceed \frac{π}{2} − β. Consequently, M locally (near p) can be represented as the graph of a convex function x_{n+1} = u(x). Since M is locally convex, we see that u is defined on a smooth strictly convex domain Ω_{p} satisfying

{φ < x_{n} < r} ⊂ Ω_{p} ⊂ {φ < x_{n} < 2r} (3.9)

with

(3.10) \underline{u} ≤ u ≤ v in \overline{Ω_{p}} and Lip_{Ω_{p}}(u) ≤ C

where C depends on r and ∥\underline{u}∥_{C^{1}(Ω′)}. For later reference we set

∂′Ω_{p} := {x_{n} = φ|0 ≤ x_{n} ≤ r} ⊂ ∂Ω_{p}

and by Γ(p) the graph of u over ∂′Ω_{p}. Note that Γ(p) ⊂ ∂M.

Step 2. Next, let q be an interior point of M. We will consider two different cases. We first assume that there exists a hyperplane P through q, which either is a local supporting hyperplane or is transversal to M at q, such that

(3.11) U_{t} ∩ ∂M = ∅ for all t > 0 sufficiently small

where U_{t} = M ∩_{q} {z ∈ ℝ^{n+1} : (z − q) · ν_{P} ≤ t}. We first note the following fact which will often be used in the sequel without being explicitly referred to.

Lemma 3.2. Suppose s > 0 such that (3.11) holds for all nonnegative t ≤ s. Then U_{s} is transversal to P_{s} := {z ∈ ℝ^{n} : (z − q) · ν_{P} = s}.

Proof. We note that U_{t} is transversal to P_{t} for all t > 0 sufficiently small. Suppose s > 0 is the first value such that (3.11) holds for all nonnegative t ≤ s while U_{s} is not transversal to P_{s} at a point p ∈ ∂U_{s}. By the local convexity of M, P_{s} is a local supporting hyperplane to M at p where M locally lies in the half space (z − q) · ν_{P} ≤ s. For ϵ > 0 small enough,

V_{ϵ} := M ∩_{p} {(z − p) · (−ν_{P}) ≤ ϵ}

is transversal to P_{s−ϵ} and is a convex disk. Moreover, ∂V_{ϵ} = ∂U_{s−ϵ}, for ∂U_{s−ϵ} is a globally convex disk as its boundary is contained in a hyperplane (see [17]). This implies M = V_{ϵ} ∪ U_{s−ϵ} and therefore is a closed convex sphere without boundary, which is a contradiction. q.e.d.

We now return to the proof of Theorem 3.1. Let t_{0} > 0 be the smallest value such that U_{t_{0}} ∩ ∂M ≠ ∅ and choose a point p ∈ U_{t_{0}} ∩ ∂M. Note that U := U_{t_{0}} is globally convex. We consider the special coordinate system at p which satisfies (3.4). Under this coordinate system, q lies in the region |x_{n+1}| ≤ x_{n} cot β. In particular, x_{n}(q) > 0. We also note that

ν_{P} = e_{n} cos θ + e_{n+1} sin θ

for some θ ∈ [β, π − β]. Moreover, M locally (near p) is given as the graph of a function u on a domain Ω_{p} as in (3.9) satisfying (3.1).

Let r > 0 be as in (3.9). We see from above that if x_{n}(q) < r/2 then q is on the graph of u over Ω_{p} and we are done. So we next consider the case that x_{n}(q) ≥ r/2. Let C_{q} = C_{q}(∂U) be the convex cone generated by ∂U with vertex q. We will show that C_{q} contains a nondegenerate cone of fixed size that contains p. This means there exists a point q_{0} ∈ ℝ^{n+1}, |q_{0} − q| = 1, and a uniform constant δ_{0} > 0 such that p ∈ C_{q}(B_{δ_{0}}(q_{0})) ⊂ C_{q} where C_{q}(B_{δ_{0}}(q_{0})) is the cone generated by B_{δ_{0}}(q_{0}) with vertex q. Since |q − p| ≥ r/2, this will complete our proof under assumption (3.11).

Choose new coordinates (y_{1}, …, y_{n+1}) in ℝ^{n+1} with origin at p such that y_{i} = x_{i} (1 ≤ i ≤ n − 1), y_{n}(q) > 0, y_{n+1}(q) = 0 and let τ_{i} denote the unit vector in the positive y_{i} direction (0 ≤ i ≤ n + 1). We have y_{n}(q) ≥ x_{n}(q) ≥ \frac{r}{2} and hence

\frac{(τ_{k} · (q − p))^{2}}{|q − p|^{2}} = \frac{(e_{k} · (q − p))^{2}}{|q − p|^{2}} ≤ 1 − (\frac{r}{2d_{M}} )^{2}, ∀ 1 ≤ k ≤ n − 1. (3.12)

From the convexity of U we see that C_{q} contains the cone generated by Γ(p) with vertex q since Γ(p) and q are separated by the hyperplane containing ∂U. By (3.6), (3.7) and (3.12) the projection of C_{q} to the hyperplane ℝ^{n} ≡ {y_{n+1} = 0} contains an n-ball B_{ρ}(0) in ℝ^{n} where ρ ≥ c_{0} for a uniform constant c_{0} > 0. To complete the proof, therefore, we only have to find a point p_{0} with

y_{n+1}(p_{0}) > 0 and \frac{y_{n}(q) − y_{n}(p_{0})}{y_{n+1}(p_{0})} ≤ C_{0} (3.13)

for some uniform constant C_{0} > 0 such that the cone generated by the convex hull of Γ(p) ∪ {p_{0}} with vertex q is contained in C_{q}. (We note that it is always possible to find such p_{0} on Γ(p) with C_{0} = C_{0}(t_{0}) depending on t_{0}; C_{0}(t_{0}) may, however, tend to infinity as t_{0} → 0.)

For 0 ≤ t ≤ y_{n}(q), let W_{t} = {y_{n} ≥ t} ∩_{q} M. If W_{y_{n}(q)} ∩ ∂U ≠ ∅ then we are done since ∂U lies in the upper half space {y_{n+1} ≥ 0}. We thus may assume W_{y_{n}(q)} ∩ ∂U = ∅. Note that then W_{y_{n}(q)} ⊂ U and is therefore a convex cap. We may find t_{1} ∈ [0, y_{n}(q)) such that W_{t} ∩ ∂M = ∅ for all t_{1} < t ≤ y_{n}(q) and W_{t_{1}} ∩ ∂M ≠ ∅. Note that W_{t_{1}} is also a convex cap and W_{t_{1}} \ U ⊂ C_{p} by convexity.

If t_{1} = 0 then

τ_{n} · Π(X, X) ≤ 0, ∀ X ∈ T_{p}∂M (3.14)

since Γ(p) lies in the half space y_{n} ≤ 0. (This implies that Γ(p) is contained in the half space x_{n+1} ≥ 0.) It follows from (3.5) that τ_{n+1} · e_{n} ≥ sin β, that is, the angle between τ_{n+1} and e_{n} does not exceed \frac{π}{2} −β. Consequently,

y_{n+1}(z) = z · τ_{n+1} ≥ x_{n}(z)e_{n} · τ_{n+1} ≥ x_{n}(z) sin β, ∀ z ∈ Γ(p)

since x_{n+1}(z) ≥ 0. We see any point p_{0} on Γ(p) with x_{n}(p_{0}) ≥ \frac{r}{2} must satisfy (3.13).

We now assume t_{1} > 0 and take an arbitrary point p_{1} ∈ W_{t_{1}} ∩ ∂M. We have p_{1} ∈ (W_{t_{1}} \ U) ∪ ∂U ⊂ C_{q} and, similarly to (3.14),

τ_{n} · X = 0 and τ_{n} · Π(X, X) ≤ 0, ∀ X ∈ T_{p}∂M. (3.15)

Moreover, since W_{t_{1}} ∩ {y_{n+1} ≤ 0} ⊂ U,

y_{n+1}(p_{1}) > 0 and 0 < y_{n}(p_{1}) = t_{1} < y_{n}(q). (3.16)

We may further assume that there exists a uniform constant ε_{0} > 0 such that

|X · τ_{n+1}| ≤ ε_{0} (3.17)

and

|Π(X, X) · τ_{n+1}| ≤ ε_{0} (3.18)

for all unit tangent vector X to ∂M at p_{1}. This can be seen as follows. Suppose there is a unit vector X ∈ T_{p_{1}}∂M which does not satisfy (3.17) or (3.18) and let γ_{X} be the geodesic on Γ(p_{1}) tangential to X at p_{0}. We can then find a point p_{0} ∈ γ_{X} near p_{1} such that, if (3.17) is violated then (3.13) holds for C_{0} = C_{0}(ε_{0}), while if (3.18) fails,

|Y · τ_{n+1}| ≥ ε_{1}

for some unit tangent vector Y to ∂M at p_{0} and some uniform constant ε_{1} > 0.

Note that (3.17) and (3.18) imply

ν_{Σ}(p_{1}) · τ_{n} ≤ 0 (3.19)

when ε_{0} is sufficiently small, since the angle between Π(X, X) and −τ_{n} is sufficiently small while that between Π(X, X) and ν_{Σ}(p_{1}) does not exceed \frac{π}{2} − 2β. By (3.16) and (3.19) we obtain

ν_{Σ}(p_{1}) · τ_{n+1} ≤ 0, (3.20)

since the segment joining p_{1} and q locally lies on the inner side of Σ near p_{1}. Finally, by (3.19), (3.20) and the local strict convexity of Σ near boundary there exists a point z ∈ Σ_{δ} ∩ V with

y_{n+1}(p_{1}) − y_{n+1}(z) ≥ c_{0}

for some uniform constant c_{0} > 0 depending on δ and κ_{min}[Σ_{δ}], where V is the vertical 2-plane (in y-coordinates) through p_{1} and q. Since z must lie above the line through q and p_{1} by the convexity of M and the assumption that Σ_{δ} does not intersect M in interior, we have y_{n+1}(z) ≥ y_{n+1}(q) = 0. Thus y_{n+1}(p_{1}) ≥ c_{0} and p_{0} := p_{1} satisfies (3.13) where C_{0} > 0 depends on δ and κ_{min}[Σ_{δ}].

Step 3. We now assume there is no hyperplane through q satisfying assumption (3.11). We will first prove that M has a unique local supporting hyperplane (thus a tangent hyperplane) at q.

Let P be a local supporting hyperplane at q to M and let E denote the set of points on ∂M that (intrinsically) belong to P ∩_{q} M. Clearly E ≠ ∅. We claim that q is contained in the convex hull of E. Indeed, if this is not the case, that is, q and E are separated by a hyperplane, we may assume P = {x_{n+1} = 0} and q lies in the region x_{n} > ε while E in x_{n} < −ε for some ε > 0. Then M ∩_{q} {x_{n+1} ≤ ax_{n}} does not intersect ∂M where a > 0 is sufficiently small, which is a contradiction.

By Caratheodory’s theorem (cf. [22]) q is contained in an l-dimen-sional simplex S with vertices in E for some 1 ≤ l ≤ n. We have S ⊆ P ∩_{q} M by the local convexity of M.

Let p be a vertex of S and consider the special coordinate system at p. We note that, by the local convexity of M, P is a local supporting hyperplane to M at every point on the segment \overline{pq} joining p and q. It follows that

ν_{P} = e_{n} cos θ + e_{n+1} sin θ (3.21)

for some θ ∈ [β, π − β]. Recall M locally near p is the graph of a convex function over a domain Ω_{p} in {x_{n+1} = 0}. Since Ω_{p} is strictly convex the segment \overline{pq} is transversal to ∂M at p. For otherwise \overline{pq} would be contained in {x_{n+1} = 0} and tangential to ∂Ω_{p} at p, resulting in a contradiction as Ω_{p} would contain points on \overline{pq} other than p. Consequently, P is the tangent hyperplane to M at p as P contains \overline{pq} and is tangential to ∂M at p.

Next, assume furthermore that \overline{pq} ⊂ \overline{pp_{1}} ⊆ S for some p_{1} ≠ q. Let Q be a local supporting hyperplane to M at q. Then \overline{pp_{1}} ⊂ Q and therefore Q is a local supporting hyperplane to M at every point on \overline{pp_{1}}. We have Q = P since both are the tangent hyperplane to M at p. This also shows that P is the tangent hyperplane of M at every point on \overline{pp_{1}} (except possibly p_{1}). Consequently, u extends along \overline{pp_{1}}.

As we can always find a point p ∈ E such that the segment \overline{pq} extends in S, we have proved the uniqueness of the local supporting plane to M at q. Using induction on l we will next prove the assertion in the Theorem at point q.

Let us first consider the case l = 1, that is, S = \overline{pp_{1}} where p, p_{1} ∈ ∂M. Suppose |p − q| ≤ |p_{1} − q| and let Ω̃ be the convex hull (in ℝ^{n} = {x_{n+1} = 0}) of {p′_{1}} ∪ Ω_{p} where p′_{1} ∈ ℝ^{n} with p_{1} = (p′_{1}, x_{n+1}(p_{1})). (Similar meaning for q′ below.) As in Step 2 we may assume x_{n}(q) ≥ \frac{r}{2} where r as in (3.9). This implies (3.12), that is the angle between \overline{pq} and e_{k} has a uniform positive lower bound for all 1 ≤ k ≤ n − 1.

Let

(3.22) v(x) = sup{L(x) : L is an affine function,

L ≤ u at p′_{1} and on Ω_{p}}, x ∈ Ω̃.

Then v is a convex function and det D^{2}v = 0 in Ω̃. We have u ≤ v where u is defined in Ω̃. Since |p_{1} − q| ≥ |p − q| ≥ \frac{r}{2}, by (3.12) there exists a uniform constant λ > 0 depending on r and max_{∂M} |Π|, such that the n-ball B_{λ}(q′) is contained in Ω̃. By the local convexity of M we see u is defined on B_{λ/2}(q′) ⊂ Ω̃ with a uniform bound on ∥ũ∥_{C^{0,1}(B_{λ/2}(q′))}. This completes the proof for l = 1.

Assume now l > 1 and suppose we have proved the assertion for any point in a simplex of dimension less than l with vertices in E. Choose p ∈ E and p_{1} on an (l − 1) dimensional face of S such that q ∈ \overline{pp_{1}}. If |p − q| ≤ |p_{1} − q| then the proof follows as exactly in case l = 1. Let us therefore assume |p − q| ≥ |p_{1} − q|. By induction, in a suitable coordinate system (y, y_{n+1}), y ∈ ℝ^{n} with origin at p_{1}, M locally near p_{1} is the graph of a convex function y_{n+1} = u(y) with a uniform C^{0,1} bound in an n-ball B_{R}(0) where R is a uniform constant. Since P is the tangent hyperplane to M at any point on \overline{pp_{1}} (except possibly p_{1}), we have ν_{P} · (0, …, 0, 1) ≥ c_{0} for some uniform constant c_{0} > 0. Thus u extends along \overline{pp_{1}}. Replacing the convex function v in (3.22) by

v(y) = sup{L(y) : L is an affine function, L ≤ u at p and on B_{R}(0)}, (3.23)

defined in the convex hull of {p} ∪ B_{R}(0), the rest of proof follows that of case l = 1. This, finally, completes our proof. q.e.d.

An important consequence of Theorem 3.1 is a compactness result (Theorem 3.4) which we will need in the next section. First, it follows immediately from Theorem 3.1 that:

Corollary 3.3. There exist uniform constants R, r > 0 depending on δ, κ_{min}[Σ_{δ}], κ_{max}[Σ_{δ}], max_{∂Σ} |Π| and d_{M} such that for any p ∈ M, M ∩_{p} B_{R}(p) is embedded and the convex body in B_{R}(p) bounded by M ∩_{p} B_{R}(p) contains a ball of radius r.

According to a convergence theorem of Alexander-Ghomi [1] we thus have:

Theorem 3.4. Let {M_{k}} be a sequence of locally convex hypersurfaces contained in a bounded region in ℝ^{n+1} with ∂M_{k} = ∂Σ for all k. Suppose each M_{k} lies on the inner side of Σ and does not intersect Σ_{δ}. Then there exists a subsequence {M_{k_{i}}} converging in Hausdorff metric to a locally convex hypersurface M with ∂M = ∂Σ. Moreover, for each i there exists a homeomorphism from M_{k_{i}} on to M with boundary fixed.

Proof. We refer to [1] (Theorem 7.1) for the major part of the proof. Here we only point out that by Corollary 3.3 the conditions of Theorem 7.1 in [1] are satisfied, and give a brief proof of the fact ∂M = ∂Σ and that M is C^{0,1} up to the boundary. Given a point p ∈ ∂Σ, we consider the special coordinates at p satisfying (3.4). Then each Σ_{k} locally near p can be represented as a convex graph x_{n+1} = u_{k}(x) over a domain Ω_{p} of form (3.9) with a uniform C^{0,1} norm bound. By compactness there exists a subsequence of {u_{k}} converging to a convex function u ∈ C^{0,1}(\overline{Ω_{p}}). Moreover, we have u = \underline{u} on \{x ∈ ∂Ω_{p} : x_{n} = φ\} since \underline{u} ≤ u_{k} ≤ v in \overline{Ω_{p}} where v is as in (3.8), φ as in (3.6) and the graph of \underline{u} represents Σ. Note that M must coincide with the graph of u near p. Consequently, M is a locally convex hypersurface of class C^{0,1} up to the boundary and ∂M = ∂Σ. q.e.d.

We next derive a priori bounds for all principal curvatures for smooth locally strictly convex K-hypersurfaces.

Theorem 3.5. Assume in addition that M is a smooth locally strictly convex hypersurface of constant Gauss curvature K > 0. Then

\frac{1}{C_{2}} ≤ κ_{min}[M] ≤ κ_{max}[M] ≤ C_{2}

where C_{2} > 0 depends on K, K^{−1}, δ, κ_{min}[Σ_{δ}], κ_{max}[Σ_{δ}], max_{∂Σ} |Π|, and d_{M}.

Proof. We first establish the estimates on the boundary. Given any point p ∈ ∂M, by Theorem 3.1 we may write M locally (near p) as a graph x_{n+1} = u(x) with an a priori gradient bound over a smooth strictly convex domain Ω_{p} where u satisfies the Gauss curvature equation (2.1). As ∂Ω_{p} is strictly convex we may appeal to the boundary estimates for |∇^{2}u| due to Caffarelli-Nirenberg-Spruck [6] (which is local in nature) to obtain

|u_{ij}(0)| ≤ C (3.24)

where C depends on ∥u∥_{C^{1}(\overline{Ω_{p}})} and geometric quantities of Σ_{δ} and ∂Σ. Since the principal curvatures of M at p are the eigenvalues of the matrix

\{(1 + |∇u|^{2})^{− \frac{1}{2}} u_{ij}\}

(with respect to {δ_{ij} + u_{i}u_{j}}, the metric of M), the desired estimates follow from (3.24) and the fact that the Gauss curvature is the product of all principal curvatures.

Turning to the global estimates, consider Λ := max κe^{ρ} where

ρ(x) = |x − x^{0}|^{2}, x ∈ ℝ^{n+1}

(x^{0} is a fixed point in ℝ^{n+1}), and the maximum is taken for all normal curvatures κ over M. As we already have estimates for principal curvatures on ∂M, we may assume Λ is attained at an interior point p ∈ M. Choose coordinates in ℝ^{n+1} with origin at p such that the tangent hyperplane of M at p is given by x_{n+1} = 0 and M locally is written as a strictly convex graph x_{n+1} = u(x) where x = (x_{1}, … x_{n}) ∈ ℝ^{n}. We may also assume the Hessian matrix {u_{ij}} to be diagonal at 0 with u_{11}(0) ≥ u_{ii}(0) > 0 for all 1 < i ≤ n. Note that, since Du(0) = 0, u_{ii}(0) (1 ≤ i ≤ n) are the principal curvatures of M at p. Thus Λ is achieved at p with respect to the normal curvature in x_{1} direction which is locally given by

κ = \frac{u_{11}}{(1 + u_{1}^{2})w} , w = (1 + |∇u|^{2})^{\frac{1}{2}} .

Since the function log u_{11}−log(1+u_{1}^{2})−log w+ρ then has a maximum at the origin where Du = 0, w = 1, Dw = 0 and w_{ii} = u_{ii}^{2} for all 1 ≤ i ≤ n, we have at 0,

\frac{u_{11i}}{u_{11}} − \frac{w_{i}}{w} − \frac{2u_{1}u_{1i}}{1 + u_{1}^{2}} + ρ_{i} = 0, 1 ≤ i ≤ n (3.25)

and

\frac{u_{11ii}}{u_{11}} − (\frac{u_{11i}}{u_{11}} )^{2} − u_{ii}^{2} − 2u_{1i}^{2} + ρ_{ii} ≤ 0, 1 ≤ i ≤ n. (3.26)

Multiplying (3.26) by u_{11}/u_{ii} and taking sum over i from 1 to n, one obtains

∑ \frac{u_{11ii}}{u_{ii}} − ∑ \frac{(u_{11i})^{2}}{u_{11}u_{ii}} − u_{11}Δu − 2u_{11}^{2} + ∑ \frac{ρ_{ii}}{u_{ii}} ≤ 0. (3.27)

Differentiating Equation (2.1) we have for 1 ≤ k ≤ n,

u^{ij}u_{ijk} = (n + 2)\frac{w_{k}}{w}

and

u^{ij}u_{ijkk} − u^{il}u^{jm}u_{ijk}u_{lmk} = (n + 2)\frac{w_{kk}}{w} − (n + 2)\frac{w_{k}^{2}}{w^{2}} ,

where {u^{ij}} is the inverse matrix of {u_{ij}}. Combining these and (3.27) we obtain

nu_{11}^{2} − u_{11}Δu + ∑ \frac{ρ_{ii}}{u_{ii}} ≤ 0. (3.28)

Next,

ρ(x) = |x − x^{0}|^{2} + (u(x) − x_{n+1}^{0})^{2}, x ∈ ℝ^{n}

where x^{0} = (x^{0}, x_{n+1}^{0}), and therefore,

ρ_{ii} = 2 + 2(u(x) − x_{n+1}^{0})u_{ii} + u_{i}^{2}.

Since Δu ≤ nu_{11}, by (3.28) one sees that at 0,

0 ≥ ∑_{i=1}^{n} \frac{ρ_{ii}}{u_{ii}} ≥ ∑_{i=2}^{n} \frac{2}{u_{ii}} − 2nx_{n+1}^{0}

≥ \frac{2}{(u_{22} … u_{nn})^{\frac{1}{n−1}}} − 2nx_{n+1}^{0}

≥ 2(\frac{u_{11}}{det u_{ij}} )^{\frac{1}{n−1}} − C

= 2(\frac{u_{11}}{K} )^{\frac{1}{n−1}} − C.

It follows that

u_{11}(0) ≤ CK.

This proves an upper bound for κ_{max}[M], from which a lower bound for κ_{min}[M] can be derived in terms of K^{−1}. The proof is complete. q.e.d.

Remark 3.6. Using an estimate of Guan-Trudinger-Wang [14] in place of that of [6], it is possible to obtain an upper bound for the principal curvatures which does not depend on the lower bound of Gauss curvature.

4. Deformation to K-hypersurfaces

The primary purpose of this section is to prove the existence part in Theorems 1.1 and 1.2. Throughout the section, let Σ be a locally convex immersed hypersurface in ℝ^{n+1} with embedded boundary ∂Σ and Gauss curvature K_{Σ} ≥ K everywhere on Σ, where K is a fixed nonnegative constant. Our idea is to deform Σ to a locally convex immersed hypersurface M with K_{M} ≡ K and ∂M = ∂Σ.

Let D ⊆ Σ be a disk on Σ which, as a hypersurface in ℝ^{n+1}, may be represented as the graph of a convex function \underline{u} defined in a domain Ω (in some hyperplane) with Lipschitz boundary. By Theorem 2.1, there is a unique function u ∈ C^{0,1}(\overline{Ω}) whose graph is a convex hypersurface D̃ of constant Gauss curvature K with ∂D̃ = ∂D. By the maximum principle, we have u ≥ \underline{u} in \overline{Ω}. Thus D̃ lies on the inner side of D.

This naturally induces a C^{0,1}-diffeomorphism Ψ_{D} : Σ → Σ̃ := D̃ ∪ (Σ \ D) which is fixed on Σ \ D. The hypersurface Σ̃ is locally convex with KΣ̃ ≥ K and ∂Σ̃ = ∂Σ. We call Σ̃ a basic lifting of Σ (by D̃ over D). A lifting of Σ is a hypersurface which is obtained by a finite number of basic liftings starting from Σ. We introduce a partial order ⪯ between liftings of Σ: Σ_{1} ⪯ Σ_{2} if and only if Σ_{2} is a lifting of Σ_{1} or Σ_{2} = Σ_{1}.

Lemma 4.1. Let Σ_{1} and Σ_{2} be any two liftings of Σ. Then there exists a unique lifting, which we denote as Σ_{1} ∨ Σ_{2}, of Σ such that Σ_{1} ⪯ Σ_{1} ∨ Σ_{2}, Σ_{2} ⪯ Σ_{1} ∨ Σ_{2}, and Σ_{1} ∨ Σ_{2} ⪯ N for any lifting N with Σ_{1} ⪯ N and Σ_{2} ⪯ N.

Proof. We first assume Σ_{1} is a basic lifting of Σ by D̃_{1} over a disk D_{1} ⊆ Σ and let A be the open region in ℝ^{n+1} bounded by D_{1} ∪ D̃_{1}. Assume Σ_{2} to be a lifting of Σ over a region D_{2}. (D_{2} is not necessarily a disk.) Intuitively, if Σ, Σ_{1} and Σ_{2} are all embedded, then it is obvious that the hypersurface

Σ_{1} ∨ Σ_{2} := (Σ_{2} \ (Σ_{2} ∩ A)) ∪ (D̃_{1} \ (D̃_{1} ∩ B))

where B is the open regions in ℝ^{n+1} bounded by Σ_{2} ∪ Σ, is a lifting of Σ with the desired properties. In the general case when some of these hypersurfaces may be immersed, we view Σ as an immersion

Φ_{0} : Σ_{0} → Σ ⊂ ℝ^{n+1} (4.1)

of a differentiable manifold Σ_{0} and let

Φ_{i} : Σ_{0} → Σ_{i} ⊂ ℝ^{n+1}, i = 1, 2 (4.2)

be the immersions induced from the liftings. (Note that Φ_{i} = Φ_{0} on Σ_{0} \ Φ_{0}^{−1}(D_{i}).) The lifting Σ_{1} ∨ Σ_{2} is then given by the immersion

Φ : Σ_{0} → Σ_{1} ∨ Σ_{2} := Φ(Σ_{0}) ⊂ ℝ^{n+1}

defined as

(4.3) Φ(p) := \begin{cases}Φ_{1}(p), if p ∈ Φ_{0}^{−1}(D_{1}) \ Φ_{0}^{−1}(D_{2}), \\ Φ_{1}(p), if p ∈ Φ_{0}^{−1}(D_{1}) ∩ Φ_{0}^{−1}(D_{2}) and Φ_{2}(p) ∈ A, \\ Φ_{2}(p), otherwise, \end{cases}

for p ∈ Σ_{0}. The general case now can be proved by induction. q.e.d.

The next lemma, which states that volume decreases under lifting, is well-known; for completeness we include a proof.

Lemma 4.2. Let Σ_{1} and Σ_{2} be liftings of Σ. If Σ_{1} ⪯ Σ_{2} then Vol(Σ_{1}) ≥ Vol(Σ_{2}). Moreover, the equality holds if and only if Σ_{1} = Σ_{2}.

Proof. Obviously we may assume Σ_{2} is a basic lifting of Σ_{1} over a disk D_{1} ⊂ Σ_{1}. Suppose D_{1} and its lifting D_{2} ⊂ Σ_{2} are the graphs of convex functions u_{1} and u_{2} over a domain Ω ⊂ ℝ^{n}, respectively. We have u_{1} ≤ u_{2} on \overline{Ω} and u_{1} = u_{2} ∂Ω.

Let

N(x, z) = \frac{(∇u_{2}, −1)}{\sqrt{1 + |∇u_{2}|^{2}}} , (x, z) ∈ \overline{Ω} × ℝ.

denote the downward unit normal vector to D_{2} at (x, u_{2}(x)). Thus div N(x, z), the distributional mean curvature of D_{2} at the point (x, u_{2}(x)) with respect to the upward normal vector, is nonnegative almost everywhere since u_{2} is a convex function. Let

ω = {(x, z) ∈ ℝ^{n+1} : u_{1}(x) < z < u_{2}(x), x ∈ Ω}.

By the divergence theorem we have

0 ≤ ∫_{ω} divNdv = ∫_{D_{1}} N · ν_{1}dσ − ∫_{D_{2}} dσ
= Vol(D_{1}) − Vol(D_{2}) + ∫_{D_{1}}(N · ν_{1} − 1)dσ

where ν_{1} is the downward unit normal vector to D_{1}. Since 0 ≤ N ·ν_{1} ≤ 1 on D_{1} we have Vol(D_{1})−Vol(D_{2}) ≥ 0; obviously, the equality holds only when D_{1} = D_{2}. q.e.d.

We need one more lemma which states that volume is continuous under uniform convergence of uniformly Lipschitz convex functions.

Lemma 4.3. Let w_{k} be a sequence of uniformly Lipschitz convex functions on Ω converging uniformly to w. Then

∫_{Ω} \sqrt{1 + |∇w|^{2}} dx = lim_{k→∞} ∫_{Ω} \sqrt{1 + |∇w_{k}|^{2}} dx. (4.4)

Proof. Let W_{k} = \sqrt{1 + |∇w_{k}|^{2}} ; then |∇W_{k}| ≤ |∇^{2}w_{k}| ≤ Δw_{k} a.e. in Ω since w_{k} is convex. Therefore,

∫_{Ω} |∇W_{k}| dx ≤ sup_{Ω} |∇w_{k}||∂Ω|.

Hence W_{k} are uniformly bounded in W^{1,1} and so converge in L^{1} to \sqrt{1 + |∇w|^{2}}. q.e.d.

We are now ready to prove the main result of this section. Let L be the collection of liftings of Σ and set

μ := inf_{L∈L} Vol(L).

Theorem 4.4. Suppose Σ_{δ} is C^{2} and locally strictly convex up to the boundary for some fixed δ > 0. There exists a locally convex hypersurface M in ℝ^{n+1} of class C^{0,1} up to the boundary with ∂M = ∂Σ and K_{M} ≡ K. Moreover, M is homeomorphic to Σ and Vol(M) = μ.

Proof. For each k ≥ 1 choose Σ_{k} ∈ L such that

Vol(Σ_{k}) ≤ μ − \frac{1}{k} .

By Lemmas 4.1 and 4.2 we may assume Σ_{k} ⪯ Σ_{k+1} for all k ≥ 1. According to Theorem 3.4 after passing to a subsequence we may assume {Σ_{k}} converges in Hausdorff metric to a locally convex hypersurface M which, in addition, is homeomorphic to each Σ_{k}. Clearly ∂M = ∂Σ. It remains to show Vol(M) = μ and K_{M} ≡ K.

Consider a point p ∈ M. There exists a sequence p_{k} ∈ Σ_{k}, k = 1, 2, …, converging to p (in ℝ^{n+1}) such that Σ_{k} ∩_{p_{k}} B_{R}(p_{k}) converges to M ∩_{p} B_{R}(p) in Hausdorff metric where R > 0. According to Theorem 3.1, when R is chosen sufficiently small each Σ_{k} ∩_{p_{k}} B_{R}(p_{k}) can be represented as the graph of a convex function w_{k} with a uniform C^{0,1} norm bound (independent of k). By compactness we may choose a coordinate system in ℝ^{n+1} such that, after possibly passing to subsequences, all the functions w_{k} are defined in a fixed domain Ω ∈ ℝ^{n} satisfying

(4.5) ∥w_{k}∥_{C^{0,1}(\overline{Ω})} ≤ C_{0} independent of k

and w_{k} converges uniformly to a function w ∈ C^{0,1}(\overline{Ω}) whose graph obviously locally represents M. Hence by Lemma 4.3 and a covering argument, Vol(M) = μ.

Consider now the Dirichlet problem for the Gauss curvature equation (2.1) in \overline{Ω}. Using w_{k} as a subsolution for each k ≥ 1, by Theorem 2.1 we obtain a unique convex solution u_{k} ∈ C^{0,1}(\overline{Ω}) of (2.1) satisfying u_{k} = w_{k} on ∂Ω. We have u_{k} ≥ w_{k} on \overline{Ω} and by (4.5)

∥u_{k}∥_{C^{0,1}(\overline{Ω})} ≤ C_{0} independent of k.

Thus there exists a subsequence, which we still denote by {u_{k}}, converging to a convex function u in C^{0,1}(\overline{Ω}). We see u satisfies (2.1) and u ≥ w on \overline{Ω} with u = w on ∂Ω.

On the other hand, for each k ≥ 1 let Σ̃_{k} be the lifting of Σ_{k} obtained by replacing D_{k} with D̃_{k}, where D_{k} and D̃_{k} are the graphs of w_{k} and u_{k} over Ω, respectively. Similarly, let M̃ be the locally convex hypersurface obtained from M by replacing the graph of w over Ω by that of u. Clearly Σ̃_{k} converges to M̃ as u_{k} converges uniformly to u on \overline{Ω}. Since by Lemma 4.2 μ ≤ Vol(Σ̃_{k}) ≤ Vol(Σ_{k}) for each k it follows that Vol(M̃) = μ and therefore Vol(D̃) = Vol(D). As both u and w are convex functions, this implies u ≡ w on \overline{Ω} by the proof of Lemma 4.2. Since u satisfies (2.1), M has constant Gauss curvature K in a neighborhood of p. q.e.d.

5. Regularity

In this section we study the regularity of the hypersurface M constructed in the previous section to complete our proof of Theorems 1.1 and 1.2. Throughout this section, we assume that Σ is a locally convex immersed hypersurface which is C^{2} and locally strictly convex along its boundary ∂Σ. Thus Σ is C^{2} and locally strictly convex in a neighborhood of, and up to, ∂Σ. In addition, we assume ∂Σ to be embedded and smooth. Let K ≤ min K_{Σ} be a nonnegative constant and let M the locally convex hypersurface with K_{M} ≡ K and ∂M = ∂Σ constructed in Section 4. By Theorem 4.4, M is C^{0,1} up to the boundary.

Theorem 5.1. If K > 0 then M is smooth up to the boundary and locally strictly convex.

Proof. Consider an interior point p ∈ M which we assume to be the origin of ℝ^{n+1}. Since M is of class C^{0,1}, M locally near p can be represented as a convex graph x_{n+1} = u(x) ≥ 0 over a domain Ω_{1} ⊂ ℝ^{n} ≡ {x_{n+1} = 0} with a C^{0,1} norm bound

∥u∥_{C^{0,1}(\overline{Ω_{1}})} ≤ C_{1}.

It follows that u satisfies the inequalities in the viscosity sense

K ≤ det(u_{ij}) ≤ K(1 + C_{1}^{2})^{\frac{n+2}{2}} in Ω_{1}.

We may assume M ∩_{p} {x_{n+1} = 0} ⊂ Ω_{1}. By a theorem of Caffarelli [2], the nodal set {u = 0} either is a single point, in which case M is smooth and strictly convex at p (see [4]), or does not contain any interior extreme points. So we will be done if we can show that {u = 0} = {0}. Suppose this is not the case. Then we can find two points q_{1}, q_{2} ∈ ∂M such that \overline{q_{1}q_{2}} ⊆ M ∩ \{x_{n+1} = 0\} and x_{n+1} = 0 is a local supporting plane of M at every point on \overline{q_{1}q_{2}}. By the proof (Step 3) of Theorem 3.1, \overline{q_{1}q_{2}} is transversal to ∂M at the endpoints. Without loss of generality we may assume

q_{i} = (0, …, 0, (−1)^{i}a, 0), i = 1, 2,

where a > 0. Consequently, there exists a constant δ > 0 such that, in a neighborhood of \overline{q_{1}q_{2}}, M is given as a convex graph x_{n+1} = u(x) ≥ 0 over a domain

Ω_{0} := {x := (x′, x_{n}) ∈ ℝ^{n}|φ_{1}(x′) < x_{n} < φ_{2}(x′) for |x′| < δ}

where φ_{1}, φ_{2} are smooth functions since ∂M is smooth and transversal to \overline{q_{1}q_{2}}. Let ψ be a smooth function defined on ∂B_{r}, where B_{r} ⊂ Ω_{0} is the n-ball of radius r ≤ δ centered at the origin, satisfying ψ(0, ±r) = 0 and

ψ(x′, x_{n}) ≥ max{u(x′, φ_{1}(x′)), u(x′, φ_{2}(x′))}, ∀ (x′, x_{n}) ∈ ∂B_{r}.

This is possible since both u(x′, φ_{1}(x′)) and u(x′, φ_{2}(x′)) are smooth in x′ as ∂M is smooth and tangential to x_{n+1} = 0. By [6] there exists a unique strictly convex solution v ∈ C^{∞}(\overline{B_{r}}) to the Dirichlet problem of the Monge-Ampère equation

det(v_{ij}) = K in \overline{B_{r}}, v = ψ on ∂B_{r}.

Since det(v_{ij}) = K ≤ det(u_{ij}) in \overline{B_{r}} and, by the convexity of u,

u(x′, x_{n}) ≤ max{u(x′, φ_{1}(x′)), u(x′, φ_{2}(x′))}, ∀ (x′, x_{n}) ∈ Ω_{0}

which implies v ≥ u on ∂B_{r}, we have v ≥ u ≥ 0 on \overline{B_{r}} by the comparison principle. Since v is strictly convex and v(0, a) = v(0, −a) = 0, however, we have v(0) < 0 which is a contradiction. This proves that M is strictly convex and smooth in any interior point. Finally the boundary regularity follows from [6]. The proof is thus complete. q.e.d.

This completes the proof of Theorem 1.1. Turning to the case K = 0 we first prove the following lemma.

Lemma 5.2. Let N be a locally convex hypersurface with K_{N} ≡ 0. Let p be an interior point of N and P a local supporting hyperplane to N at p. Then p is contained in a k-dimensional subsimplex of N ∩_{p} P with vertices on ∂N for some 1 ≤ k ≤ n.

Proof. This follows form the argument in Step 3 of the proof of Theorem 3.1 as there is no hyperplane through p satisfying assumption (3.11). We redo the proof here for the reader’s convenience. Since N is locally convex, P is a local supporting hyperplane to N at every point on N ∩_{p} P. Let D be the set of points on ∂N that (intrinsically) belong to N ∩_{p}P. It suffices to show that any point in N ∩_{p}P is contained in the convex hull of D. If this is not the case, there is a point q ∈ N ∩_{p}P which is separated by a hyperplane from D. We may assume P = {x_{n+1} = 0} and q lies in x_{n} > ε while D lies in x_{n} < −ε for some ε > 0. It then follows that N ∩_{q} {x_{n+1} < δx_{n}} is contained in the interior of N when δ is sufficiently small. This is a contradiction as the Gauss curvature of N ∩_{q} {x_{n+1} < δx_{n}} is zero everywhere while its boundary is contained in the hyperplane x_{n+1} = δx_{n}. q.e.d.

Theorem 5.3. If K = 0, then M is C^{1,1} up to the boundary.

Proof. Let p be an interior point of M. From Step 3 of the proof of Theorem 3.1 we see that M has a tangent hyperplane at p. Suppose M locally (near p) is written as a convex graph x_{n+1} = u(x) with u ≥ 0 over T_{p}M := {x_{n+1} = 0}. Since T_{p}M is the tangent hyperplane to M at every point on M ∩_{p} T_{p}M, u is defined in a domain Ω ⊂ ℝ^{n} such that M ∩_{p} T_{p}M ⊂ \{(x, 0) : x ∈ \overline{Ω}\}. By [7], in order to prove that M is C^{1,1} it suffices to show that there exists a constant C, depending only on ∂M, and ϵ = ϵ(p) > 0 such that

u(x) ≤ C|x|^{2} (5.1)

for all x ∈ B_{ϵ}(x^{0}) ⊂ ℝ^{n} where p = (x^{0}, 0).

By Lemma 5.2, p is contained in a k-dimensional subsimplex, which we denote as S, of M ∩_{p} T_{p}M with vertices on ∂M for some 1 ≤ k ≤ n. According to [7], in order to prove (5.1) it suffices to consider the case k = 1. Suppose now that S is a segment with end points q_{1} := (x^{1}, 0), q_{2} := (x^{2}, 0) on ∂M. By the proof of Theorem 3.1, S is transversal to ∂Ω at the end points and both ∂Ω and u|_{∂Ω} are smooth in a neighborhood of x^{i} (i = 1, 2). Of the two end points, suppose that q_{2} is the closer to p. We may assume x_{2} = 0 and e_{n} to be the interior unit normal to ∂Ω at 0 where e_{k} (1 ≤ k ≤ n + 1) is the unit vector in the positive x_{k}-axis direction. Since x_{n+1} = 0 is a local supporting hyperplane to M at q_{2}, e_{n+1} · Π(X, X) ≥ 0 for any X ∈ T_{q_{2}}∂M. On the other hand, from the proof of Theorem 3.1 we see that the angle between Π(X, X) and ν_{Σ} at q_{2} does not exceed \frac{π}{2} − 2β for some uniform constant β > 0. It follows that

ν_{Σ}(q_{2}) = e_{n} cos α + e_{n+1} sin α (5.2)

where 2β − \frac{π}{2} ≤ α ≤ \frac{π}{2} . We distinguish two cases: (i) α ≤ β and (ii) α > β.

If α ≤ β, then for any X ∈ T_{p}∂M, the angle between Π(X, X) and e_{n} is less than or equal to \frac{π}{2} −β and, therefore, e_{n} ·Π(X, X) ≥ sin β > 0. This implies that ∂Ω ∩ B_{δ}(0) is uniformly strictly convex where δ > 0 is a uniform constant. We therefore may follow the proof of [7] to derive (5.1).

We now suppose α > β. Then locally Σ is a strictly convex graph x_{n+1} = \underline{u}(x) over Ω ∩ B_{δ}(0) for some uniform constant δ > 0. To prove (5.1) we then can follow the proof of Theorem 3.2 in [11]. This proves Theorem 5.3. q.e.d.

Remark 5.4. Theorem 1.2 follows from Lemma 5.2 and Theorem 5.3. If min K_{Σ} > 0, Theorem 5.3 may be proved by approximation as follows. For any positive constant ε ≤ min K_{Σ}, by Theorem 5.1 there exists a smooth locally strictly convex hypersurface M^{ε} with constant Gauss curvature ε and ∂M^{ε} = ∂Σ. By Theorem 3.5 (see Remark 3.6) we obtain a subsequence ε_{k} → 0 such that {M^{ε_{k}}} is convergent in local C^{1,1} norms. Clearly, the limiting hypersurface must be M. Consequently, M is C^{1,1} up to the boundary.

Remark 5.5. We have the following characterization of M at boundary (for K = 0). Let p ∈ ∂M and choose coordinates of ℝ^{n+1} with origin at p such that e_{n+1} and e_{n} are the unit normal and interior conormal to T_{p}M, respectively, where e_{k} as before is the unit vector in the positive direction of x_{k} axis.

Proposition 5.6. Let K = 0 and p ∈ M. Suppose M ∩_{p} T_{p}M does not contain any point in ∂M ∩ {x_{n} > 0}. Then there exists some unit vector X ∈ T_{p}∂M such that θ(X) = 0 where θ(X) is defined by

(5.3) Π(X, X) = |Π(X, X)|(e_{n} cos θ(X) + e_{n+1} sin θ(X)), X ∈ T_{p}∂Σ, X ≠ 0.

Proof. Suppose

min{θ(X) : X ∈ T_{p}∂Σ, |X| = 1} > 0.

Then M ∩_{p}{0 ≤ x_{n+1} ≤ λx_{n}} does not contain any point on ∂M ∩{x_{n} > 0} when λ > 0 is sufficiently small. By Lemma 5.2 this implies

(M ∩_{p} {0 ≤ x_{n+1} ≤ λx_{n}}) ∩ {x_{n} > 0} = ∅,

contradicting the fact that T_{p}M = {x_{n+1} = 0}. q.e.d.

6. Locally convex hypersurfaces with extreme boundary

We first give a brief proof of Corollary 1.3.

Proof of Corollary 1.3. By Theorem 1.2 we obtain a globally convex hypersurface M with K_{M} ≡ 0 and ∂M = Γ. Moreover, M is on the inner side of Σ along the boundary. Consider now an arbitrary interior point q ∈ Σ and let P be a local supporting hyperplane to Σ at q. Since Σ is C^{2} and locally strictly convex near the boundary, Σ ∩_{q} P does not (intrinsically) contain points on ∂Σ. Let t_{0} > 0 be the smallest value such that Σ^{t_{0}} contains a point p on ∂Σ, where

Σ^{t} := Σ ∩_{q} {z ∈ ℝ^{n+1} : (z − q) · ν_{P} ≤ t}, t ≥ 0.

We see that ∂Σ locally near p lies in the half space {z ∈ ℝ^{n+1} : (z − p) · ν_{P} ≥ 0}. Since M is globally convex, ∂M lies on one side of T_{p}M. Let us assume T_{p}M = {x_{n+1} = 0} and that ∂M lies in x_{n+1} ≥ 0.

We now choose coordinates in ℝ^{n+1} such that e_{n+1} and e_{n} are the unit normal and interior conormal to ∂M at p, respectively. We claim that

(6.1) ν_{P} = e_{n} cos α + e_{n+1} sin α, for some 0 ≤ α ≤ \frac{π}{2} .

Note that this implies Σ^{t_{0}} ⊂ {x_{n+1} ≤ 0} and thus completes the proof.

For any 0 < t < t_{0}, since Σ^{t} does not contain points on ∂M, it is easy to see that (Σ\Σ^{t})∪D_{t} is a lifting of Σ with respect to K = 0, where D_{t} is the region on the hyperplane P_{t} := {z ∈ ℝ^{n+1} : (z−q)·ν_{P} = t} bounded by Σ^{t} ∩ P_{t}. Thus M ∩_{p} {z ∈ ℝ^{n+1} : (z − q) · ν_{P} ≤ t_{0}} does not intersect the region bounded by Σ^{t} ∪ D_{t} for any 0 < t < t_{0}. Consequently, T_{p}M does not intersect the interior of Σ^{t_{0}}. This proves (6.1). q.e.d.

We next construct a smooth locally strictly convex, non-embedded, surface M of Gauss curvature one in ℝ^{3} such that ∂M is strictly extreme. Let S_{1} be the unit sphere centered at (\frac{1}{2}, 0, 0). Cut off a small cap from the top of S_{1} using a plane perpendicular to the line through (0, 0, 2) and the center of S_{1}. Let Σ_{1} be the resulting spherical cap and Σ_{2} the reflection of Σ_{1} with respect to x_{1} = 0. Now, connecting the boundary circles of Σ_{1} and Σ_{2} by a thin convex bridge, we obtain a locally strictly convex surface Σ with self-intersection. Moreover, ∂Σ is strictly extreme. According to the bridge principle of Hauswirth [16] there exists a locally strictly convex surface M of constant Gauss curvature one with the same boundary. It follows from [16] that M is a small perturbation of Σ and therefore has self-intersection.

If we start with cutting a small cap from the top of S_{1} using a horizontal plane and repeat the rest of the above procedure, we get a nonembedded locally strictly convex K-surface M such that ∂M is extreme while the boundary of the convex hull of ∂M has interior singularities along the bottom edges.

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