On Weyl problem with nonnegative Gauss curvature

Pengfei Guan^{∗} Department of Mathematics McMaster University Hamilton, Ontario L8S 4K1, Canada YanYan Li^{†} Department of Mathematics Rutgers University New Brunswick, NJ 08903

1 Introduction

H. Weyl posed the following problem in 1916 ([22]): Consider the two-sphere S^{2} and suppose g^{0} is a Riemannian metric on S^{2} whose Gauss curvature is everywhere positive. Does there exist a global C^{2} isometric embedding X : (S^{2}, g^{0})→(R^{3}, δ) where δ is the standard flat metric on R^{3}? The first attempt to solve the problem was made by Weyl himself. He suggested the continuity method and obtained apriori estimates upto second derivatives. Later H. Lewy [13] solved the problem in the case of g^{0} being analytic. The complete solution was given in 1953 by Nirenberg in a beautiful paper [16] under very mild hypothesis that the metric g^{0} has continuous fourth derivatives. His result depended on the strong apriori estimates he had derived for uniformly elliptic equations in dimension two ([17]). The result was extended to the case of continuous third derivatives of the metric by Heinz ([9]) in 1962. In a completely different approach to the problem, Alexandroff ([1]) obtained a generalized solution of Weyl′s problem as a limit of polyhedra. The regularity of this generalized solution was proved by Pogorelov ([19], [20]).

The uniqueness question was considered by Weyl in [22]. The first proof of the uniqueness of a solution of the problem (within rigid motion and a possible reflection), i.e., a proof of the theorem that two closed isometric convex surfaces are congruent (within a reflection) was given by Cohn-Vossen [6] in 1927, under the assumption that the surfaces are analytic. It was later shortened considerably by Zhitomirsky [23]. In 1943 Herglotz [10] gave a very short proof of the uniqueness, assuming that the surfaces are three times continuously differentiable. Finally in 1962 it was extended to surfaces having merely two times continuously differentiable metrics by Sacksteder [21]. Notice that the rigidity results in [21] hold under more general hypotheses than the Gauss curvature of g^{0} being positive everywhere.

∗Partially supported by NSERC Grant OGP-0046732

†Partially supported by NSF grant DMS-9104293

A natural question to ask is if the Gauss curvature of the metric is nonnegative instead of positive everywhere, do we still have a smooth isometric embedding if we assume the smoothness of the metric? Examples given in [11] and [2] show that for some analytic metrics with Gauss curvature positive on S^{2} except at one point, there exist no C^{3} global isometric embedding into (R^{3}, δ).

It is proved by Iaia in [11] that if g^{0} is a C^{4} Riemannian metric on S^{2} with Gauss curvature K satisfying, for some P ∈ S^{2}, that

(i) K(P) = 0, K(Q) > 0 for Q ≠ P,

(ii) Δ_{g^{0}}K(Q) ≥ 0 for all Q near P.

Then there exists a C^{1,1} isometric embedding X : (S^{2}, g^{0})→(R^{3}, δ) where δ is the standard flat metric on R^{3}.

The loss of strict positivity of K leads to degenerate Monge-Ampère equations. Certain types of degenerate Monge-Ampère equations have been studied by Caffarelli, Kohn, Nirenberg and Spruck ([3], [4]).

In this note we prove the following result.

Theorem: Suppose that g^{0} is a C^{4} Riemannian metric on S^{2} with Gauss curvature K^{0} ≥ 0. Then there exists a C^{1,1} isometric embedding X : (S^{2}, g^{0})→(R^{3}, δ) where δ is the standard flat metric on R^{3}.

The proof of this result contains the following proposition which is of independent interest.

Proposition: Let M be any closed convex surface in R^{3}, normalized so that the smallest ball containing M is centered at the origin. Set

a_{1} = min_{x∈M} |x|, a_{2} = max_{x∈M} |x|.

Then we have

max_{x∈M} |H(x)| ≤ a_{2}e^{\frac{(a_{2})^{2}−(a_{1})^{2}}{6(a_{2})^{2}}} \sqrt{max_{x∈M}(K(x)^{2} − \frac{3}{2}Δ_{g}K(x))},

where g denotes the metric on M induced from (R^{3}, δ), K(x) denotes the Gauss curvature at x ∈ M with respect to g and H(x) denotes the mean curvature of M at x ∈ M .

There are many interesting questions which still need to be studied. We mention a few in the following. Question 1. Does there exist some smooth Riemannian metric on S^{2} with nonnegative Gauss curvature which can never be C^{2} isometrically embedded into (R^{3}, δ)? Or conversely, is it possible to improve our theorem to C^{2} embedding (Even C^{2,α}(0 < α < 1), or C^{2,1}? ) instead of C^{1,1} embedding?

Question 2. What are the sufficient conditions ( even necessary and sufficient conditions ) on the metric with nonnegative Gauss curvature which give rise to a smooth isometric embedding into (R^{3}, δ) ?

Notice that the problems above are global since Lin proved (see [14]) that for any smooth 2-dimensional Riemannian metric with nonnegative Gauss curvature, there always exits some smooth local isometric embedding into (R^{3}, δ).

Acknowledgment. We would like to thank the referee for some helpful suggestions in the presentation of the results in this paper and clearing up many grammatical errors.

2 Proof of the Theorem

We first approximate g^{0} in C^{4} by a sequence of C^{∞} metrics g^{ϵ} , with corresponding Gauss curvature {K^{ϵ}} being positive everywhere. This can be achieved very easily, for instance, by first setting

g̃^{ϵ} = e^{2ϵw}g^{0}

with ϵ > 0 and w ∈ C^{4,α}(S^{2})(0 < α < 1). It is well known that K̃^{ϵ} satisfies

−ϵΔ_{g^{0}}w + K = K̃^{ϵ}e^{2ϵw}.

Clearly if we choose w such that −Δ_{g^{0}}w = 1 in {x ∈ S^{2} | K^{0}(x) = 0 } (notice that K^{0} has to be positive somewhere due to the Gauss-Bonnet theorem, therefore it is easy to find such w), and then choose ϵ to be very small, g̃^{ϵ} will be close to g^{0} in C^{4} and K̃^{ϵ} will be positive everywhere. Next we fix w, ϵ and hence g̃^{ϵ}, then we choose C^{∞} metrics g^{ϵ} which is C^{4} as close to g̃^{ϵ} as we want (hence K^{ϵ} > 0 everywhere).

Now we can apply Nirenberg′s theorem to g^{ϵ} (since K^{ϵ} > 0 everywhere and g^{ϵ} is smooth) and therefore obtain a C^{∞} isometric embedding X^{ϵ} : (S^{2}, g^{ϵ})→(R^{3}, δ).

It is not difficult to see that there exist constants α_{1}, β_{1} > 0 (independent of ϵ), such that, for all ϵ > 0, we have

0 < K^{ϵ}(ω) ≤ β_{1} ∀ ω ∈ S^{2},  (1)
R(X^{ϵ}) = min_{Y ∈R^{3}} max_{ω∈S^{2}} |X^{ϵ}(ω) − Y | ≤ α_{1}.  (2)

¿From now on we can simply assume that the origin is the center of the smallest ball containing the surface. It is elementary to see that R(X^{ϵ}) = ∥X^{ϵ}∥_{C^{0}(S^{2})}. With this normalization and (1) we immediately have

∥X^{ϵ}∥_{C^{0}} = R(X^{ϵ}) ≤ C.  (3)

where C is some constant independent of ϵ.

Write g^{0} and g^{ϵ} in local coordinates,

g^{0} = E^{0}du^{2} + 2F^{0}dudv + G^{0}dv^{2},
g^{ϵ} = E^{ϵ}du^{2} + 2F^{ϵ}dudv + G^{ϵ}dv^{2}.

We already know that X^{ϵ} : (S^{2}, g^{ϵ})→(R^{3}, δ) is an isometric embedding, so we have dX^{ϵ} · dX^{ϵ} = g^{ϵ}, namely,

X_{u}^{ϵ} · X_{u}^{ϵ} = E^{ϵ},
X_{u}^{ϵ} · X_{v}^{ϵ} = F^{ϵ},
X_{v}^{ϵ} · X_{v}^{ϵ} = G^{ϵ}.

It follows easily from the above that

∥▽_{g^{0}}X^{ϵ}∥_{C^{0}} ≤ C.  (4)

Where C is some constant independent of ϵ.

The following will be devoted to establishing a bound on ∥▽_{g^{0}}^{2}X^{ϵ}∥_{C^{0}}. Once we obtain such a bound, the limit of X^{ϵ}(along a subsequence) as ϵ→0 will be a C^{1,1} isometric embedding of g^{0}. For convenience, we drop the dependence on ϵ in our notation in the following.

Let us recall some elementary differential geometry formulae. Let X : (S^{2}, g)→(R^{3}, δ) be an isometric embedding where δ = dx^{2} + dy^{2} + dz^{2} is the standard flat metric on R^{3}. Let (u, v) denote local coordinates on part of the sphere and write

g = g_{11}du^{2} + 2g_{12}dudv + g_{22}dv^{2} = Edu^{2} + 2F dudv + Gdv^{2}.

Suppose X(u, v) = (x(u, v), y(u, v), z(u, v)), then the first fundamental form is given by

I = dX · dX = g.

Namely,

X_{u} · X_{u} = E = g_{11},
X_{u} · X_{v} = F = g_{12},
X_{v} · X_{v} = G = g_{22}.

Let the orientation be chosen so that the inner unit normal is given by

X̄ = \frac{X_{u} × X_{v}}{|X_{u} × X_{v}|} = \frac{X_{u} × X_{v}}{\sqrt{EG − F^{2}}} ,

where | · | denotes the Euclidean norm and × the standard cross product between vectors in R^{3}. The second fundamental form is then given by

II = −dX · dX̄ = Ldu^{2} + 2Mdudv + Ndv^{2},

where

L = −X_{u} · X̄_{u} = X_{uu} · X̄,
M = −X_{v} · X̄_{u} = −X_{u} · X̄_{v} = X_{uv} · X̄,
N = −X_{v} · X̄_{v} = X_{vv} · X̄.

Therefore the Gauss and mean curvature are ( Determinant and \frac{1}{2} trace of the second fundamental form with respect to the first fundamental form ):

K = \frac{LN − M^{2}}{EG − F^{2}} ,  (5)
H = \frac{1}{2} \frac{GL − 2F M + EN}{EG − F^{2}} .  (6)

Furthermore the Gauss theorem egregium asserts that K can be expressed in terms of up to second derivatives of E, F, G only.

The Gauss equation takes the form

\{ \begin{matrix}X_{uu} = Γ_{11}^{1}X_{u} + Γ_{11}^{2}X_{v} + LX̄, \\ X_{uv} = Γ_{12}^{1}X_{u} + Γ_{12}^{2}X_{v} + MX̄, \\ X_{vv} = Γ_{22}^{1}X_{u} + Γ_{22}^{2}X_{v} + NX̄, \end{matrix}  (7)

where Γ_{ij}^{k} are called Christoffel symbols associated with the metric tensor g_{ij} which are given by

Γ_{ij}^{k} = \frac{1}{2}g^{kl}(∂_{i}g_{jl} + ∂_{j}g_{il} − ∂_{l}g_{ij})  (8)

with ∂_{1} = ∂_{u} and ∂_{2} = ∂_{v}.

The Weingarten equation takes the form

\{ \begin{matrix}−X̄_{u} = L_{1}^{1}X_{u} + L_{1}^{2}X_{v}, \\ −X̄_{v} = L_{2}^{1}X_{u} + L_{2}^{2}X_{v}. \end{matrix}  (9)

where {L_{j}^{i}} are expressions involving L, M, N and E, F, G.

The Mainardi-Codazzi equation takes the form

\{ \begin{matrix}L_{v} − M_{u} = LΓ_{12}^{1} + M(Γ_{12}^{2} − Γ_{11}^{1}) − NΓ_{11}^{2}, \\ M_{v} − N_{u} = LΓ_{22}^{1} + M(Γ_{22}^{2} − Γ_{12}^{1}) − NΓ_{12}^{2}. \end{matrix}  (10)

Let Δ_{g} denote the Laplace-Beltrami operator associated with the metric g, namely,

Δ_{g} = \frac{1}{\sqrt{detg}} ∂_{j}(\sqrt{detg}g^{ij}∂_{i}) = g^{ij}▽_{i}▽_{j},

where (g^{ij}) is the inverse matrix of (g_{ij}) and ▽_{i} the covariant differentiation with respect to g in the direction ∂_{i}. It follows from (7) that

Δ_{g}X = 2HX̄.  (11)

We remark that in order to obtain a C^{2} bound on X it is enough to obtain a L^{∞} bound on H. The reason is the second fundamental form is positive definite, therefore a L^{∞} bound on H (\frac{1}{2}trace of the second fundamental form with respect to the first fundamental form) will give rise to a L^{∞} bound of L, M, N. Now the desired L^{∞} bound on X_{uu}, X_{uv} and X_{vv} will follow immediately from the Gauss equation (7) and the gradient estimate obtained in (4).

Therefore the only thing we need to do is to establish a L^{∞} bound on H. Notice that with the choice of inner unit normal, H ≥ 0.

Define

ρ(u, v) = \frac{1}{2}X · X.

The above function has been used by Darboux ( See [7], §707, see also [16] ).

Differentiating ρ(u, v) and using (7), we have

\{ \begin{matrix}ρ_{u} = X · X_{u}, ρ_{v} = X · X_{v} \\ ρ_{uu} = X · X_{uu} + E = Γ_{11}^{1}ρ_{u} + Γ_{11}^{2}ρ_{v} + LX · X̄ + E, \\ ρ_{uv} = X · X_{uv} + F = Γ_{12}^{1}ρ_{u} + Γ_{12}^{2}ρ_{v} + MX · X̄ + F, \\ ρ_{vv} = X · X_{vv} + G = Γ_{22}^{1}ρ_{u} + Γ_{22}^{2}ρ_{v} + NX · X̄ + G. \end{matrix}  (12)

We consider the following function on S^{2},

f = e^{αρ}H  (13)

with constant α > 0 being chosen later. Consider the following maximization

f_{max} = max_{S^{2}} f.

There exists some point P ∈ S^{2}, such that,

f(P) = f_{max}.

Let us write the metric g = g^{ϵ} near P in conformal coordinates:

g = e^{2h}(du^{2} + dv^{2}),  (14)

where

(u, v) = (0, 0) corresponds to P,

and

h = ∂_{u}h = ∂_{v}h = 0 at (0, 0).  (15)

It is not difficult to calculate in the uv coordinates, using (8), that

Γ_{11}^{1} = Γ_{12}^{2} = ∂_{1}h,
Γ_{11}^{2} = −∂_{2}h,
Γ_{12}^{1} = Γ_{22}^{2} = ∂_{2}h,
Γ_{22}^{1} = −∂_{1}h.

Here and in the following, h_{1} = ∂_{1}h = ∂_{u}h and h_{2} = ∂_{2}h = ∂_{v}h.

The Gauss and mean curvature formula ((5) and (6)) becomes

K = \frac{LN − M^{2}}{e^{4h}} ,  (16)

H = \frac{1}{2} \frac{L + N}{e^{2h}} .  (17)

K = −\frac{Δ̃h}{e^{2h}}.  (18)

where Δ̃ = ∂_{11} + ∂_{22}. The Mainardi-Codazzi equation becomes

\{ \begin{matrix}L_{2} − M_{1} = h_{2}(L + N) = 2He^{2h}h_{2} \\ M_{2} − N_{1} = −h_{1}(N + L) = −2He^{2h}h_{1} \end{matrix}  (19)

Clearly (see (14))

Δ_{g}K = e^{−2h}Δ̃K.  (20)

Differentiating (16) , we have

K_{1} = \frac{L_{1}N + LN_{1} − 2MM_{1}}{e^{4h}} − 4h_{1} \frac{LN − M^{2}}{e^{4h}}
= \frac{L_{1}N + LN_{1} − 2MM_{1}}{e^{4h}} − 4h_{1}K  (21)

K_{11} = \frac{L_{11}N + LN_{11} + 2N_{1}L_{1} − 2M_{1}^{2} − 2MM_{11}}{e^{4h}}

−\frac{4h_{1}(L_{1}N + LN_{1} − 2MM_{1})}{e^{4h}} − 4h_{11}K − 4h_{1}K_{1}

= \frac{L_{11}N + LN_{11} + 2N_{1}L_{1} − 2M_{1}^{2} − 2MM_{11}}{e^{4h}}
−8h_{1}K_{1} − 4h_{11}K − 16h_{1}^{2}K  (22)

The last equality follows from (21).

K_{2} = \frac{L_{2}N + LN_{2} − 2MM_{2}}{e^{4h}} − 4h_{2} \frac{LN − M^{2}}{e^{4h}}
= \frac{L_{2}N + LN_{2} − 2MM_{2}}{e^{4h}} − 4h_{2}K  (23)

K_{22} = \frac{L_{22}N + LN_{22} + 2N_{2}L_{2} − 2M_{2}^{2} − 2MM_{22}}{e^{4h}}
−8h_{2}K_{2} − 4h_{22}K − 16h_{2}^{2}K  (24)

Applying ∂_{2} to the first equation in the Mainardi-Codazzi equation (19), ∂_{1} to the second and then add them together, we have

(−L_{2} + 2He^{2h}h_{2})_{2} = (−N_{1} + 2He^{2h}h_{1})_{1},

which yields

N_{11} = L_{22} + 2H_{1}h_{1}e^{2h} − 2H_{2}h_{2}e^{2h} + 2(h_{11} − h_{22})He^{2h}
+4(h_{1}^{2} − h_{2}^{2})He^{2h}  (25)

Differentiating (17), we have

2H_{1} = \frac{L_{1} + N_{1}}{e^{2h}} − 2h_{1} \frac{L + N}{e^{2h}}
= \frac{L_{1} + N_{1}}{e^{2h}} − 4h_{1}H  (26)

2H_{11} = \frac{L_{11} + N_{11}}{e^{2h}} − 2h_{1} \frac{L_{1} + N_{1}}{e^{2h}} − 4h_{1}H_{1} − 4h_{11}H
= \frac{L_{11} + N_{11}}{e^{2h}} − 2h_{1}(2H_{1} + 4h_{1}H) − 4h_{1}H_{1} − 4h_{11}H
= \frac{L_{11} + N_{11}}{e^{2h}} − 8h_{1}H_{1} − 8h_{1}^{2}H − 4h_{11}H  (27)
2H_{2} = \frac{L_{2} + N_{2}}{e^{2h}} − 4h_{2}H  (28)

2H_{12} = \frac{L_{12} + N_{12}}{e^{2h}} − 4h_{2}H_{1} − 4h_{12}H − 4h_{1}H_{2} − 8h_{1}h_{2}H  (29)

2H_{22} = \frac{L_{22} + N_{22}}{e^{2h}} − 8h_{2}H_{2} − 8h_{2}^{2}H − 4h_{22}H  (30)

Applying ∂_{1} to the first equation in the Mainardi-Codazzi equation (19) and ∂_{2} to the second equation, we obtain

M_{11} = (L_{2} − 2h_{2}He^{2h})_{1}
= L_{12} − 2H_{1}e^{2h}h_{2} − 2He^{2h}h_{12} − 4Hh_{1}h_{2}e^{2h}  (31)
M_{22} = (N_{1} − 2h_{1}He^{2h})_{2}
= N_{12} − 2H_{2}e^{2h}h_{1} − 2He^{2h}h_{12} − 4Hh_{1}h_{2}e^{2h}  (32)

Using (22) and (24), we have

e^{6h}Δ_{g}K = e^{4h}Δ̃K
= (L_{11}N + LN_{11} + 2L_{1}N_{1} − 2M_{1}^{2} − 2MM_{11})

+(L_{22}N + LN_{22} + 2L_{2}N_{2} − 2M_{2}^{2} − 2MM_{22}) −e^{4h}(8h_{1}K_{1} + 8h_{2}K_{2} + 4(h_{11} + h_{22})K + 16(h_{1}^{2} + h_{2}^{2})K)

= N(L_{11} + L_{22}) + L(N_{11} + N_{22}) − 2M(M_{11} + M_{22})

+2(L_{1}N_{1} + L_{2}N_{2} − M_{1}^{2} − M_{2}^{2}) −e^{4h}(8h_{1}K_{1} + 8h_{2}K_{2} + 4(h_{11} + h_{22})K + 16(h_{1}^{2} + h_{2}^{2})K)

In the above formula we replace L_{22}, N_{11}, M_{11} + M_{22} by using (25) and (31). We have

e^{6h}Δ_{g}K = N(L_{11} + N_{11} − 2H_{1}h_{1}e^{2h} + 2H_{2}h_{2}e^{2h} − 2(h_{11} − h_{22})He^{2h} − 4(h_{1}^{2} − h_{2}^{2})He^{2h})

+L(L_{22} + N_{22} + 2H_{1}h_{1}e^{2h} − 2H_{2}h_{2}e^{2h} + 2(h_{11} − h_{22})He^{2h} + 4(h_{1}^{2} − h_{2}^{2})He^{2h}) −2M(L_{12} + N_{12} − 2(H_{1}h_{2} + H_{2}h_{1})e^{2h} − 4He^{2h}h_{12} − 8Hh_{1}h_{2}e^{2h}) +2(L_{1}N_{1} + L_{2}N_{2} − M_{1}^{2} − M_{2}^{2}) −e^{4h}(8h_{1}K_{1} + 8h_{2}K_{2} + 4(h_{11} + h_{22})K + 16(h_{1}^{2} + h_{2}^{2})K)

Now we use (27), (29) and (30) to replace L_{11} + N_{11}, L_{12} + N_{12}, L_{22} + N_{22} by H_{11}, H_{12}, H_{22} plus some lower order terms, we have

e^{6h}Δ_{g}K = N(2e^{2h}H_{11} + 6H_{1}h_{1}e^{2h}

+2H_{2}h_{2}e^{2h} + 2(h_{11} + h_{22})He^{2h} + 4(h_{1}^{2} + h_{2}^{2})He^{2h}) +L(2e^{2h}H_{22} + 6h_{2}H_{2}e^{2h} +2H_{1}h_{1}e^{2h} + 2(h_{11} + h_{22})He^{2h} + 4(h_{1}^{2} + h_{2}^{2})He^{2h}) −2M(2e^{2h}H_{12} + 2(H_{1}h_{2} + H_{2}h_{1})e^{2h}) +2(L_{1}N_{1} + L_{2}N_{2} − M_{1}^{2} − M_{2}^{2}) −e^{4h}(8h_{1}K_{1} + 8h_{2}K_{2} + 4(h_{11} + h_{22})K + 16(h_{1}^{2} + h_{2}^{2})K)

We now regroup the terms and , using (18), replace h_{11} + h_{22} by −e^{2h}K.

e^{6h}Δ_{g}K = 2e^{2h}(NH_{11} − 2MH_{12} + LH_{22})

+2(L_{1}N_{1} + L_{2}N_{2} − M_{1}^{2} − M_{2}^{2}) +N(6H_{1}h_{1}e^{2h} +2H_{2}h_{2}e^{2h} + 2(h_{11} + h_{22})He^{2h} + 4(h_{1}^{2} + h_{2}^{2})He^{2h}) +L(6h_{2}H_{2}e^{2h} +2H_{1}h_{1}e^{2h} + 2(h_{11} + h_{22})He^{2h} + 4(h_{1}^{2} + h_{2}^{2})He^{2h}) −4M(H_{1}h_{2} + H_{2}h_{1})e^{2h}

−e^{4h}(8h_{1}K_{1} + 8h_{2}K_{2} − 4K^{2}e^{2h} + 16(h_{1}^{2} + h_{2}^{2})K)  (33)

Notice that we derive from (26) and (28) that

\{ \begin{matrix}L_{1}N_{1} ≤ \frac{1}{4}(L_{1} + N_{1})^{2} = e^{4h}(H_{1} + 2h_{1}H)^{2} \\ L_{2}N_{2} ≤ \frac{1}{4}(L_{2} + N_{2})^{2} = e^{4h}(H_{2} + 2h_{2}H)^{2} \end{matrix}  (34)

Use the relation h_{11} + h_{22} = −e^{2h}K, (34) and evaluate (33) at P (So h = h_{1} = h_{2} = 0), we have (noticing that K ≤ H^{2} )

Δ_{g}K ≤ 2(NH_{11} − 2MH_{12} + LH_{22}) + 2(H_{1}^{2} + H_{2}^{2}).  (35)

It follows from (13) that the following holds at P ( Using f_{i}(P) = 0 ),

H_{1} = −αρ_{1}H, H_{2} = −αρ_{2}H,

H_{11} = −αHρ_{11} + α^{2}ρ_{1}^{2}H + f_{11}e^{−αρ}, H_{12} = −αHρ_{12} + α^{2}ρ_{1}ρ_{2}H + f_{12}e^{−αρ}, H_{22} = −αHρ_{22} + α^{2}ρ_{2}^{2}H + f_{22}e^{−αρ}.

Since f has P as the maximum point, we also have

(f_{ij}(P)) ≤ 0,  (36)

and therefore

\{ \begin{matrix}& (LH_{22} − 2MH_{12} + NH_{11}) \\ ≤ & H(L(−αρ_{22} + α^{2}ρ_{2}^{2}) − 2M(−αρ_{12} + α^{2}ρ_{1}ρ_{2}) + N(−αρ_{11} + α^{2}ρ_{1}^{2})) \end{matrix}  (37)

(H_{1}^{2} + H_{2}^{2}) = H^{2}α^{2}(ρ_{1}^{2} + ρ_{2}^{2})  (38)

Using (35), (37) and (38) we have, at point P, that

\{ \begin{matrix}& Δ_{g}K \\ ≤ & −2αH(Lρ_{22} − 2Mρ_{12} + Nρ_{11}) + 2α^{2}H(Lρ_{2}^{2} − 2Mρ_{1}ρ_{2} + Nρ_{1}^{2}) + 2(H_{1}^{2} + H_{2}^{2}) \end{matrix}  (39)

Notice that H_{i}^{2} = α^{2}ρ_{i}^{2}H^{2} and L, M, N ≤ 2H, we have at P that

\{ \begin{matrix}& Δ_{g}K \\ ≤ & −2αH(Lρ_{22} − 2Mρ_{12} + Nρ_{11}) + 2α^{2}H(L + N)(ρ_{1}^{2} + ρ_{2}^{2}) + 2H^{2}α^{2}(ρ_{1}^{2} + ρ_{2}^{2}) \\ = & −2αH(Lρ_{22} − 2Mρ_{12} + Nρ_{11}) + 6H^{2}α^{2}(ρ_{1}^{2} + ρ_{2}^{2}) \end{matrix}

Notice that we are in a geodesic normal coordinate, so it follows from (12) that, at P,

|X · X̄|^{2} + ρ_{1}^{2} + ρ_{2}^{2} = |X|^{2},

\{ \begin{matrix}ρ_{11} = & LX · X̄ + E, \\ ρ_{12} = & MX · X̄ + F, \\ ρ_{22} = & NX · X̄ + G. \end{matrix}

Therefore we have at P that,

Δ_{g}K
≤ −4αH(LN − M^{2})X · X̄ − 2αH(LG − 2MF + NE) + 6H^{2}α^{2}(ρ_{1}^{2} + ρ_{2}^{2})
= −4αHKX · X̄ − 2αH(L + N) + 6H^{2}α^{2}(ρ_{1}^{2} + ρ_{2}^{2})
≤ 4αHK|X · X̄| − 4αH^{2} + 6H^{2}α^{2}(ρ_{1}^{2} + ρ_{2}^{2})
≤ −4αH^{2} + 6α^{2}H^{2}|X|^{2} + \frac{2}{3}K^{2},

Choose α = \frac{1}{3}R^{−2}, where R = R(X) is as in (2), we have at P that

H^{2} ≤ R^{2}(K^{2} − \frac{2}{3}Δ_{g}K)

It follows immediately that

H(P) ≤ C.

It follows easily that

max_{S^{2}} f = f(P) = e^{αρ(P)}H(P) ≤ C,

namely,

max_{S^{2}} H = max_{S^{2}} e^{−αρ}f ≤ max_{S^{2}} Ce^{−αρ} ≤ C.

We have obtained the desired estimate on H.

Notice that if we keep track of the above constants in the above computation, we immediately obtain the the proof of the Proposition.

References

[1] A.D. Alexandroff, Intrinsic geometry of convex surfaces, OGIZ, Moscow-Leningrad, 1948.

[2] Y.D. Burago and S.Z. Shefel, The geometry of surfaces in Euclidean spaces, in Geometry III, Y.D. Burago and V.Z. Zalgaller (Eds.), Springer-Verlag, 1992.

[3] L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for the degenerate Monge-Ampère equation, Revista Matemática Iberoamericana, Vol. 2, No. 1 y 2 (1986), pp. 19-27.

[4] L. Caffarelli, J.J. Kohn, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations II. Complex Monge-Ampère, and uniformly elliptic, equations. Comm. Pure Appl. Math. 38 (1985), pp. 209-252.

[5] S.Y. Cheng and S.T. Yau, On the regularity of the solution of n-dimensional Minkowski problem, Comm. Pure Appl. Math., Vol. 29 (1976), pp. 495-516.

[6] E. Cohn-Vossen, Zwei Sätze über die Starrheit der Eiflächen, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1927), pp. 125-134.

[7] G. Darboux, Théorie des surfaces, Volume 3, 1894.

[8] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, 2^{ed} edition, Grundlehren 224, Springer, Berlin-Heidelberg-New York-Tokyo, 1983.

[9] E. Heinz, On Weyl′s embedding problem, J. Math. Mech., Vol. 11, No. 3 (1962), pp. 421-454.

[10] G. Herglotz, Über die Starrheit der Eiflächen, Abhandlungen des mathematischen Seminars der Hansischen Universität, 15 (1943), pp. 127-129.

[11] J.A. Iaia, Isometric embeddings of surfaces with nonnegative curvature in R^{3}, Duke Math. J., Vol. 67, No. 2 (1992), pp. 423-459.

[12] H. Lewy, A priori limitations for solutions of Monge-Ampère equations I, Trans. Amer. Math. Soc., 37 (1935), pp. 417-434.

[13] H. Lewy, On the existence of a closed convex surface realizing a given Riemannian metric, Proceedings of the National Academy of Sciences, U.S.A., Vol. 24, No. 2 (1938), pp. 104-106.

[14] C.S. Lin, The local isometric embedding in R^{3} of 2-dimensional Riemannian manifolds with nonnegative curvature, J. Diff. Geom. 21 (1985), pp. 213-230.

[15] F.H. Lin, On Darboux′s system, preprint.

[16] L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure and Appl. Math., 6 (1953), pp. 337-394.

[17] L. Nirenberg, On nonlinear elliptic and partial differential equations and Hölder continuity, in the large, Comm. Pure and Appl. Math., 6 (1953), pp. 103-156.

[18] F. Schulz, Regularity theory for quasilinear elliptic systems and Monge-Ampère equations in two dimensions, Lecture Notes in Mathematics, Vol. 1445, Springer-Verlag, 1990.

[19] A.V. Pogorelov, On the regularity of convex surfaces with regular metric, Doklady, Akademia Nauk, S.S.S.R., (N.S.) 66 (1949), pp. 1051-1053.

[20] A.V. Pogorelov, On convex surfaces with regular metric, Doklady, Akademia Nauk, S.S.S.R., (N.S.) 67 (1949), pp. 791-794.

[21] R. Sacksteder, The rigidity of hypersurfaces, J. Math. Mech., Vol. 11, No. 6, 1962, pp. 929-939.

[22] H. Weyl, Über die Bestimmung einer geschlossen konvexen Fläche durch ihr Linienelement, Vierteljahrsschrift der naturforschenden Gesellschaft, Zürich, 61 (1916), pp.40-72.

[23] O.K.Zhitomirsky, Sur la non-flexibilité des ovaloides, Akademia Nauk, S.S.S.R., Doklady, (N.S.) 25 (1939), pp. 347-349.