AN ESTIMATE FOR THE GAUSS CURVATURE OF MINIMAL

SURFACES IN ℝ^{m} WHOSE GAUSS MAP OMITS A SET OF HYPERPLANES

ROBERT OSSERMAN AND MIN RU

Abstract. We give an estimate of the Gauss curvature for minimal surfaces in ℝ^{m} whose Gauss map omits more than m(m + 1)/2 hyperplanes in ℙ^{m−1}(ℂ).

1. Introduction

The purpose of this paper is to prove the following theorem.

Theorem 1.1 (Main Theorem). Let x : M → ℝ^{m} be a minimal surface immersed in ℝ^{m}. Suppose that its generalized Gauss map g omits more than \frac{m(m+1)}{2} hyperplanes in ℙ^{m−1}(ℂ), located in general position. Then there exists a constant C, depending on the set of omitted hyperplanes, but not the surface, such that

|K(p)|^{1/2}d(p) ≤ C (1)

where K(p) is the Gauss curvature of the surface at p, and d(p) is the geodesic distance from p to the boundary of M.

This theorem provides a considerable sharpening of an earlier result of the same type:

Theorem 1.2. (Osserman [O1]) An inequality of the form (1) holds for all minimal surfaces in ℝ^{m} whose Gauss map omits a neighborhood of some hyperplane in ℙ^{m−1}(ℂ).

Also, Theorem 1.1 implies the earlier result:

Theorem 1.3. (Ru[1]) Let x : M → ℝ^{m} be a complete minimal surface immersed in ℝ^{m}. Suppose that its generalized Gauss map g omits more than \frac{m(m+1)}{2} hyperplanes in ℙ^{m−1}(ℂ), located in general position. Then g is constant and the minimal surface must be a plane.

In fact, given any point p on a complete surface satisfying the hypotheses, inequality (1) must hold with d(p) arbitrarily large, so that K(p) = 0. But a minimal surface in ℝ^{m} with K ≡ 0 must lie on a plane (see [HO1]) and hence its Gauss map g is constant.

Research supported in part by NSF grant DMS-9506424, and at MSRI by NSF grant DMS-9022140.

Theorem 1.3 had been proved earlier by Fujimoto [F2] in the case where the Gauss map g was assumed nondegenerate. Fujimoto (see [F3]) also showed that the number m(m + 1)/2 was optimal in that for every odd dimension m, there exist complete minimal surfaces whose Gauss map omits m(m+1)/2 hyperplanes in general position. It follows that Theorem 1.1 is also an optimal result of its type, since with any smaller number of omitted hyperplanes, a universal inequality of the form (1) cannot be valid, at least in odd dimensions.

When m = 3, we may consider the classical Gauss map into the unit sphere. Fujimoto [F1] showed that an inequality of type (1) holds whenever the Gauss map omits 5 given points. In fact he obtained an explicit expression for C in that case, depending on the given points. Ros [Ro1] gave a different proof that does not yield an explicit value for the constant C, but which allows the extension to higher dimension that we give here.

2. Some theorems and lemmas

In this section, we recall some results which will be used later.

We first recall the following construction theorem of minimal surfaces.

Theorem 2.1. (see [CO1]) Let M be an open Riemann surface and let ω_{1}, ω_{2}, …, ω_{m} be holomorphic forms on M having no common zero, no real periods and locally satisfying the identity

f_{1}^{2} + f_{2}^{2} + ⋯ + f_{m}^{2} = 0

for holomorphic functions f_{i} with ω_{i} = f_{i}dz. Set

x_{i} = 2Re ∫_{z_{0}}^{z} ω_{i},

for an arbitrary fixed point z_{0} of M. Then the surface x = (x_{1}, …, x_{m}) : M → ℝ^{m} is a minimal surface immersed in ℝ^{m} such that the Gauss map is the map g = [ω_{1} : ⋯: ω_{m}] : M → Q_{m−2}(ℂ) and the induced metric is given by

ds^{2} = 2(|ω_{1}|^{2} + ⋯ + |ω_{m}|^{2}).

The following is the general version of Hurwitz’s theorem:

Theorem 2.2 (Hurwitz’s theorem). Let f_{j} : M → N be a sequence of holomorphic maps between two connected complex manifolds converging uniformly on every compact subset of M to a holomorphic map f. If the image of each map f_{j} misses a divisor D of N then either the image of f misses D or it lies entirely in D.

GAUSS CURVATURE OF CERTAIN MINIMAL SURFACES IN ℝ^{m} 3

Proof. Assume first that D = {z|g(z) = 0} for some holomorphic function g. Then g ∘ f_{j} is a sequence of holomorphic functions converging to the holomorphic function g∘f. Since g∘f_{j} is non-vanishing, by the classical Hurwitz theorem the limit function is either identically zero or non-vanishing. In other words the image of f either lies entirely in D or misses D completely.

In the general case, if f does not miss D entirely then there exists a point q in D and a point p in M such that f(p) = q. There exists a neighborhood U of q and a holomorphic function g on U so that D ∩ U = {z|g(z) = 0}. Applying the previous argument to the restriction of the sequence of maps to the open set V = f^{−1}(U) in U, we conclude that f(V) is contained in D ∩ U. Since M is connected, the principle of analytic continuation implies that the image f(M) is contained in D.

Lemma 2.1. Let D_{r} be the disk of radius r, 0 < r < 1, and let R be the hyperbolic radius of D_{r}. Let

ds^{2} = λ(z)^{2}|dz|^{2}

be any conformal metric on D_{r} with the property that the geodesic distance from z = 0 to |z| = r is greater than or equal to R. If the Gauss curvature K of the metric ds^{2} satisfies

−1 ≤ K ≤ 0

then the distance of any point to the origin in the metric ds^{2} is greater than or equal to the hyperbolic distance.

Remark 2.1. The hyperbolic metric in the unit disk is given by

dŝ^{2} = λ̂(z)^{2}|dz|^{2}, λ̂(z) = \frac{2}{1 − |z|^{2}} ,

and has curvature K̂ ≡ −1. The relation between the quantities R and r is therefore given by

R = ∫_{0}^{r} λ̂(z)|dz| = ∫_{0}^{r} \frac{2}{1 − t^{2}} dt = log \frac{1 + r}{1 − r}

and the conclusion of the theorem is that

ρ(z) ≥ ρ̂(z) = log \frac{1 + |z|}{1 − |z|}

where ρ and ρ̂ represent the distances from the point z to the origin in the metric ds^{2} and the hyperbolic metric, respectively.

Remark 2.2. Lemma 2.1 and its proof are basically geometric reformulations of lemma 6 of Ros[Ro 1]. The lemma may be viewed as a kind of dual to the Ahlfors form of the Schwarz-Pick lemma [A 1].

Proof of lemma 2.1. Note first, that in the relation above between R and r, we have

\frac{dR}{dr} = \frac{2}{1 − r^{2}} > 0

and we may solve for r in terms of R:

r = \frac{e^{R} − 1}{e^{R} + 1}, (2)

or in general

|z| = \frac{e^{ρ̂(z)} − 1}{e^{ρ̂(z)} + 1}, (3)

where the right-hand side is monotone increasing in ρ̂(z). We may apply a comparison theorem of Greene and Wu ([GW 1], Prop. 2.1, p. 26) to the two metrics, ds^{2} and the hyperbolic metric dŝ^{2}, on the disk |z| ≤ r. The comparison theorem states that for any smooth monotone increasing function f, one has

△(f ∘ ρ) ≤ ˆ△(f ∘ ρ̂)

where ρ and ρ̂ are the distances to the origin in the metrics ds^{2} and dŝ^{2} respectively, △ and ˆ△ are the Laplacians with respect to the two metrics, and the two sides are evaluated at points of the same level sets of the two metrics, i.e. ρ = c on the left and ρ̂ = c on the right, provided in two dimensions that the Gauss curvatures K and K̂ satisfy 0 ≥ K ≥ K̂, with a similar condition on Ricci curvature in higher dimension. In our case we have 0 ≥ K ≥ −1 = K̂, and so we may apply the theorem. We note that the function

log |z| = log \frac{e^{ρ̂(z)} − 1}{e^{ρ̂(z)} + 1}

is harmonic with respect to z and is therefore also harmonic with respect to any conformal metric on 0 < |z| < 1. In other words, if we set

f(t) = log \frac{e^{t} − 1}{e^{t} + 1}

we have

ˆ△(f ∘ ρ̂) ≡ 0

for 0 < |z| < 1. Since f is monotone increasing, we may apply the Greene-Wu comparison theorem to conclude that

△(f ∘ ρ) ≤ 0

for 0 < |z| < r, i.e. f ∘ ρ is superharmonic. For z near 0, we have ρ(z) ∼ λ(0)|z|, and we may apply the minimum principle to the function

log \frac{1}{|z|}f(ρ(z)) = log \frac{1}{|z|} \frac{e^{ρ(z)} − 1}{e^{ρ(z)} + 1},

which is superharmonic in 0 < |z| < r and bounded near the origin, to conclude that it takes on its minimum on the boundary |z| = r. But since ρ(z) ≥ R on |z| = r, we have for |z| < r that

log \frac{1}{|z|} \frac{e^{ρ(z)} − 1}{e^{ρ(z)} + 1} ≥ log \frac{1}{r} \frac{e^{R} − 1}{e^{R} + 1} = 0,

by (2). Hence

\frac{e^{ρ(z)} − 1}{e^{ρ(z)} + 1} ≥ |z| = \frac{e^{ρ̂(z)} − 1}{e^{ρ̂(z)} + 1},

by (3), which implies ρ(z) ≥ ρ̂(z), proving the lemma.

As an application of lemma 2.1, we have the following lemma:

Lemma 2.2. Let ds_{n}^{2} be a sequence of conformal metrics on the unit disk D whose curvatures satisfy −1 ≤ K_{n} ≤ 0. Suppose that D is a geodesic disk of radius R_{n} with respect to the metric ds_{n}^{2}, where R_{n} → ∞, and that the metrics ds_{n}^{2} converge, uniformly on compact sets, to a metric ds^{2}. Then all distances to the origin with respect to ds^{2} are greater than or equal to the corresponding hyperbolic distances in D. In particular, ds^{2} is complete.

Proof. For any point z in D, let ρ_{n}(z) be the distance from 0 to z in the metric ds_{n}^{2} and let ρ(z) be the distance in the limit metric ds^{2}. Let |z| = r_{n} be the circle in D of hyperbolic radius R_{n}. Explicitly, by Remark 2.1 above,

R_{n} = log \frac{1 + r_{n}}{1 − r_{n}} .

If we make the change of parameter w = r_{n}z, we may apply lemma 2.1 to the induced metric in |w| < r_{n} and conclude that

ρ_{n}(z) ≥ log \frac{1 + |w|}{1 − |w|} = log \frac{1 + r_{n}|z|}{1 − r_{n}|z|}.

As n → ∞ we have R_{n} → ∞ and r_{n} → 1. Hence, by uniform convergence on compact sets, we have

ρ(z) = lim_{n→∞} ρ_{n}(z) ≥ lim_{r_{n}→1} log \frac{1 + r_{n}|z|}{1 − r_{n}|z|} = log \frac{1 + |z|}{1 − |z|},

which proves the lemma.

Note: Although we shall not make use of it, we remark that lemma 2.1 also implies another dual form of the Ahlfors-Schwarz-Pick lemma, closer in form to the original:

Lemma 2.3. Let S be a simply-connected surface with a complete metric ds^{2} whose Gauss curvature satisfies −1 ≤ K ≤ 0. If S is mapped conformally onto the unit disc, then the distance between any two points of S is greater than or equal to the hyperbolic distance between the corresponding points in the disk.

Proof. Given two points p, q of S, we may map p onto the origin, and let z be the image of the point q. Then the distance between p and q on S is given by ρ(z) in terms of the pull-back of the metric on S onto the disk. For any r such that |z| < r < 1, let ρ̂(z) be the hyperbolic distance from 0 to z and let ρ_{r}(w) be the pullback of the metric on S to |w| < r under the map z = w/r. Then, since S is complete, we may apply lemma 2.1 to conclude that

ρ̂(z) ≤ ρ_{r}(w) = ρ_{r}(rz).

But as r → 1, ρ_{r}(rz) → ρ(z), which proves the lemma.

Note that Lemma 2.3 combined with the standard Ahlfors-Schwarz-Pick lemma implies a generalization of Ahlfors’ lemma due to Yau ([Y1]; see also Troyanov [T1]): Let S_{1} be a simply-connected Riemann surface with a complete metric ds^{2} whose Gauss curvature satisfies −1 ≤ K ≤ 0, and let S_{2} be a Riemann surface with Gauss curvature bounded above by −1. Let f : S_{1} → S_{2} be a holomorphic map. Then f is distance decreasing.

We also need the following more precise version of theorem 1.3; the proof follows exactly as in Ru[1].

Theorem 2.3. (cf. Ru[1]) Let x : M → ℝ^{m} be a complete minimal surface immersed in ℝ^{m}. Suppose that its generalized Gauss map g omits the hyperplanes H_{1}, …, H_{q} in ℙ^{m−1}(ℂ) and g(M) is contained in some ℙ(V), where V is a subspace of ℂ^{m} of dimension k. Assume that H_{1} ∩ ℙ(V), …, H_{q} ∩ ℙ(V) are in general position in ℙ(V) and q > k(k + 1)/2. Then g must be constant.

The following theorem due to M. Green (see [G1]) shows that the complement of 2m + 1 hyperplanes in general position in ℙ^{m}(ℂ) is complete Kobayashi hyperbolic.

Theorem 2.4. Let H_{1}, …, H_{q} be hyperplanes in ℙ^{m}(ℂ), located in general position. If q ≥ 2m + 1, then X = ℙ^{m}(ℂ) − ∪_{j=1}^{q}H_{j} is complete hyperbolic and hyperbolically imbedded in ℙ^{m}(ℂ). Hence, if D ⊂ ℂ is the unit disc, and Φ is a subset of Hol(D, X), then Φ is relatively locally compact in Hol(D, ℙ^{m}(ℂ)), i.e. given a sequence {f_{n}} in Φ there exists a subsequence which converges uniformly on every compact subset of D to an element of Hol(D, ℙ^{m}(ℂ)).

For the notions of “complete Kobayashi hyperbolicity” and “hyperbolically imbedded in ℙ^{m}(ℂ)”, see Lang [L1].

Before going to the next section, we recall here a standard definition.

Definition 2.1. Let f : M → ℙ^{n}(ℂ) be a holomorphic map. Let p ∈ M. A local reduced representation of f around p is a holomorphic map f̃: U → ℂ^{n+1}−{0}, such that ℙ(f̃) = f, where U is a neighborhood of p, and ℙ is the projection map of ℂ^{n+1} − {0} onto ℙ^{n}(ℂ).

3. Proof of the main theorem

Let x : M → ℝ^{m} be a minimal surface, where M is a connected, oriented, realdimension 2 manifold without boundary, and x = (x_{1}, …, x_{m}) is an immersion. Then M is a Riemann surface in the induced structure defined by local isothermal coordinates (u, v). The generalized Gauss map of the minimal surface, g = [\frac{∂x_{1}}{∂z} : ⋯: \frac{∂x_{m}}{∂z}] : M → Q_{m−2}(ℂ) ⊂ ℙ^{m−1}(ℂ) is a holomorphic map, where z = u + iv. The metric ds^{2} on M, induced from the standard metric in ℝ^{m}, is ds^{2} = ∑_{j=1}^{m} |\frac{∂x_{j}}{∂z} |^{2}dzdz̄, and the Gauss curvature K is given by ([HO1] p.37)

K = −4\frac{|g̃ ∧ g̃′|^{2}}{|g̃|^{6}} = −4\frac{∑_{j<k} |g_{j}g′_{k} − g_{k}g′_{j}|^{2}}{(∑_{j=1}^{m} |g_{j}|^{2})^{3}} (4)

where g̃ = (g_{1}, …, g_{m}), g_{j} = \frac{∂x_{j}}{∂z} , 1 ≤ j ≤ m.

We will need the following lemma:

Lemma 3.1. Let M be a Riemann surface. Let f_{n} : M → ℙ^{m}(ℂ) be a sequence of holomorphic maps converging uniformly on every compact subset of M to a holomorphic map f : M → ℙ^{m}(ℂ). Given a, b ∈ ℙ^{m}(ℂ^{∗}), let f_{a,b} be the meromorphic function (called coordinate function) defined by

f_{a,b}|_{U} = \frac{α(f̃)}{β(f̃)},

where f̃ is a reduced representation of f on U, and α, β ∈ ℂ^{m+1^{∗}} such that a = ℙ(α), b = ℙ(β). Assume that β(f̃) ≢ 0 on some U (i.e. the image of f is not contained in the hyperplane defined by b). Let p ∈ M be such that β(f̃)(p) ≠ 0, and U_{p} be a neighborhood of p such that β(f̃)(z) ≠ 0 for z ∈ U_{p}; then {f_{n_{a,b}}} converges uniformly on U_{p} to the meromorphic function f_{a,b}.

Proof. Since the image of f is not contained in the hyperplane defined by b, the image of f_{n} is also not contained in the hyperplane defined by b for n large enough. Since \frac{a(x)}{b(x)} is a rational function on ℙ^{m}(ℂ) and f_{n} converges uniformly on every compact subset of M to f, the composition functions also converge compactly. This concludes the proof.

Lemma 3.2. Let x^{(n)} = (x_{1}^{(n)}, …, x_{m}^{(n)}) : M → ℝ^{m} be a sequence of minimal immersions, and g^{(n)} : M → Q_{m−2}(ℂ) ⊂ ℙ^{m−1}(ℂ) the sequence of their (generalized) Gauss maps. Suppose that {g^{(n)}} converges uniformly on every compact subset of M to a non-constant holomorphic map g : M → Q_{m−2}(ℂ) ⊂ ℙ^{m−1}(ℂ) and that there is some p_{0} ∈ M such that for each j, 1 ≤ j ≤ m, {x_{j}^{(n)}(p_{0})} converges. Assume also that {|K_{n}|} is uniformly bounded, where K_{n} is the Gauss curvature of the minimal surface x^{(n)}. Then

(i) either a subsequence {K_{n′}} of {K_{n}} converges to zero or

(ii) a subsequence {x^{(n′)}} of {x^{(n)}} converges to a minimal immersion, x : M → ℝ^{m}, whose Gauss map is g.

Proof. By assumption, g is not constant and we may assume that |K_{n}| ≤ 1 in M, for each n ∈ N. For every point p ∈ M let (U_{p}, z) be a complex local coordinate centered

at p. Let g̃^{(n)} = (g_{1}^{(n)}, …, g_{m}^{(n)}) where g_{i}^{(n)} = \frac{∂x_{i}^{(n)}}{∂z} , 1 ≤ i ≤ m, and let g̃ = (g_{1}, …, g_{m}) be a local reduced representation of g on U_{p}. Since some g_{i}(z) is non-zero for each z, we know that g(M) is not contained in some coordinate hyperplane. Without loss of generality, we assume that g(M) is not contained in the first coordinate hyperplane H_{1} = {[y_{1} : ⋯: y_{m}] ∈ ℙ^{m−1}(ℂ)|y_{1} = 0}. Let

M_{1} = {p ∈ M|g(p) ∉ H_{1}, g̃(p) ∧ g̃′(p) ≠ 0}.

Note that M − M_{1} is a discrete set: namely, it consists of the zeros of g_{1} (which are isolated, since g(M) ⊄ H_{1}, which is equivalent to g_{1} ≢ 0) together with the common zeros of the components of g̃ ∧ g̃′, which are the holomorphic functions g_{j}g′_{k} − g_{k}g′_{j}. In particular,

g_{1}g′_{k} − g_{k}g′_{1} = g_{1}^{2}(\frac{g_{k}}{g_{1}} )′

so that g̃ ∧ g̃′ ≡ 0 implies that g_{k}/g_{1} = c_{k}, a constant for each k, so that g̃ = g_{1}(1, c_{2}, …, c_{m}) and the map g would be constant, contrary to assumption. Thus, the zeros of g̃ ∧ g̃′ are isolated and the points of M − M_{1} are also isolated.

Let p ∈ M_{1}. Since g(p) ∉ H_{1}, there is a neighborhood U_{p} of p such that g_{1}(z) ∉ H_{1}, and g^{(n)}(z) ∉ H_{1} for n large enough, for every z ∈ U_{p}. Choosing U_{p} sufficiently small, we have that g_{2}/g_{1}, …, g_{n}/g_{1} are holomorphic and

4\frac{|g̃ ∧ g̃′|^{2}/|g_{1}|^{4}}{(1 + ∑_{j=2}^{m} |g_{j}/g_{1}|^{2})^{3}} = 4\frac{∑_{j<k} |\frac{g_{j}}{g_{1}} (\frac{g_{k}}{g_{1}} )′ − \frac{g_{k}}{g_{1}} (\frac{g_{j}}{g_{1}} )′|^{2}}{(1 + ∑_{j=2}^{m} |g_{j}/g_{1}|^{2})^{3}} ≥ 2c_{1},

in U_{p}, where c_{1} is some positive constant. Since g^{(n)} → g uniformly, by lemma 3.1, {g_{j}^{(n)}/g_{1}^{(n)}} converges uniformly to g_{j}/g_{1} on U_{p}, 1 ≤ j ≤ m. So we have

4\frac{∑_{j<k} |\frac{g_{j}^{(n)}}{g_{1}^{(n)}} (\frac{g_{k}^{(n)}}{g_{1}^{(n)}} )′ − \frac{g_{k}^{(n)}}{g_{1}^{(n)}} (\frac{g_{j}^{(n)}}{g_{1}^{(n)}} )′|^{2}}{(1 + ∑_{j=2}^{m} |g_{j}^{(n)}/g_{1}^{(n)}|^{2})^{3}} ≥ c_{1},

in U_{p} and so, by (4),

\frac{c_{1}}{|g_{1}^{(n)}|^{2}} ≤ 4\frac{∑_{l<k} |\frac{g_{l}^{(n)}}{g_{1}^{(n)}} (\frac{g_{k}^{(n)}}{g_{1}^{(n)}} )′ − \frac{g_{k}^{(n)}}{g_{1}^{(n)}} (\frac{g_{l}^{(n)}}{g_{1}^{(n)}} )′|^{2}}{|g_{1}^{(n)}|^{2}(1 + ∑_{j=2}^{m} |g_{j}^{(n)}/g_{1}^{(n)}|^{2})^{3}} = |K_{n}| ≤ 1,

in U_{p}. Therefore

c_{1} ≤ |g_{1}^{(n)}|^{2}

in U_{p}, for large n. Then {g_{1}^{(n)}} is relatively compact in M(U_{p}). Noticing that M −M_{1} is discrete, by taking a subsequence, if necessary, we can assume that the globally defined holomorphic 1-forms {g_{1}^{(n)}dz} converge on M_{1}, to a holomorphic 1-form h_{1}dz or to infinity, uniformly on every compact subset of M_{1}. We consider each case below:

Case 1. {g_{1}^{(n)}dz} converges to infinity uniformly on every compact subset of M_{1}.

For p ∈ M_{1}, we have, by (4),

K_{n}(p) = −4\frac{∑_{j<k} |\frac{g_{j}^{(n)}}{g_{1}^{(n)}} (\frac{g_{k}^{(n)}}{g_{1}^{(n)}} )′ − \frac{g_{k}^{(n)}}{g_{1}^{(n)}} (\frac{g_{j}^{(n)}}{g_{1}^{(n)}} )′|^{2}}{|g_{1}^{(n)}|^{2}(1 + ∑_{j=2}^{m} |g_{j}^{(n)}/g_{1}^{(n)}|^{2})^{3}} → 0. (5)

Let p be a point such that p ∉ M_{1} but also g(p) ∉ H_{1}; then in a small disc of U_{p}, D(2ϵ), g^{(n)}(z) ∉ H_{1} for n large enough, z ∈ D(2ϵ). This means that g_{1}^{(n)} is non-vanishing on D(2ϵ) and g_{1}^{(n)} converges to infinity on ∂D(ϵ). From the maximum principle we conclude that {g_{1}^{(n)}} converges to infinity on D(ϵ). Therefore we again have K_{n}(p) → 0 by (4).

Finally suppose that g(p) ∈ H_{1}, i.e. g_{1}(p) = 0. Since g(p) is not contained in some coordinate hyperplane, we assume that g(p) ∉ H_{2}, where H_{2} is the second coordinate hyperplane, H_{2} = {[y_{1} : ⋯: y_{n}] ∈ ℙ^{n−1}(ℂ)|y_{2} = 0}. Therefore, on a small disc, D(2ϵ), g^{(n)}(z) ∉ H_{2} for n large enough, i.e. g_{2}^{(n)}(z) ≠ 0, for z ∈ D(2ϵ), and g_{1}^{(n)}, g_{1} have no zeros other than the point p on D(2ϵ) for n large enough. By lemma 3.1,

\{\frac{g_{2}^{(n)}}{g_{1}^{(n)}} \}, as a sequence of non-vanishing holomorphic functions, converges uniformly

on ∂D(ϵ). Clearly, \{\frac{g_{2}^{(n)}}{g_{1}^{(n)}} g_{1}^{(n)}\} converges uniformly to infinity on ∂D(ϵ), and therefore g_{2}^{(n)} converges uniformly to infinity on ∂D(ϵ). Again from the maximum principle, we conclude that g_{2}^{(n)} converges to infinity on D(ϵ). By (4), noticing that

|g̃^{(n)} ∧ g̃^{(n)′}|^{2}/|g_{2}^{(n)}|^{4} = ∑_{j<k} |\frac{g_{j}^{(n)}}{g_{2}^{(n)}} (\frac{g_{k}^{(n)}}{g_{2}^{(n)}} )′ − \frac{g_{k}^{(n)}}{g_{2}^{(n)}} (\frac{g_{j}^{(n)}}{g_{2}^{(n)}} )′|^{2},

we have

K_{n}(p) = −4\frac{∑_{j<k} |\frac{g_{j}^{(n)}}{g_{2}^{(n)}} (\frac{g_{k}^{(n)}}{g_{2}^{(n)}} )′ − \frac{g_{k}^{(n)}}{g_{2}^{(n)}} (\frac{g_{j}^{(n)}}{g_{2}^{(n)}} )′|^{2}}{|g_{2}^{(n)}|^{2}(∑_{j=1}^{n} |g_{j}^{(n)}/g_{2}^{(n)}|^{2})^{3}} → 0.

Thus, we have proved that K_{n}(p) → 0 for all p ∈ M. This corresponds to case (i) of the lemma.

Case 2. {g_{1}^{(n)}dz} converges to a holomorphic 1-form, h_{1}dz, on M_{1}.

Let p ∈ M −M_{1}. If D(2ϵ) is a small disc contained in U_{p}, as {g_{1}^{(n)}} → h_{1} uniformly on ∂D(ϵ) and g_{1}^{(n)} are holomorphic, using the maximum principle, we see that {g_{1}^{(n)}} is relatively compact on D(ϵ). Therefore h_{1}dz extends to a holomorphic 1-form on M and the global 1-forms {g_{1}^{(n)}dz} converge to h_{1}dz on M.

We now prove that, for every integer j, 2 ≤ j ≤ m, the global 1-forms {g_{j}^{(n)}dz} converge to a holomorphic form h_{j}dz on M. Let p ∈ M such that g(p) ∉ H_{1}; then there is a neighborhood U_{p} of p such that g_{1}, g_{1}^{(n)} have no zeros for n large enough, z ∈ U_{p}. Since g_{j}^{(n)} = \frac{g_{j}^{(n)}}{g_{1}^{(n)}} g_{1}^{(n)}, and by lemma 3.1, \{\frac{g_{j}^{(n)}}{g_{1}^{(n)}} \} converges uniformly on U_{p}, and g_{1}^{(n)} also converges uniformly on U_{p}, {g_{j}^{(n)}} must converge uniformly on U_{p}. For the points p such that g(p) ∈ H_{1}, if D(2ϵ) ⊂ U_{p} is small enough so that g_{1}, g_{1}^{(n)} have no zeros other than p for n large enough on D(2ϵ), then we just proved that {g_{j}^{(n)}} is uniformly convergent on ∂D(ϵ). Since g_{j}^{(n)} are holomorphic, by the maximum principle, we have that g_{j}^{(n)} converges uniformly on D(ϵ). Therefore the globally defined holomorphic 1-forms g_{j}^{(n)}dz converge to ω_{j} = h_{j}dz, 1 ≤ j ≤ m. Obviously these ω_{j} satisfy the conditions in Theorem 2.1 since the g_{j}^{(n)} satisfy the conditions in Theorem 2.1 for every n. So they define a minimal surface x : M → ℝ^{m} whose Gauss map is g.

We now prove the main theorem.

Proof of the Main Theorem.

Suppose the theorem is not true. We will construct a nonflat complete minimal surface whose Gauss map omits a set of hyperplanes in general position, thus getting a contradiction with theorem 2.3. So suppose the conclusion of the theorem is not true; then there is a sequence of (non complete) minimal surfaces x^{(n)} : M_{n} → ℝ^{m} and points p_{n} ∈ M_{n} such that |K_{n}(p_{n})|d_{n}^{2}(p_{n}) → ∞, and such that the Gauss map g^{(n)} of x^{(n)} omits a fixed set of q hyperplanes in general position, with q > m(m + 1)/2.

We claim that the surfaces M_{n} can be chosen so that

K_{n}(p_{n}) = −1, −4 ≤ K_{n} ≤ 0 on M_{n} for all n and d_{n}(p_{n}) → ∞. (6)

We now prove the claim. Without loss of generality, we can assume that M_{n} is a geodesic disk centered at p_{n}. Let M′_{n} = {p ∈ M_{n} : d_{n}(p, p_{n}) ≤ d_{n}(p_{n})/2}. Then K_{n} is uniformly bounded on M′_{n} and d′_{n}(p) = distance of p to the boundary of M′_{n} tends to zero as p → ∂M′_{n}. Hence |K_{n}(p)|(d′_{n}(p))^{2} has a maximum at a point p′_{n} interior to M′_{n}. Therefore

|K_{n}(p′_{n})|d′_{n}(p′_{n})^{2} ≥ |K_{n}(p_{n})|d′_{n}(p_{n})^{2} = \frac{1}{4}|K_{n}(p_{n})|d_{n}^{2}(p_{n}) → ∞.

So we can replace the M_{n} by the M′_{n}, with |K_{n}(p′_{n})|d′_{n}(p′_{n})^{2} → ∞. We rescale M′_{n} to make K_{n}(p′_{n}) = −1. By the invariance under scaling of the quantity K(p)d(p)^{2}, we will have d′_{n}(p′_{n}) → ∞; here, without causing confusion, we use the same notation d′_{n} to denote the geodesic distance with respect to the rescaled metric. Again we can assume that M′_{n} is a geodesic disc centered at p′_{n}, and let

M′′_{n} = \{p ∈ M′_{n}|d_{n}(p, p′_{n}) < \frac{d′_{n}(p′_{n})}{2}\}.

Then p ∈ M′′_{n} implies that d′_{n}(p) ≥ \frac{d′_{n}(p′_{n})}{2} and

|K_{n}(p)|\frac{d′_{n}(p′_{n})^{2}}{4} ≤ |K_{n}(p)|d′_{n}(p)^{2} ≤ |K_{n}(p′_{n})|d′_{n}(p′_{n})^{2} = d′_{n}(p′_{n})^{2}.

Therefore |K_{n}(p)| ≤ 4 on M′′_{n}. Furthermore, d′′_{n}(p′_{n}) = d(p′_{n}, ∂M′′_{n}) = d′_{n}(p′_{n})/2 → ∞. This proves the claim.

By translations of ℝ^{m} we can assume that x^{(n)}(p_{n}) = 0. We can also assume that M_{n} is simply connected, by taking its universal covering, if necessary. By the uniformization theorem, M_{n} is conformally equivalent to either the unit disc D or the complex plane ℂ, and we can suppose that p_{n} maps onto 0 for each n. But the case that M_{n} is conformally equivalent to ℂ is impossible because the condition that g^{(n)} misses more than m(m + 1)/2 hyperplanes in general position in ℙ^{m−1}(ℂ), implies, by Picard’s theorem, that g^{(n)} is constant, so K_{n} ≡ 0, which contradicts the condition that |K_{n}(0)| = 1. So we have constructed a sequence of minimal surfaces, x^{(n)} : D → ℝ^{3}, satisfying (6). Since, by theorem 2.4, ℙ^{m−1}(ℂ) minus 2m − 1 hyperplanes is complete Kobayashi hyperbolic, and m(m + 1)/2 ≥ 2m − 1, a subsequence of generalized Gauss maps g^{(n)} of x^{(n)} exists−without of loss generality we assume g^{(n)} itself−such that g^{(n)} : D → ℙ^{m−1}(ℂ) converges uniformly on every compact subset of D to a map g : D → ℙ^{m−1}(ℂ).

We now claim that g is non-constant. Suppose not, i.e. g is a constant map, and g maps the disk D onto a single point P. Let H be any hyperplane not containing the point P, and let U, V be disjoint neighborhoods of H and P respectively. Let C be the constant in theorem 1.2 such that

|K(p)|^{1/2}d(p) ≤ C

for any minimal surface in ℝ^{m} whose Gauss map omits the neighborhood U of H, where p is a point of S and d(p) is the geodesic distance of p to the boundary of S. Choose r < 1 such that the hyperbolic distance R of z = 0 to |z| = r satisfies R > C. Since g^{(n)} converges uniformly to g on |z| ≤ r, the image of |z| = r lies in the neighborhood V of q for sufficiently large n, say n ≥ n_{0}. It follows that for n ≥ n_{0}, the image of the disk |z| ≤ r under g^{(n)} omits the neighborhood U of H and we may therefore apply the above inequality to conclude

|K_{n}(0)|^{1/2}d_{n}(r) ≤ C

where d_{n}(r) is the geodesic distance from the origin to the boundary of the surface x^{(n)} : D(r) → ℝ^{m}. But |K_{n}(0)| = 1 for all n, and hence d_{n}(r) ≤ C for n ≥ n_{0}. On the other hand, we get a lower bound for d_{n}(r) from lemma 2.1. The surface x^{(n)} : {|z| < 1} → ℝ^{m} is a geodesic disk of radius R_{n}. If we reparametrize by w = r_{n}z where |w| = r_{n} has hyperbolic radius R_{n}, then the circle |z| = r corresponds to |w| = r_{n}r, and by lemma 2.1, the distance in the surface metric from the origin to any point on the circle |z| = r, or equivalently, |w| = r_{n}r, is greater than or equal to the hyperbolic distance from 0 to |w| = r_{n}r. But as n → ∞, R_{n} → ∞ and r_{n} → 1, so that the hyperbolic radius of |w| = r_{n}r tends to the hyperbolic radius of |w| = r, which is R. Since by assumption R > C we have for n sufficiently large that the surface distance from z = 0 to |z| = r is greater than C, contradicting the earlier bound d_{n}(r) ≤ C. Thus we conclude that the limit function g can not be constant.

Therefore the hypotheses of lemma 3.2 are satisfied. Since |K_{n}(0)| = 1, the possibility (i) of lemma 3.2 cannot happen. Thus, a subsequence {x^{(n′)}} of {x^{(n)}} converges to a minimal immersion x : D → ℝ^{m}, whose Gauss map is g. By (6) and by lemma 2.2, x is complete. By assumption, g^{(n)} omits hyperplanes H_{1}, …, H_{q} in ℙ^{m−1}(ℂ), located in general position, q > m(m + 1)/2. By Hurwitz’s theorem(theorem 2.2), either g omits these hyperplanes, or the image of g lies in some of these hyperplanes. Say g(M) ⊂ ∩_{j=1}^{k}H_{j} = ℙ(V), where V is a subspace of ℂ^{m} of dimension m − k, and g(M) : M → ℙ(V) omits the hyperplanes H_{k+1} ∩ (∩_{j=1}^{k}H_{j}), …, H_{q} ∩ (∩_{j=1}^{k}H_{j}) in ℙ(V). Since the hyperplanes H_{k+1} ∩ (∩_{j=1}^{k}H_{j}), …, H_{q} ∩ (∩_{j=1}^{k}H_{j}) in ℙ(V) are still in general position in ℙ(V) because H_{1}, …, H_{q} are in general position in ℙ^{m−1}(ℂ), and q − k > m(m + 1)/2 − k ≥ (m − k)(m − k + 1)/2, it follows from theorem 2.3 that g is constant. But we have just proved that g is not constant. This leads to a contradiction. Therefore the main theorem is proved.

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Robert Osserman Department of Mathematics, Stanford University, Stanford, CA 94305; Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720.

E-mail address: ro@msri.org

Min Ru, Department of Mathematics, University of Houston, Houston, TX 77204-3476; Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720.

  E-mail address: minru@math.uh.edu, minru@msri.org