Page 150 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Generalities about surfaces 137

Remark 5.14 (i) Note that for a nonparametric surface

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Σ = v (x, y)= (x, y, u (x, y)) , (x, y) ∈ Ω
2
2
we have E = G =1 + u =1 + u and F = u x u y =0 only if u x = u y =0.
x y
(ii) Enneper surface (Example 5.10) is globally parametrized with isothermal
coordinates.
One of the remarkable aspects of minimal surfaces is that they can be globally
parametrized by such coordinates as the above Enneper surface. We have the
following result that we will not use explicitly. We will, in part, give an idea of
the proof in the next section (for a proof, see Nitsche [78], page 175).
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Theorem 5.15 Let Ω = (x, y) ∈ R : x + y < 1 .Let Σ be a minimal sur-
¡ ¢ ¡ ¢
face of the type of the disk (i.e., there exists e ∈ C 2 Ω; R 3 so that Σ = e Ω )
v
v
such that ∂Σ = Γ is a Jordan curve. Then there exists a global isothermal rep-
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resentation of the surface Σ. This means that Σ = v (x, y):(x, y) ∈ Ω with v
satisfying
¡ ¢ ¡ ¢
(i) v ∈ C Ω; R 3 ∩ C ∞ Ω; R 3 and ∆v =0 in Ω;
2 2
(ii) E = G> 0 and F =0 (i.e., |v x | = |v y | > 0 and hv x ; x y i =0);
(iii) v maps the boundary ∂Ω topologically onto the Jordan curve Γ.

Remark 5.16 The second result asserts that Σ is a regular surface (i.e., v x ×
√ 2
v y 6=0)since |v x × v y | = EG − F = E = |v x | > 0.
2
To conclude we point out the deep relationship between isothermal coordi-
nates of minimal surfaces and harmonic functions (see also Theorem 5.15) which
is one of the basic facts in the proof of Douglas.
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Theorem 5.17 Let Σ = v (x, y) ∈ R :(x, y) ∈ Ω be a regular surface (i.e.
2
v x × v y 6=0)ofclass C globally parametrized by isothermal coordinates; then
2
1
3
Σ is a minimal surface ⇔ ∆v =0 (i.e., ∆v = ∆v = ∆v =0).
2 2
Proof. We will show that if E = G = |v x | = |v y | and F =0,then
(5.4)
∆v =2EHe 3 =2Hv x × v y
where H is the mean curvature and e 3 =(v x × v y ) / |v x × v y |. The result will
readily follow from the fact that H =0.