Page 146 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Generalities about surfaces 133

(iv) Note that we always have (see Exercise 5.2.1)
p
2
|v x × v y | = EG − F .
We are now in a position to define the notion of minimal surface.
2
Definition 5.5 A regular surface of class C is said to be minimal if H =0 at
every point.
We next give several examples of minimal surfaces, starting with the non-
parametric ones.
Example 5.6 The first minimal surface that comes to mind is naturally the
plane, defined parametrically by (α, β, γ being constants)
© 2 ª
Σ = v (x, y)= (x, y, αx + βy + γ):(x, y) ∈ R .
We trivially have H =0.
Example 5.7 Scherk surface is a minimal surface in nonparametric form given
by
π
n o
Σ = v (x, y)= (x, y, u (x, y)) : |x| , |y| <
2
where
u (x, y)= log cos y − log cos x.
We now turn our attention to minimal surfaces in parametric form.
2
Example 5.8 Catenoids defined, for (x, y) ∈ R ,by
x + µ
v (x, y)= (x, w (x)cos y, w (x)sin y) with w (x)= λ cosh ,
λ
where λ 6=0 and µ are constants, are minimal surfaces. We will see that they
are the only minimal surfaces of revolution (here around the x axis).
2
Example 5.9 The helicoid given, for (x, y) ∈ R ,by
v (x, y)= (y cos x, y sin x, ax)
with a ∈ R is a minimal surface (see Exercise 5.2.2).

2
Example 5.10 Enneper surface defined, for (x, y) ∈ R ,by
µ 3 3 ¶
x 2 y 2 2 2
v (x, y)= x − + xy , −y + − yx ,x − y
3 3
is a minimal surface (see Exercise 5.2.2).