Page 147 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

134 Minimal surfaces

As already said we have the following characterization for surfaces of revolu-
tion.
Proposition 5.11 The only regular minimal surfaces of revolution of the form

v (x, y)= (x, w (x)cos y, w (x)sin y) ,

are the catenoids, i.e.
x + µ
w (x)= λ cosh
λ
where λ 6=0 and µ are constants.
Proof. We have to prove that Σ given parametrically by v is minimal if and
only if
w (x)= λ cosh ((x + µ) /λ) .

Observe first that
02
0
v x =(1,w cos y, w sin y) ,v y =(0, −w sin y, w cos y) ,E =1+w ,F =0,G = w 2
0
w (w , − cos y, − sin y)
0
v x × v y = w (w , − cos y, − sin y) ,e 3 = √
0
|w| 1+ w 02
v xx = w (0, cos y, sin y) ,v xy = w (0, − sin y, cos y) ,v yy = −w (0, cos y, sin y)
0
00
w −w 00 |w|
L = √ ,M =0,N = √ .
|w| 1+ w 02 1+ w 02
2
Since Σ is a regular surface, we must have |w| > 0 (because |v x × v y | = EG −
2
F > 0). We therefore deduce that (recalling that |w| > 0)
¡ ¡ 02 ¢¢
00
H =0 ⇔ EN + GL =0 ⇔ |w| ww − 1+ w =0
02
00
⇔ ww =1 + w . (5.1)
Any solution of the differential equation necessarily satisfies
" #
d w (x)
p =0 .
dx 1+ w (x)
02
The solution of this last differential equation (see the corrections of Exercise
5.2.3) being either w ≡ constant (which however does not satisfy (5.1)) or of the
form w (x)= λ cosh ((x + µ) /λ),wehavethe result.
We now turn our attention to the relationship between minimal surfaces and
surfaces of minimal area.