Page 145 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

132 Minimal surfaces

(i) The mean curvature of Σ,denoted by H,at a point p ∈ Σ (p = v (x, y))
is given by
1 EN − 2FM + GL
H = .
2 EG − F 2
(ii) The Gaussian curvature of Σ, denoted by K,at a point p ∈ Σ (p = v (x, y))
is by definition
LN − M 2
K = .
EG − F 2
(iii) The principal curvatures, k 1 and k 2 ,are defined as
p p
2
2
k 1 = H + H − K and k 2 = H − H − K
so that H =(k 1 + k 2 ) /2 and K = k 1 k 2 .
2
Remark 5.4 (i) We always have H ≥ K.
(ii) For a nonparametric surface v (x, y)= (x, y, u (x, y)), we have
2
2
2
2
E =1 + u ,F = u x u y ,G =1 + u ,EG − F =1 + u + u 2
x y x y
(−u x , −u y , 1) u xx
= ,
e 3 q ,L = q
2
2
1+ u + u 2 y 1+ u + u 2 y
x
x
u xy u yy
M = q ,N = q
2
2
1+ u + u 2 1+ u + u 2
x y x y
and hence
¡ 2 ¢ ¡ 2 ¢ 2
1+ u y u xx − 2u x u y u xy + 1+ u x u yy u xx u yy − u xy
H = ¢ .
¡ ¢ 3/2 and K = ¡ 2
2
2
2 1+ u + u 2 1+ u + u 2
x y x y
(iii) For a nonparametric surface in R n+1 given by x n+1 = u (x 1 ,..., x n ),we
have that the mean curvature is defined by (cf. (A.14) in Gilbarg-Trudinger [49])
⎡ ⎤
n
1 X ∂ u x i
H = ⎣ q ⎦
n ∂x i 2
i=1 1+ |∇u|
⎡ ⎤
n
1 ³ 2 ´ − 2 2 ´ X
3 ³
= 1+ |∇u| ⎣ 1+ |∇u| ∆u − u x i x j x i x j ⎦ .
u u
n
i,j=1
In terms of the operator M defined in the introduction of the present chapter,
we can write
³ ´ 3
2 2
Mu = n 1+ |∇u| H.