Seminar in Geometry- Chapter 14- Isotherman Parameters

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by  Emma Carberry

Chapter 14
Isotherman Parameters
Let x : U S be a regular surface. Let
→
∂x ∂x
k k
φ (z) = i ,z = u +iu . (14.1)
k 1 2
∂u − ∂u
1 2
Recall from last lecture that
a)φ is analytic in z x is harmonic in u and u .
k 1 2
⇔
b)u and u are isothermal parameters
1 2
⇔
n
φ2(z) = 0 (14.2)
k
k=1
X
c) If u ,u are isothermal parameters, then S is regular
1 2
⇔
n
φ (z) 2 = 0 (14.3)
k
| | 6
k=1
X
We start by stating a lemma that summarizes what we did in the last lecture:
Lemma 4.3 in Osserman: Let x(u) define a minimal surface, with
u ,u isothermal parameters. Then the functions φ (z) are analytic and
1 2 k
they satisfy the eqns in b) and c). Conversely if φ ,φ ,..,φ are analytic
1 2 n
functions satisfying the eqns in b) and c) in a simply connected domain D
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then there exists a regular minimal surface defined over domain D, such that
the eqn on the top of the page is valid.
Now we take a surface in non-parametric form:
x = f (x ,x ),k = 3,...,n (14.4)
k k 1 2
and we have the notation from the last time:
∂f ∂f ∂2f ∂2f ∂2f
f = (f ,f ,...,f ),p = ,q = ,r = ,s = ,t = (14.5)
3 4 n ∂x ∂x ∂x2 ∂x ∂x ∂x2
1 2 1 1 2 2
Then the minimal surface eqn may be written as:
∂p ∂p ∂q ∂q
(1+ q 2) (p.q)( + )+(1+ p 2) = 0 (14.6)
| | ∂x − ∂x ∂x | | ∂x
1 2 1 2
equivalently
(1+ q 2)r 2(p.q)s+(1+ p 2)t = 0 (14.7)
| | − | |
One also has the following:
detg = 1+ p 2 + q 2 + p 2 q 2 (p.q)2 (14.8)
ij
| | | | | | | | −
Define
W = detg (14.9)
ij
p
Below we’ll do exactly the same things with what we did when we showed
that the mean curvature equals 0 if the surface is minimizer for some curve.
Now we make a variation in our surface just like the one that we did before
(the only difference is that x and x are not varied.)
1 2
˜
f = f +λh ,k = 3,...,n, (14.10)
k k k
where λ is a real number, and h C1 in the domain of definition D of the
k
∈
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f We have
k
∂h ∂h
˜
f = f +λh,p˜= p+λ ,q˜= q +λ (14.11)
∂x ∂x
1 2
One has
W˜ 2 = W2 +2λX +λ2Y (14.12)
where
∂h ∂h
X = [(1+ q 2)p (p.q)q]. +[(1+ p 2)q (p.q)p]. (14.13)
| | − ∂x | | − ∂x
1 2
and Y is a continuous function in x and x . It follows that
1 2
X
W˜ = W +λ +O(λ2) (14.14)
W
as λ 0 Now we consider a closed curve Γ on our surface. Let ∆ be the
| | →
region bounded by Γ If our surface is a minimizer for ∆ then for every choice
of h such that h = 0 on Γ we have
˜
Wdx dx Wdx dx (14.15)
1 2 1 2
≥
Z Z∆ Z Z∆
which implies
X
= 0 (14.16)
W
Z Z∆
Substituting for X, integrating by parts, and using the fact that h = 0 on Γ
, we find
∂ 1+ q 2 p.q ∂ 1+ p 2 p.q
| | p q + | | q p hdx dx = 0
1 2
∂x W − W ∂x W − W
Z Z∆· 1 · ¸ 2 · ¸¸
(14.17)
must hold everywhere. By the same reasoning that we used when we found
the condition for a minimal surface the above integrand should be zero.
∂ 1+ q 2 p.q ∂ 1+ p 2 p.q
| | p q + | | q p = 0 (14.18)
∂x W − W ∂x W − W
1 · ¸ 2 · ¸
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Once we found this equation it makes sense to look for ways to derive it
from the original equation since after all there should only be one constraint
for a minimal surface. In fact the LHS of the above eqn can be written as
the sum of three terms:
1+ q 2 ∂p p.q ∂q ∂p 1+ p 2 ∂q
| | ( + )+ | | (14.19)
W ∂x − W ∂x ∂x W ∂x
· 1 1 2 2¸
∂ 1+ q 2 ∂ p.q
+ ( | | ) ( ) p (14.20)
∂x W − ∂x W
· 1 2 ¸
∂ 1+ p 2 ∂ p.q
+ ( | | ) ( ) q (14.21)
∂x W − ∂x W
· 2 1 ¸
The first term is the minimal surface eqn given on the top of the second
page. If we expand out the coefficient of p in the second term we find the
expression:
1
[(p.q)q (1+ q 2)p].[(1+ q 2)r 2(p.q)s+(1+ p 2)t] (14.22)
W3 − | | | | − | |
which vanishes by the second version of the minimal surface eqns. Similarly
the coefficient of q in third term vanishes so the while expression equals zero.
In the process we’ve also shown that
∂ 1+ q 2 ∂ p.q
| | = (14.23)
∂x W ∂x W
1 µ ¶ 2
³ ´
∂ 1+ p 2 ∂ p.q
| | = (14.24)
∂x W ∂x W
2 µ ¶ 1
³ ´
Existence of isothermal parameters or Lemma 4.4 in Osserman
Let S be a minimal surface. Every regular point of S has a neighborhood in
whichthereexistsareparametrizationofSintermsofisothermalparameters.
Proof: Since the surface is regular for any point there exists a neighbor-
hood of that point in which S may be represented in non-parametric form.
In particular we can find a disk around that point where the surface can be
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represented in non parametric form. Now the above eqns imply the existence
of functions F(x ,x ) G(x ,x ) defined on this disk, satisfying
1 2 1 2
∂F 1+ p 2 ∂F p.q
= | | , = ; (14.25)
∂x W ∂x W
1 2
∂G p.q ∂G 1+ q 2
= , = | | (14.26)
∂x W ∂x W
1 2
If we set
ξ = x +F(x ,x ),ξ = x +G(x ,x ), (14.27)
1 1 1 2 2 2 1 2
we find
∂(x ,x ) 2+ p 2 + q 2
1 2
J = = 2+ | | | | 0 (14.28)
∂(x ,x ) W ≥
1 2
Thus the transformation (x ,x ) (ξ ,ξ ) has a local inverse (ξ ,ξ )
1 2 1 2 1 2
→ →
(x ,x ). We find the derivative of x at point (ξ ,ξ ):
1 2 1 2
Dx = J−1[x ,x ,f ,...,f ] (14.29)
1 2 3 n
It follows that with respect to the parameters ξ , ξ we have
1 2
∂x ∂x
g = g = 2 = 2 (14.30)
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|∂ξ | |∂ξ |
1 2
∂x ∂x
g = . = 0 (14.31)
12
∂ξ ∂ξ
1 2
so that ξ , ξ are isothermal coordinates.
1 2
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