Parametric surfaces 183
2.2. Conformal parametrizations and the H -system
In problems concerning mean curvature, it is convenient to use conformal parametri- zations, since this leads to an equation for the mean curvature that can be handle with
powerful tools in functional analysis.
D
EFINITION
1. Let M be a two-dimensional regular surface in R
3
and let u : O →
M be a local parametrization, O being a connected open set in R
2
. The parametriza-
tion u is said to be conformal if and only if for every z ∈ O the linear map
duz : R
2
→ T
uz
M preserves angles and consequently multiplies lengths by a con- stant factor, that is there exists λz 0 such that
10 hduzh, duzki
R
3
= λzhh, ki
R
2
for every h, k ∈ R
2
. In other words, u is conformal if and only if for every z
∈ O duz is the product of an isometry and a homothety from R
2
into R
3
. Note also that the condition of conformality 10 can be equivalently written as:
11 |u
x
|
2
− |u
y
|
2
= 0 = u
x
· u
y
at every point z ∈ O. In what will follow, an important role is played by the Hopf
differential, which is the complex-valued function: ω
= |u
x
|
2
− |u
y
|
2
− 2iu
x
· u
y
. In particular, u is conformal if and only if ω
= 0. R
EMARK
2. If the target space of a conformal map u has dimension two, then u is analytical. This follows by the fact that, given a domain O in R
2
, a mapping u
∈ C
1
O, R
2
is conformal if and only if u is holomorphic or anti-holomorphic we identify R
2
with the complex field C. However for conformal maps u : O → R
k
with k
≥ 3 there is no such as regularity result. We turn now to the expression of H for conformal parametrizations. If u is confor-
mal, then E
= |u
x
|
2
= u
y 2
= G F
= u
x
· u
y
= 0, so that
12 2H u
= 1
u · −
→ n
|u
x
|
2
on O . On the other hand, deriving conformality conditions 11 with respect to x and y, we
can deduce that 1u is orthogonal both to u
x
and to u
y
. Hence, recalling the expression 6 of the normal vector −
→ n , we infer that 1u and −
→ n are parallel. Moreover, by 11,
|u
x
∧ u
y
| = |u
x
|
2
= |u
y
|
2
, and then, from 12 it follows that 13
1 u
= 2H uu
x
∧ u
y
on O .
184 F. Bethuel - P. Caldiroli - M. Guida
Let us emphasize that 13 is a system of equations, often called H -system, or also H - equation, and for this system the scalar coefficient H u has the geometric meaning
of mean curvature for the surface M parametrized by u at the point uz provided that u is conformal and uz is a regular point, i.e., u
x
z ∧ u
y
z 6= 0.
2.3. Some geometric problems involving the H -equation
Kategori