Parametric surfaces 209
iv p sequences ε
1 n
, . . . , ε
p n
in R
+
with lim
n →+∞
ε
n i
= 0 for any i = 1, . . . , p such that, up to a subsequence, we have
u
n
− u −
p
X
i =1
v
i
· − a
i n
ε
i n
H
1
→ 0 Z
D
2
|∇u
n
|
2
= Z
D
2
|∇u|
2
+
p
X
i =1
Z
R
2
|∇v
i
|
2
+ o1 E
H
u
n
= E
H
u +
p
X
i =1
¯ E
H
v
i
+ o1 , where in general ¯
E
H
v =
1 2
R
R
2
|∇v|
2
+
2H 3
R
R
2
v · v
x
∧ v
y
. In case p = 0 any sum
P
p i
=1
is zero and u
n
→ ¯u strongly in H
1
. R
EMARK
7. The conformal invariance is reflected in the concentrated maps v
i ·−a
i n
ε
i n
. This theorem also emphasizes the role of solutions of the H -equation on
whole R
2
, which are completely known see below.
6.3. Characterization of solutions on R
2
The solutions to the H -equation on the whole plane R
2
are completely classified in the next theorem. It basically asserts that all solutions of the problem
48 1
u = 2H
u
x
∧ u
y
on R
2
R
R
2
|∇u|
2
+∞ are conformal parametrizations of the sphere of radius R
= 1|H |.
Note first that, if u is a solution to 48, defining ω = |u
x
|
2
− |u
y
|
2
− 2iu
x
· u
y
the usual defect of conformality for u, it holds that
∂ω ∂
¯z
= 0 by the equation, and R
R
2
|ω| +∞ by the summability condition on ∇u. Hence ω ≡ 0, that is, u is conformal.
Pushing a little further the analysis, Brezis and Coron obtained the following result see [14].
T
HEOREM
15. Let u ∈ L
1 loc
R
2
, R
3
be a solution to 48 with H 6= 0. Then u
has the form uz
= 1
H 5
Pz Qz
+ C, where C is a constant vector in R
3
, P and Q are irreducible polynomials in the complex variable z
= x, y = x +iy and 5: C → S
2
is the stereographic projection.
210 F. Bethuel - P. Caldiroli - M. Guida
Moreover Z
R
2
|∇u|
2
= 8π k
H
2
, ¯E
H
u =
4π k 3H
2
,
where k = max{deg P, deg Q} is the number of coverings of the sphere S
2
by the parametrization u.
We point out that problem 48 is invariant with respect to the conformal group. For instance, if u is a solution to 48, then u
λ
z = uλz is also a solution. Note that
u
λ
→ const as λ → +∞, or as λ → 0.
6.4. Existence of the large solution
In this subsection, taking advantage from the results stated in the previous subsections, we will sketch the conclusion of the proof of theorem 12.
Let us recall that the functional F defined by 44 admits a mountain pass level c 0. In view of the result on the Palais-Smale sequences stated in Theorem 14, it is
useful also an upper bound for c, and precisely: L
EMMA
11. c
4π 3H
2
. This estimate is obtained by evaluating the functional E
H
along an explicit moun- tain pass path which, roughly speaking, is constructed by attaching in a suitable way a
sphere to the small solution.
Let now u
n
⊂ H
1 g
be the Palais-Smale sequence for E
H
introduced at the end of the subsection 6.1. We have already seen that, up to a subsequence, u
n
converges weakly to a solution
¯u to D
H
. If u
n
→ ¯u strongly in H
1
then 49
E
H
u = E
H
u + c E
H
u because c 0.
On the contrary, if no subsequence of u
n
converges strongly in H
1
, then we use theorem 14 on the characterization of Palais smale sequences. In particular, with
the same notation of theorem 14, we have p ≥ 1 and, denoting by S the set of all
Parametric surfaces 211
nonconstant solutions to 48, E
H
u = E
H
u + c −
p
X
i =1
¯E
H
v
i
≤ E
H
u + c − p inf
v ∈S
¯ E
H
v ≤ E
H
u + c − inf
ω ∈S
¯ E
H
ω ≤ E
H
u + c −
4π 3H
2
E
H
u 50
according to 46, theorem 15 and lemma 11. Thus, either from 49 or from 50, it follows that u
6= u and the conclusion of theorem 12 is achieved.
6.5. The second solution for variable H
