Conformally Invariant Variational Problems. - SAM

Combining (II.18) and (II.19) implies that
∀x ∈ U (A | x−x 0 | 2 +B)(C | u(x)−y 0 | 2 +D) ≡ 1
(II.20)
As a consequence of (II.20) we obtain in particular that the
image of a sphere of radius R centered at x 0 and included in U
is contained in a sphere of center y 0 and radius ρ such that
cρ 2 +D =
1
AR 2 +B
(Applying the same argument to u −1 gives that
u(∂B R (x 0 )) = ∂B ρ (y 0 ))
Let V 0 be a unit vector of R n and consider the line passing by
n 0 and oriented along V 0 . Its parametric equation is given by
x(t) = tV 0 +x 0 . Since U is conformal and since it is sending the
sphere centered at x 0 onto the spheres centered at
y 0 , y(t) = U(x(t))is then a path that remainsorthogonalto this
foliation of spheres and it has to be then contained is a straight
line passing by y 0 .
∀s,< τ s.t. y(t) ∈ u(U) for t ∈ [s,τ] we have
| y(τ)−y(s) |=
∫ τ
s
| ẏ | (+)dt =
=
=
∫ τ
∫s
τ
∫s
τ
Combining (II.20) and (II.21) we have
s
e λ | ẋ | dt
1
A | x−x 0 | 2 +B | ẋ | dt
dt
(II.21)
At 2 +B
(∫ τ
C
s
) 2
dt
+D =
At 2 +B
1
A | τ −s | 2 +B
(II.22)
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