Conformally Invariant Variational Problems. - SAM

and denote
C ∗ C(Γ) := {u ∈ C ∗ (Γ) s. t. E(u) ≤ C} .
The lemma is equivalent to the following claim.
∀ε > 0 ∃δ > 0 s.t. ∀u ∈ C ∗ C(Γ) ∀p,q ∈ ∂D 2
|p−q| < δ =⇒ |u(p)−u(q)| < ε
(V.30)
Since Γ is the image of a continuous and injective map γ from
S 1 into R m the following claim holds
∀ε > 0 ∃η > 0 s.t. ∀ 0 < θ 1 < θ 2 ≤ 2π
|γ(e iθ 2
)−γ(e iθ 1
)| < η =⇒ ‖γ(e iθ )−γ(e iθ 1
)‖ L∞ ([θ 1 ,θ 2 ]) < ε
(V.31)
This claim can be proved by contradiction and we leave the
details of the argument to the reader.
We are heading now to the proof of (V.30). Let ε > 0 such
that
2ε < inf |Q i −Q j | (V.32)
i≠j
where the Q i are the 3 fixed points on Γ appearing in the definition
(V.25) of C ∗ (Γ). We are then considering ε small enough
in such a way that each ball of radius ǫ contains at most one Q i .
ε > 0 being fixed and satisfying (V.32), we consider η > 0
given by (V.31).
Consider 1 > δ > 0 to be fixed later but satisfying at least
2 √ δ < inf
i≠j |P i −P j |
(V.33)
Take an arbitrary pair of points p and q in ∂D 2 such that
|p−q| < δ. Let p 0 be the point on the small arc pq ⊂ ∂D 2 right
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