Conformally Invariant Variational Problems. - SAM

If we adjust ε 0 sufficiently small as to have 3 C 0 |H| ε 0 < 1/4,
it follows that
∫
|∇u| 2 ≤ 3 ∫
|∇u| 2 . (VII.18)
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B ρ/2 (p)
B ρ (p)
Iterating this inequality gives the existence of a constant α > 0
such that for all p ∈ B 1/2 (0) and all r < ρ, there holds
∫ ( ) α ∫ r
|∇u| 2 ≤ |∇u| 2 ,
ρ 0 D 2
B r (p)
which implies (VII.9). Accordingly, the solution u of the CMC
equation is Hölder continuous.
Next, we infer from (VII.9) and (VII.1) the bound
∫
sup ρ −α |∆u| < +∞ . (VII.19)
ρ (2 − α)/(1 − α).
Substituted back into (VII.1), this fact implies that ∆u ∈ L r
for some r > 1. The equation the becomes subcritical, and
a standard bootstrapping argument eventually yields that u ∈
C ∞ . This concludes the proof of the regularity of solutions of
the CMC equation.
VII.2 Harmonic maps with values in the sphere S n
When the target manifold N n has codimension 1, the harmonic
map equation (VI.26) becomes (cf. (VI.30))
−∆u = ν(u) ∇(ν(u))·∇u ,
(VII.20)
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