Conformally Invariant Variational Problems. - SAM

VIII A proof of Heinz-Hildebrandt’s regularity
conjecture.
The methods which we have used up to now to approach Hildebrandt’s
conjecture and obtain the regularity of W 1,2 solutions
of the generic system
∆u+A(u)(∇u,∇u)= H(u)(∇ ⊥ u,∇u)
(VIII.1)
rely on two main ideas:
i) recast, as much as possible, quadratic nonlinear terms as
linear combinations of Jacobians or as null forms ;
ii) project equation (VIII.1) on a moving frame (e 1···e n ) satisfying
the Coulomb gauge condition
∀i,j = 1···m div((e j ,∇e i )) = 0 .
Both approaches can be combined to establish the Hölder continuity
of W 1,2 solutions of (VIII.1) when the target manifold N n
is C 2 , and when the prescribed mean curvature H is Lipschitz
continuous (see [Bet1], [Cho], and [He]). Seemingly, these are
the weakest possible hypotheses required to carry out the above
strategy.
However, to fully solve Heinz-Hildebrandt’s conjecture, one
must replace the Lipschitzean condition on H by its being an
element of L ∞ . This makes quite a difference!
Despite its evident elegance and verified usefulness, Hélein’s
moving frames method suffers from a relative opacity: 20 what
20 Yet another drawback of the moving frames method is that it lifts an N n -valued
harmonic map, with n > 2, to another harmonic map, valued in a parallelizable manifold
(S 1 ) q of higher dimension. This procedure requires that N n have a higher regularity
than the “natural” one (namely, C 5 in place of C 2 ). It is only under this more stringent
assumption that the regularity of N n -valued harmonic maps was obtained in [Bet2] and
[He]. The introduction of Schrödinger systems with antisymmetric potentials in [RiSt]
enabled to improve these results.
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