Conformally Invariant Variational Problems. - SAM
Conformally Invariant Variational Problems. - SAM
Conformally Invariant Variational Problems. - SAM
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for any perturbation V ⃗ which is equivalent to (X.64) and theorem<br />
X.5 is proved.<br />
✷<br />
As mentioned in the introduction of this course questions we<br />
are interested in are analysis questions for conformallyinvariant<br />
lagrangians.<br />
In this part of the course devoted to Willmore Lagrangianwe<br />
shall look at the following problems :<br />
i) DoesthereexistsaminimizerofWillmorefunctionalamong<br />
all smooth immersions for a fixed 2-dimensional surface Σ 2<br />
? and, if yes, can one estimate the energy and special properties<br />
of such a minimizer ?<br />
ii) DoesthereexistsaminimizersofWillmorefunctionalamong<br />
a more restricted class of immersionssuch as conformal immersionsfor<br />
a fixed chosen conformalclasscon Σ ? or does<br />
there exist a minimizing immersion of Willmore functional<br />
among all immersions into R 3 enclosing a domain of given<br />
volume and realizing a fixed area...<br />
iii) What happens to a sequence of weak Willmore immersions<br />
of a surface Σ 2 having a uniformly bounded energy at the<br />
limit? does it convergencein some sense to a surface which<br />
is still Willmore and if not what are the possible ”weak<br />
limits” of Willmore surfaces ?<br />
iv) How stable is the Willmore equation ? that means : following<br />
a sequence of ”almostWillmore”surfaces”solvingmore<br />
and more” the Willmore equation - Willmore Palais Smale<br />
sequences for instance - does such a sequence converges to<br />
a Willmore surface ?<br />
v) Can one apply fundamental variational principles such as<br />
Ekeland’s variational Principles or Mountain pass lemma<br />
to the Willmore functional ?<br />
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