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A proof of the multiplicity one conjecture for min-max minimal surfaces in arbitrary codimension
Given any admissible -dimensional family of immersions of a given closed
oriented surface into an arbitrary closed Riemannian manifold, we prove that
the corresponding min-max width for the area is achieved by a smooth (possibly
branched) immersed minimal surface with multiplicity one and Morse index
bounded by .Comment: 34 pages (major improvements in the presentation: several typos
fixed, few proofs expanded, added explanatory Subsection 5.1
Willmore Spheres in Compact Riemannian Manifolds
The paper is devoted to the variational analysis of the Willmore, and other
L^2 curvature functionals, among immersions of 2-dimensional surfaces into a
compact riemannian m-manifold (M^m,h) with m>2. The goal of the paper is
twofold, on one hand, we give the right setting for doing the calculus of
variations (including min max methods) of such functionals for immersions into
manifolds and, on the other hand, we prove existence results for possibly
branched Willmore spheres under various constraints (prescribed homotopy class,
prescribed area) or under curvature assumptions for M^m. To this aim, using the
integrability by compensation, we develop first the regularity theory for the
critical points of such functionals. We then prove a rigidity theorem
concerning the relation between CMC and Willmore spheres. Then we prove that,
for every non null 2-homotopy class, there exists a representative given by a
Lipschitz map from the 2-sphere into M^m realizing a connected family of
conformal smooth (possibly branched) area constrained Willmore spheres (as
explained in the introduction, this comes as a natural extension of the minimal
immersed spheres in homotopy class constructed by Sacks and Uhlembeck in
\cite{SaU}, in situations when they do not exist). Moreover, for every A>0 we
minimize the Willmore functional among connected families of weak, possibly
branched, immersions of the 2-sphere having prescribed total area equal to A
and we prove full regularity for the minimizer. Finally, under a mild curvature
condition on (M^m,h), we minimize the sum of the area with the square of the
L^2 norm of the second fundamental form, among weak possibly branched
immersions of the two sphere and we prove the regularity of the minimizer.Comment: 58 page
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