129 research outputs found
The Classification of Branched Willmore Spheres in the -Sphere and the -Sphere
We extend the classification of Robert Bryant of Willmore spheres in to
variational branched Willmore spheres and show that they are inverse
stereographic projections of complete minimal surfaces with finite total
curvature in and vanishing flux. We also obtain a classification
of variational branched Willmore spheres in , generalising a theorem of
Seb\'{a}stian Montiel. As a result of our asymptotic analysis at branch points,
we obtain an improved regularity of the unit normal of variational
branched Willmore surfaces in arbitrary codimension. We also prove that the
width of Willmore sphere min-max procedures in dimension and , such as
the sphere eversion, is an integer multiple of .Comment: 74 pages, 1 figur
Weak closure of Singular Abelian -bundles in dimensions
We prove the closure for the sequential weak -topology of the class of
vectorfields on having integer flux through almost every sphere. We show
how this problem is connected to the study of the minimization problem for the
Yang-Mills functional in dimension higher than critical, in the abelian case.Comment: 29 pages, some typing errors fixe
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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