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Small surfaces of Willmore type in Riemannian manifolds
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abstract
In this paper we investigate the properties of small surfaces of Willmore type in Riemannian manifolds. By \emph{small} surfaces we mean topological spheres contained in a geodesic ball of small enough radius. In particular, we show that if there exist such surfaces with positive mean curvature in the geodesic ball $B_r(p)$ for arbitrarily small radius $r$ around a point $p$ in the Riemannian manifold, then the scalar curvature must have a critical point at $p$. As a byproduct of our estimates we obtain a strengthened version of the non-existence result of Mondino \cite{Mondino:2008} that implies the non-existence of certain critical points of the Willmore functional in regions where the scalar curvature is non-zero.
Forward citations
Cited by 2 Pith papers
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Concentration of Small Hawking Type Surfaces
For small area, minimizers of Hawking type functionals are embedded spheres, and small concentrating sequences for the Hawking energy are shown to accumulate only at critical points of Sc + (3/5)trK² + (1/5)|K|².
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Minimizers of Generalized Willmore Functionals
A generalized Willmore framework yields existence of area-constrained, and area-volume-constrained, minimizers among haunted bubble trees, with partial regularity for critical points.
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