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Small surfaces of Willmore type in Riemannian manifolds

MPG-Autoren

Lamm,  Tobias
Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Metzger,  Jan
Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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0909.0590v1.pdf
(Preprint), 185KB

IMRN19_3786.pdf
(beliebiger Volltext), 141KB

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Zitation

Lamm, T., & Metzger, J. (2010). Small surfaces of Willmore type in Riemannian manifolds. International mathematics research notices, 2010(19), 3786-3813. doi:https://doi.org/10.1093/imrn/rnq048.


Zitierlink: http://hdl.handle.net/11858/00-001M-0000-0013-60DA-5
Zusammenfassung
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.