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Small surfaces of Willmore type in Riemannian manifolds

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.

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0909.0590v1.pdf (Preprint), 185KB
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### Creators

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Creators:
Lamm, Tobias1, Author
Metzger, Jan1, Author
Affiliations:
1Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, ou_24012

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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.

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Dates: 2010
Publication Status: Published in print
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Identifiers: eDoc: 436260
arXiv: 0909.0590
DOI: https://doi.org/10.1093/imrn/rnq048
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Title: International mathematics research notices
Source Genre: Journal
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Pages: - Volume / Issue: 2010 (19) Sequence Number: - Start / End Page: 3786 - 3813 Identifier: -