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European Mathematical Society Publishing House
2016-09-19 17:05:49
Rendiconti del Seminario Matematico della Università di Padova
Rend. Sem. Mat. Univ. Padova
RSMUP
0041-8994
2240-2926
General
10.4171/RSMUP
http://www.ems-ph.org/doi/10.4171/RSMUP
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2013)
127
2012
0
A Convergence Theorem for Immersions with $L^2$-Bounded Second Fundamental Form
Cheikh Birahim
Ndiaye
Universität Tübingen, TÜBINGEN, GERMANY
Reiner
Schätzle
Universität Tübingen, TÜBINGEN, GERMANY
In this short note, we prove a convergence theorem for sequences of immersions from some closed surface $\Sigma$ into some standard Euclidean space $\mathbb{R}^n$ with $L^2$-bounded second fundamental form, which is suitable for the variational analysis of the famous Willmore functional, where $n\geq 3$. More precisely, under some assumptions which are automatically verified (up to subsequence and an appropriate Möbius transformation of $\mathbb{R}^n$) by sequences of immersions from some closed surface $\Sigma$ into some standard Euclidean space $\mathbb{R}^n$ arising from an appropriate stereographic projection of $\mathbb{S}^n$ into $\mathbb{R}^n$ of immersions from $\Sigma$ into $\mathbb{S}^n$ and minimizing the $L^2$-norm of the second fundamental form with $n\geq 3$, we show that the varifolds limit of the image of the measures induced by the sequence of immersions is also an immersion with some minimizing properties.
General
235
247
10.4171/RSMUP/127-12
http://www.ems-ph.org/doi/10.4171/RSMUP/127-12