Rendiconti del Seminario Matematico della Università di Padova


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Volume 127, 2012, pp. 235–247
DOI: 10.4171/RSMUP/127-12

A Convergence Theorem for Immersions with $L^2$-Bounded Second Fundamental Form

Cheikh Birahim Ndiaye[1] and Reiner Schätzle[2]

(1) Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076, Tübingen, Germany
(2) Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076, 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.

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Ndiaye Cheikh Birahim, Schätzle Reiner: A Convergence Theorem for Immersions with $L^2$-Bounded Second Fundamental Form. Rend. Sem. Mat. Univ. Padova 127 (2012), 235-247. doi: 10.4171/RSMUP/127-12