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

### Cheikh Birahim Ndiaye

Universität Tübingen, Germany### Reiner Schätzle

Universität Tübingen, Germany

## Abstract

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.

## Cite this article

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

DOI 10.4171/RSMUP/127-12