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

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 $n\ge 3$. 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 $n\ge 3$, 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.