JournalsrsmupVol. 127falsepp. 235–247

A Convergence Theorem for Immersions with L2L^2-Bounded Second Fundamental Form

  • Cheikh Birahim Ndiaye

    Universität Tübingen, Germany
  • Reiner Schätzle

    Universität Tübingen, Germany
A Convergence Theorem for Immersions with $L^2$-Bounded Second Fundamental Form cover

Abstract

In this short note, we prove a convergence theorem for sequences of immersions from some closed surface Σ\Sigma into some standard Euclidean space Rn\mathbb{R}^n with L2L^2-bounded second fundamental form, which is suitable for the variational analysis of the famous Willmore functional, where n3n\geq 3. More precisely, under some assumptions which are automatically verified (up to subsequence and an appropriate Möbius transformation of Rn\mathbb{R}^n) by sequences of immersions from some closed surface Σ\Sigma into some standard Euclidean space Rn\mathbb{R}^n arising from an appropriate stereographic projection of Sn\mathbb{S}^n into Rn\mathbb{R}^n of immersions from Σ\Sigma into Sn\mathbb{S}^n and minimizing the L2L^2-norm of the second fundamental form with n3n\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 L2L^2-Bounded Second Fundamental Form. Rend. Sem. Mat. Univ. Padova 127 (2012), pp. 235–247

DOI 10.4171/RSMUP/127-12