Lower Semi-continuity of the Index in the Viscosity Method for Minimal Surfaces

Abstract

Abstract The goal of the present work is two-fold. First we prove the existence of a Hilbert manifold structure on the space of immersed oriented closed surfaces with three derivatives in $L^2$ in an arbitrary compact submanifold $M^m$ of an Euclidian space ${{\mathbb{R}}}^Q$. Second, using this Hilbert manifold structure, we prove a lower semi-continuity property of the index for sequences of conformal immersions, critical points to the viscous approximation of the area satisfying a Struwe entropy estimate and a bubble tree strongly converging in $W^{1,2}$ to a limiting minimal surface as the viscous parameter is going to zero.