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

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 L2 in an arbitrary compact submanifold Mm of an Euclidian space RQ. 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 W1,2 to a limiting minimal surface as the viscous parameter is going to zero.