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The goal of the present work is twofold. First we prove the existence of anHilbert Manifold structure on the space of immersed oriented closed surfaceswith three derivatives in $L^2$ in an arbitrary sub-manifold $M^m$ of aneuclidian space $R^Q$. Second, using this Hilbert manifold structure, we provea lower semi continuity property of the index for sequences of conformalimmersions, critical points to the viscous approximation of the area satisfyingStruwe entropy estimate and bubble tree strongly converging in $W^{1,2}$ to alimiting minimal surface as the viscous parameter is going to zero.

Lower Semi-Continuity of the Index in the Visosity Method for Minimal Surfaces