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I'm refering to article of R.Schoen 'Estimates for stable minimal surfaces in three dimensional manifolds' (1983). In the first paragraph of the proof of theorem 2 the author wants to apply the methods of article 4 (i think theorem 1 of article 4) to a stable immersed surface M in a three dimensional manifold N with non negative scalar curvature. This immersed surface is in general not complete, while the methods of Fischer colbrie-schoen are developed for complete manifolds. Now, as i've said above, i think that the particular theorem that author wants to use is theorem 1 of article 4 and i'm not sure (actually i think no) that this theorem can be generalized to non complete case. Moreover why is the universal covering of M conformally equivalent to the unit disc? (this fact is use in the same first paragraph). Thank you

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I suspect that "by the methods" means the consideration on page 206 of F-C&S. // Also, the stability is determined by testing against compactly supported perturbations, so maybe the completeness is not really crucial there. –  user53153 Feb 6 '13 at 6:19

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