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We prove that there are no complete one-sided stable minimal surfaces in the Euclidean 3-space. We classify least area surfaces in the quotient of R 3 by one or two linearly independent translations and we give sharp upper bounds of the genus of compact twosided index one minimal surfaces in non-negatively curved ambient spaces. Finally we estimate from below the index of complete minimal surfaces in flat spaces in terms of the topology of the surface.doi:10.4310/jdg/1175266182 fatcat:guc5xdngyza35byjctb2gzcs5i