Abstract
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 $\mathbb{R}^3$ by one or two linearly independent translations and we give sharp upper bounds of the genus of compact two-sided 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
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