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We study the topology of (properly) immersed complete minimal surfaces $P^2$in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature,using some isoperimetric inequalities satisfied by the extrinsic balls in thesesurfaces, (see \cite{Pa}). We present an alternative and partially unifiedproof of the Chern-Osserman inequality satisfied by these minimal surfaces, (in$\erre^n$ and in $\Han$), based in the isoperimetric analysis above alluded.Finally, we show a Chern-Osserman type equality attained by complete minimalsurfaces in the Hyperbolic space with finite total extrinsic curvature.

Extrinsic isoperimetry and compactification of minimal surfaces in Euclidean and Hyperbolic spaces