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

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 these surfaces (see [10]). We present an alternative and unified proof of the Chern-Osserman inequality satisfied by these minimal surfaces (in ℝ n and in ℕ n (b)), based in the isoperimetric analysis mentioned above. Finally, we show a Chern-Osserman-type equality attained by complete minimal surfaces in the Hyperbolic space with finite total extrinsic curvature