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

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

2010-11-24

arxiv.org/abs/1011.5380v2

Mathematics

Differential Geometry

10 pages

Scientific paper

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 \cite{Pa}). We present an alternative and partially unified proof 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 minimal surfaces in the Hyperbolic space with finite total extrinsic curvature.

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