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

TL;DR

This paper investigates the topology of complete minimal surfaces with finite extrinsic curvature in Euclidean and Hyperbolic spaces, providing new proofs of isoperimetric inequalities and establishing a Chern-Osserman type equality.

Contribution

It offers an alternative, unified proof of the Chern-Osserman inequality and introduces a Chern-Osserman type equality for minimal surfaces in Hyperbolic space.

Findings

Unified proof of Chern-Osserman inequality in Euclidean and Hyperbolic spaces

Establishment of a Chern-Osserman type equality in Hyperbolic space

Analysis of isoperimetric inequalities for extrinsic balls in minimal surfaces

Abstract

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.