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

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 [12]). We present an alternative and partially unified proof of the Chern-Osserman inequality satisfied by these minimal surfaces, (in R n and in H n (b)), 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.2000 Mathematics Subject Classification. Primary 53C20 ; Secondary 53C42, 49Q05.

Volume growth of submanifolds and the Cheeger isoperimetric constant

“…(see [1], [6] and [8]. For an alternative proof, see [9]). Let P 2 be an complete minimal surface immersed in a simply connected real space form with constant sectional curvature b ≤ 0, K n (b).…”

On the Fundamental Tone of Minimal Submanifolds with Controlled Extrinsic Curvature

“…But we can apply the Main Theorem under the assumptions of the Theorems A, B and C because of the following three theorems that we recall here: Theorem 2.4 (See [1], [6] and [11]). Let M n be a minimal submanifold immersed in the Euclidean space R m .…”

“…Theorem 2.5 (See [7], [11] and [16]). Let M 2 be a minimal surface immersed in the hyperbolic space H m (κ) of constant sectional curvature κ < 0 or in the Euclidean space R m .…”

“…Our approach to the problem makes use of our previous results about the influence of the extrinsic curvature restrictions on the volume growth of the submanifold (see [11], [10], [12] or §2), and also the relation between the finite volume growth and the fundamental tone as a new tool (see the Main Theorem).…”

Density and spectrum of minimal submanifolds in space forms

“…see also [28]. However, in the higher dimensional case we found no analogous of (1.14), (1.17) in the literature, and adapting the proof of (1.14) to the hyperbolic ambient space seems to be subtler than what we expected.…”