On properness of minimal surfaces with bounded curvature

Abstract

We show that immersed minimal surfaces in the euclidean 3-space with bounded curvature and proper self intersections are proper. We also show that restricted to wide components the immersing map is always proper, regardless the map being proper or not. Prior to these results it was only known that injectively immersed minimal surfaces with bounded curvature were proper.

Half-Space Theorems for Minimal Surfaces with Bounded Curvature

“…A main fact used in the proof of the previous theorem is the monotonicity of the mean curvature of the parallel surfaces along normal geodesics issuing from S µ z , see (3). A sufficient condition for this monotonicity to hold is given by non-negativeness of the Ricci curvature of the ambient space, which in our case is guaranteed by the hypothesis Ric M ě 0 (cf.…”

“…If a projection point z lies on N it is contained in a neighborhood V z Ă N which is locally embedded and has radius uniformly bounded from below. On the other hand, if the projection point z lies on the limit set of N , then since our analysis is local, we can consider a uniform bound from below for the injective radius of the ambient space in order to guarantee the existence of a minimal neighborhood V z , with radius uniformly bounded from below, contained in the limit set of N , see the proof of [4, Thm.1.5] and [3]. As in the proof of Theorem 1.1, by Lemma 2.1, at each projection point z P V z , for every µ ą 0, we can choose an embedding supporting surface S µ z at z satisfying (see [10,Lemm.1]) y R cutpS µ z q and H µ z pzq ą ´µ.…”

“…Our next example is a complete recurrent minimal surface which is neither properly immersed in R 3 The Gaussian curvature of χ is given by Kpzq "…”