Complete Surfaces with Ends of Non Positive Curvature

TL;DR

This paper extends Efimov's Theorem to complete surfaces in three-dimensional space with negative curvature outside a compact set, showing they have finite total curvature, finite area, and specific asymptotic behavior.

Contribution

It proves new properties of complete surfaces with non-positive curvature outside compact sets and addresses a partial case of Milnor's conjecture regarding isometric immersions.

Findings

Complete surfaces have finite total curvature and area.

Ends of such surfaces are asymptotic to half-lines.

Provides partial solution to Milnor's conjecture.

Abstract

In this paper we extend Efimov's Theorem by proving that any complete surface in $\mathbb{R}^3$ with Gauss curvature bounded above by a negative constant outside a compact set has finite total curvature, finite area and is properly immersed. Moreover, its ends must be asymptotic to half-lines. We also give a partial solution to Milnor's conjecture by studying isometric immersions in a space form of complete surfaces which satisfy that outside a compact set they have non positive Gauss curvature and the square of a principal curvature function is bounded from below by a positive constant.