Isometric Immersion of Complete Surfaces with Slowly Decaying Negative Gauss Curvature

TL;DR

This paper establishes conditions under which complete surfaces with slowly decaying negative Gauss curvature can be smoothly immersed in three-dimensional Euclidean space, advancing understanding in differential geometry.

Contribution

It introduces a new approach using weighted Riemann invariants and a comparison principle to prove global isometric immersions under slow decay conditions.

Findings

Global smooth isometric immersion possible with slow decay of negative Gauss curvature.

Introduction of weighted Riemann invariants for the Gauss-Codazzi system.

Novel decay analysis leading to new immersion results.

Abstract

The isometric immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the isometric immersion problem is considered in this paper through the Gauss-Codazzi system for the second fundamental forms. It is shown that if the Gauss curvature satisfies an integrability condition, the surface has a global smooth isometric immersion in the three-dimensional Euclidean space even if the Gauss curvature decays very slowly at infinity. The new idea of the proof is based on the novel observations on the decay properties of the Riemann invariants of the Gauss-Codazzi system. The weighted Riemann invariants are introduced and a comparison principle is applied with properly chosen control functions.