The Isometric Immersion of Negatively Curved Surfaces with Finite Total Curvature

TL;DR

This paper establishes conditions under which complete negatively curved surfaces with finite total curvature can be smoothly immersed in three-dimensional space, addressing longstanding mathematical challenges involving oscillating and slowly decaying curvature.

Contribution

It introduces a new sufficient condition involving finite total curvature and curvature oscillations, proving the existence of a global smooth isometric immersion for such surfaces.

Findings

Finite total Gauss curvature is necessary for immersion.

New techniques handle slow decay and oscillations of curvature.

Reformulation as a symmetric hyperbolic system enables analysis.

Abstract

In this paper, we study the smooth isometric immersion of a complete, simply connected surface with a negative Gauss curvature into the three-dimensional Euclidean space. A fundamental and longstanding problem is to find a sufficient condition for a complete negatively curved surface to be isometrically embedded in R^3 [67]. It can be described as an initial and/or boundary value problem for a hyperbolic system of nonlinear partial differential equations derived from the Gauss-Codazzi equations. The mathematical theory associated with this system is largely incomplete. The global smooth isometric immersion has been proven in the literature when the Gauss curvature decays rapidly and monotonically. However, when the Gauss curvature oscillates or decays slowly, the problem becomes much more challenging and little is known. In our paper, we find a sufficient condition, consisting of a finite total Gauss curvature and appropriate oscillations of the Gauss curvature. Under this condition we prove the global existence of a smooth solution to the Gauss-Codazzi system, achieving a global smooth isometric immersion of the surface into R^3. Furthermore, we show that the finite total Gauss curvature is necessary for the existence of a solution in a special case of the Gauss-Codazzi system. New techniques are developed to overcome the difficulties posed by the slow decay and oscillations of the Gauss curvature. By observing that certain combinations of the Riemann invariants decay faster than others, we reformulate the Gauss-Codazzi equations as a symmetric hyperbolic system and uncover a crucial structure of partial dampings. These partial dampings, along with the finite total curvature and appropriate oscillations of the Gauss curvature, enable us to obtain a global smooth solution through delicate analysis, and consequently establish a global smooth isometric immersion of such surfaces.