The semi-global isometric embedding of surfaces with curvature changing signs stably

TL;DR

This paper develops a method for semi-global isometric embedding of surfaces with Gaussian curvature that changes signs, using solutions to the Darboux equation, extending embedding techniques to more complex curvature scenarios.

Contribution

It introduces a new approach to semi-global isometric embedding for surfaces with curvature sign changes, which was not previously achievable.

Findings

Successfully embeds surfaces with curvature sign changes using Darboux equation

Extends isometric embedding theory to surfaces with finite order curvature sign changes

Provides a framework for stable embeddings in complex curvature conditions

Abstract

A semi-global isometric embedding of abstract surfaces with Gaussian curvature changing signs of any finite order is obtained through solving the Darboux equation.