Isometric Immersions and Weak Solutions to the Darboux Equation

TL;DR

This paper investigates the Darboux equation related to isometric immersions of 2D manifolds into 3D space, extending classical results to low-regularity solutions with Hölder continuous metrics.

Contribution

It introduces a weak solution framework for the Darboux equation with low regularity and proves the classical correspondence persists in this regime.

Findings

Classical correspondence between solutions and immersions holds for $C^{1, heta}$ with $ heta>1/2$.

Extension of flatness criterion to Hölder continuous metrics.

Analysis of weak Gaussian curvature in low-regularity settings.

Abstract

We study the Darboux equation, a fundamental PDE arising in the theory of isometric immersions of two-dimensional Riemannian manifolds into $\mathbb{R}^3$, in the low-regularity regime. We introduce a notion of weak solution for $u\in C^{1,\theta}$ with $\theta>1/2$, and show that the classical correspondence between solutions of the Darboux equation and isometric immersions remains valid in this regime. The key ingredient is an extension of the classical flatness criterion to H\"older continuous metrics, achieved via an analysis of a weak notion of Gaussian curvature.