Finite element methods for isometric embedding of Riemannian manifolds

TL;DR

This paper develops a finite element framework for numerically approximating isometric embeddings of 2D Riemannian manifolds with positive curvature into 3D space, including Ricci flow visualization, with proven convergence and error estimates.

Contribution

It introduces a new weak formulation and finite element scheme for Weyl's problem, establishing well-posedness, convergence, and error bounds, advancing numerical analysis in differential geometry.

Findings

Proved well-posedness and convergence of the finite element scheme.

Demonstrated effective numerical approximation of isometric embeddings.

Extended the framework to Ricci flow visualization with error estimates.

Abstract

The isometric embedding problem for Riemannian manifolds, which connects intrinsic and extrinsic geometry, is a central question in differential geometry with deep theoretical significance and wide-ranging applications. Despite extensive analytical progress, the nonlinear and degenerate nature of this problem has hindered the development of rigorous numerical analysis in this area. As the first step toward addressing this gap, we study the numerical approximation of Weyl's problem, i.e., the isometric embedding of two-dimensional Riemannian manifolds with positive Gaussian curvature into $\mathbb{R}^3$, by establishing a new weak formulation that naturally leads to a numerical scheme well suited for high-order finite element discretization, and conducting a systematic analysis to prove the well-posedness of this weak formulation, the existence and uniqueness of its numerical solution, as well as its convergence with error estimates. This provides a foundational framework for computing isometric embeddings of Riemannian manifolds into Euclidean space, with the goal of extending it to a broader range of cases and applications in the future. Our framework also extends naturally to the isometric embedding of the Ricci flow, with rigorous error estimates, enabling the visualization of geometric evolutions in intrinsic curvature flows. Numerical experiments support the theoretical analysis by demonstrating the convergence of the method and its effectiveness in simulating isometric embeddings of given Riemannian manifolds as well as Ricci flows.