[论文解读] Finite element methods for isometric embedding of Riemannian manifolds

Guangwei Gao, Kaibo Hu+2 more|arXiv (Cornell University)|Feb 21, 2026

Model Reduction and Neural Networks被引用 0

一句话总结

论文提出了一种新的弱形式和用于 Weyl 的等距嵌入问题的高阶有限元离散化，证明了良定性和收敛性，并展示了数值收敛性和对 Ricci 流可视化的适用性。

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.

研究动机与目标

• 解决正高斯曲率的二维黎曼流形等距嵌入数值分析中的空缺。
• 开发适用于高阶有限元离散化的可构造弱形式。
• 证明数值方案的良定性并提供收敛/误差估计。
• 将框架扩展到演化度量的等距嵌入以及内在曲率流（如 Ricci 流）的可视化。

提出的方法

• 从等距嵌入方程的微分导出嵌入流的速度的变分问题新表述。
• 使用正交于无穷小刚性运动的速度场 v 以克服退化，进而得到鞍点系统。
• 在流形上建立离散 Korn 类型不等式以确保良定性。
• 用高阶拉格朗日元离散速度与嵌入，并用 Regge 元离散度量，与弹性复合一致。
• 给出收敛性分析，对多项式阶数 k ≥ 5 给出误差估计。

Figure 1 . Errors and convergence rate of the numerical scheme in ( 3.15 )

实验结果

研究问题

• RQ1 是否可以为等距嵌入问题设计一个适用于高阶有限元离散的良定性弱形式？
• RQ2 在何种条件下离散 Korn 型不等式能确保数值解的存在性和唯一性？
• RQ3 所提出的离散化是否获得收敛近似，并对 Weyl 问题给出可证明的误差估计？
• RQ4 该框架是否可扩展以可视化几何演化，如内在曲率流（如 Ricci 流）？

主要发现

• 一种新颖的弱形式产生一个鞍点系统，其良定性来自流形 Korn 型不等式。
• 半离散有限元方案对多项式阶数 k ≥ 5 给出带有误差估计的收敛性。
• 离散 Korn 不等式刻画离散无穷小刚性并支撑数值解的存在性与唯一性。
• 由于在法向方向处理退化，需要较高阶有限元（k ≥ 5）以实现收敛性。
• 该框架支持数值实验，验证收敛性并具备模拟等距嵌入与 Ricci 流演化的能力。

(a) $t=0$