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Finite element methods for isometric embedding of Riemannian manifolds

Guangwei Gao, Kaibo Hu, Buyang Li, Ganghui Zhang

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.

Finite element methods for isometric embedding of Riemannian manifolds

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 , 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.
Paper Structure (21 sections, 9 theorems, 176 equations, 3 figures)

This paper contains 21 sections, 9 theorems, 176 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Isometric embedding: background and notations
  3. Basic notations
  4. Problems to be addressed numerically
  5. Weak formulation and numerical scheme
  6. New weak formulation
  7. Finite element method
  8. Well-posedness and error estimates
  9. Discrete Korn inequality
  10. Well-posedness and error estimates of numerical scheme
  11. Well-posedness of the numerical scheme
  12. Discrete graph norm
  13. Defect estimates
  14. Error equation and continuity method
  15. Time derivative of the error
  16. ...and 6 more sections

Key Result

Theorem 2.1

Let $g(t)$ be a smooth Riemannian metric on $\mathcal{M}$, depending smoothly on $t\in[0,T]$, with smooth time derivative $\partial_t g(t)$. Assume that the Gaussian curvature of $g(t)$ remains strictly positive for all $t\in[0,T]$. Then there exists a unique smooth embedding flow $r(t):\mathcal{M}\

Figures (3)

  • Figure 1: Errors and convergence rate of the numerical scheme in \ref{['eq:num']}
  • Figure 2: Numerical simulation of the isometric embedding of the revolution metric \ref{['eq:rev_met']} at different time steps.
  • Figure 3: Visualization of the normalized Ricci flow via isometric embeddings. The surface evolves toward a round sphere corresponding to a constant-curvature metric.

Theorems & Definitions (25)

  • Theorem 2.1
  • Remark 2.1: Positive Gaussian curvature
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Remark 3.1: Evolution equation for isometric embedding
  • Remark 3.2: Graph norm on $H^1_T(\mathcal{M}; \mathbb{R}^3)$
  • Theorem 3.3
  • Remark 3.3: Isometric embedding of the Ricci flow with error estimates
  • Lemma 4.1: Discrete Korn inequality
  • ...and 15 more