Cartan's and Gauss's equations and rigidity theorems for isometric embeddings in low Sobolev regularity

Technical Summary: Cartan's and Gauss's Equations and Rigidity Theorems for Isometric Embeddings in Low Sobolev Regularity

Problem Statement
The paper addresses the validity of fundamental geometric equations—specifically Cartan's structural equations and the Gauss equation—for isometric embeddings of surfaces into $\mathbb{R}^3$ when the regularity of the embedding map is low. In the classical smooth setting, the relationship between intrinsic geometry (curvature) and extrinsic geometry (embedding) is governed by Cartan's structural equations and the Gauss equation. However, for embeddings with regularity below $C^2$, the Hessian determinant and connection 1-forms may not be well-defined in the classical sense. The authors investigate whether these equations hold in the sense of distributions for embeddings with low Sobolev regularity, specifically identifying the minimal regularity thresholds required to avoid "defect measures" (singularities that violate the equations).

Methodology
The authors employ a combination of distributional analysis, fractional Sobolev space theory, and geometric measure theory.

1. Distributional Framework for Cartan's Equations:

   • The authors define the connection 1-form $\omega$ for a coframe $\{\eta_i\}$ of regularity $C^0 \cap H^{1/2}$ using the interior product formula $\omega = \iota_{e_2}d\eta_1 - \iota_{e_1}d\eta_2$.
   • They utilize the lifting property of continuous $S^1$-valued maps (specifically, maps in $H^{1/2} \cap C^0$) to relate a rough orthonormal frame to a smooth reference frame via a rotation angle $\theta \in H^{1/2} \cap C^0$.
   • By approximating the rough frame with smooth frames and passing to the limit, they prove that the structural equations hold in the sense of distributions, provided the regularity is at least $C^0 \cap H^{1/2}$. This threshold is noted as optimal, referencing counterexamples in lower regularity (e.g., $W^{1,p}$ for $p<2$) where defect measures arise.

2. Derivation of the Gauss Equation:

   • The paper establishes a link between Cartan's second structural equation ($d\omega = K_g d\text{vol}_g$) and the Gauss equation for graphical representations $u = (\Psi, f)$.
   • A key technical step involves proving a chain rule for the distributional Jacobian determinant. For embeddings in $C^1 \cap W^{1+2/3, 3}$, the authors use standard approximation and chain rule arguments.
   • For the more delicate case of $c^{1, 1/2} \cap BV^2$, they develop a specific chain rule for the Jacobian determinant involving functions in little Hölder spaces and bounded variation. This relies on cancellation estimates for mollified derivatives and the specific structure of the Jacobian determinant as a distribution.

3. Removable Singularities:

   • The authors analyze the behavior of the Gauss equation near singular sets. They prove that if an embedding is sufficiently regular away from a set of zero $(s_0, p_0)$-capacity, the Gauss equation holds globally.

4. Rigidity and Convexity:

   • Applying the established Gauss equation, the authors adapt arguments from Conti–De Lellis–Székelyhidi and Pakzad. They utilize Pogorelov's theory of surfaces of bounded extrinsic curvature to deduce convexity and rigidity results for embeddings with non-negative Gaussian curvature ($K_g \ge 0$).

Key Contributions and Results

• Theorem 1.1 (Cartan's Equations): The authors prove that Cartan's structural equations hold in the sense of distributions for orthonormal coframes in $C^0 \cap H^{1/2}_{loc}$. This extends previous results (e.g., Giron [35] for $W^{1,p}$) and identifies $H^{1/2}$ as the critical threshold for the existence of a unique distributional connection 1-form satisfying the equations.
• Theorem 1.3 (Gauss's Equation): The paper establishes the validity of the Gauss equation $\det D^2 f = K_g(1+|Df|^2)^2$ for isometric embeddings $u$ with regularity:
  • $u \in c^{1, 1/2}_{loc} \cap BV^2_{loc}$, or
  • $u \in C^1 \cap W^{1+2/3, 3}_{loc}$.
    In both cases, the equation holds in the sense of distributions.
• Corollary 1.4: For embeddings in the $c^{1, 1/2} \cap BV^2$ class, the Gauss equation holds almost everywhere with respect to the Lebesgue measure, relating the absolutely continuous part of the Hessian measure to the curvature.
• Theorem 1.5 (Removable Singularities): The Gauss equation holds globally if the embedding is regular ($W^{1+2/3, 3}$) outside a compact set $S$ of zero $(s_0, p_0)$-capacity, where $s_0 = 2(1-s)$ and $p_0 = p/(p-2)$.
• Theorem 1.6 (Rigidity of Closed Surfaces): Any $C^1 \cap W^{1+2/3, 3}$ isometric embedding of a smooth sphere with $K_g \ge 0$ is a closed convex surface. If $K_g > 0$, the embedding is smooth ($C^\infty$). This extends Borisov's and Pakzad's rigidity theorems to lower regularity.
• Theorem 1.7 (Rigidity of Convex Caps): Similar rigidity and regularity results are proven for convex caps (domains with boundary) under planar boundary conditions, extending Han and Hong's smooth results.

Significance and Claims
The paper claims to provide the first rigorous extension of Cartan's structural equations to the $C^0 \cap H^{1/2}$ regime, a threshold known to be optimal for related Jacobian problems. By establishing the validity of the Gauss equation in low regularity classes ($C^1 \cap W^{1+2/3, 3}$ and $c^{1, 1/2} \cap BV^2$), the authors bridge a gap between the smooth theory and the highly irregular settings studied in the Nash-Kuiper theorem.

The significance lies in enabling the application of classical rigidity theorems (Cohn-Vossen, Borisov) to embeddings with lower regularity than previously proven. The authors explicitly note that their method differs from prior works (e.g., Pakzad [53]) by deriving the Gauss equation as a consequence of Cartan's equations rather than assuming it or using different approximation schemes. They modestly state that while they have identified specific regularity thresholds ($s=2/3, p=3$), the precise boundary for the validity of the Gauss equation for all $C^1$ isometric immersions in $W^{1+s, p}$ remains an open question for certain parameter ranges (e.g., $s=2/3, p \neq 3$). The work does not propose new experimental applications but rather solidifies the theoretical foundation for studying isometric embeddings in low regularity.