[Paper Review] Sobolev spaces of isometric immersions of arbitrary dimension and codimension

Robert L. Jerrard, Mohammad Reza Pakzad|arXiv (Cornell University)|May 19, 2014

Geometric Analysis and Curvature Flows13 references4 citations

TL;DR

This paper establishes the $C^{1}_{ ext{loc}}$ regularity and developability of $W^{2,p}_{ ext{loc}}$ isometric immersions of $n$-dimensional flat domains into $\mathbb{R}^{n+k}$ for $p \geq \min\{2k, n\}$, extending rigidity results to Sobolev spaces below $C^2$. It proves that such immersions are locally ruled surfaces (developable), and derives similar rigidity for scalar functions with Hessian rank bounded by $k$, including solutions to the degenerate Monge-Ampère equation $\det D^2u = 0$. The key contribution is a sharp regularity threshold for rigidity in arbitrary dimension and codimension.

ABSTRACT

We prove the $C^{1}$ regularity and developability of $W^{2,p}$ isometric immersions of $n$-dimensional flat domains into ${\mathbb R}^{n+k}$ where $p\ge \min\{2k, n\}$. Another parallel consequence of our methods is a similar regularity and rigidity result for the $W^{2,n}$ solutions of the degenerate Monge-Ampère equations in $n$ dimensions. The analysis also applies to the situations when the degeneracy is extended to $(k+1) imes (k+1)$ minors of the Hessian matrix and the solution is $W^{2,p}$, with $p\ge \min\{2k, n\}$.

Motivation & Objective

• To determine the critical regularity threshold at which rigidity in isometric immersions transitions from flexibility (à la Nash-Kuiper) to developability.
• To extend the theory of developable surfaces to higher-dimensional flat domains and higher codimensions using Sobolev regularity.
• To establish rigidity and regularity for scalar functions whose Hessian has rank bounded by $k$ a.e., particularly in the context of the degenerate Monge-Ampère equation $\det D^2u = 0$.
• To prove that $W^{2,p}_{\text{loc}}$ isometric immersions with $p \geq \min\{2k,n\}$ are locally $C^1$ and developable, even in arbitrary codimension.

Proposed method

• Uses $W^{2,p}_{\text{loc}}$ regularity and the structure of the Hessian matrix to analyze the rank of the second derivative in Sobolev spaces.
• Applies techniques from geometric measure theory, including $p$-capacity and quasicontinuity, to control the size of singular sets.
• Employs induction on codimension and foliation arguments to prove local developability via existence of flat affine planes on which the immersion is constant a.e.
• Utilizes the concept of dense weak flat foliation to link the behavior of the immersion to the rank of the Hessian matrix.
• Applies a generalized notion of weak developability through the use of connected components of level sets of the derivative and their projections.
• Relies on contradiction arguments involving capacity estimates and the structure of level sets to prove continuity of the blow-up limit $\overline{w}$.

Experimental results

Research questions

• RQ1 What is the sharp regularity threshold $p$ for $W^{2,p}$ isometric immersions to be $C^1_{\text{loc}}$ and developable in arbitrary dimension and codimension?
• RQ2 Can the rigidity of isometric immersions into $\mathbb{R}^{n+k}$ be extended from $C^2$ to $W^{2,p}$ for $p < 2$ when $k < n$?
• RQ3 To what extent does the condition $\text{rank}(D^2u) \leq k$ a.e. imply regularity and developability for scalar functions $u$?
• RQ4 Does the degenerate Monge-Ampère equation $\det D^2u = 0$ imply developability for $W^{2,p}$ solutions with $p \geq \min\{2(n-1), n\}$?
• RQ5 How does the $p$-capacity of singular sets relate to the regularity of the blow-up limit $\overline{w}$ in the context of weak developability?

Key findings

• For $W^{2,p}_{\text{loc}}$ isometric immersions $U: \Omega \to \mathbb{R}^{n+k}$ with $p \geq \min\{2k, n\}$, the immersion is $C^1_{\text{loc}}$ regular and locally developable, meaning it is foliated by straight lines in the direction of the first fundamental form.
• The result holds for arbitrary $n$ and $k$, generalizing classical developability results from $n=2$ and $k=1$ to higher dimensions and codimensions.
• The paper proves that scalar functions $u \in W^{2,p}_{\text{loc}}(\Omega)$ with $\text{rank}(D^2u) \leq k$ a.e. are $C^1_{\text{loc}}$ and developable under the same $p$-threshold, extending rigidity to Hessian rank constraints.
• The degenerate Monge-Ampère equation $\det D^2u = 0$ implies $C^1_{\text{loc}}$ regularity and developability for $u \in W^{2,p}_{\text{loc}}(\Omega)$ when $p \geq \min\{2(n-1), n\}$, corresponding to $k = n-1$.
• The proof relies on a contradiction argument using $p$-capacity: if the blow-up limit $\overline{w}$ were discontinuous, the capacity of a ball in the flat direction would be zero, contradicting the dimension $d \geq n - 2k + 1 > n - p$.
• The continuity of $\overline{w}$ on the set $F_{\geq j}$ is established via induction and the use of $p$-quasicontinuity, showing that limits along sequences in different level sets converge consistently.