[Paper Review] On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg target

Noel DeJarnette, Piotr Hajłasz+2 more|arXiv (Cornell University)|Sep 21, 2011

Geometric Analysis and Curvature Flows31 references4 citations

TL;DR

This paper establishes that Lipschitz mappings are dense in the Sobolev space $W^{1,p}(M, /mathbb{H}^n)$ if and only if $\dim M \leq n$, proving nondensity when $\dim M \geq n+1$ and $n \leq p < n+1$. The key mechanism involves constructing horizontal bi-Lipschitz embeddings of $\mathbb{S}^n$ into the Heisenberg group $\mathbb{H}^n$, which obstruct Lipschitz approximation due to nontrivial topology captured by Lipschitz homotopy groups.

ABSTRACT

We study the question: when are Lipschitz mappings dense in the Sobolev space $W^{1,p}(M,\mathbf{H}^n)$? Here $M$ denotes a compact Riemannian manifold with or without boundary, while $\mathbf{H}^n$ denotes the $n$th Heisenberg group equipped with a sub-Riemannian metric. We show that Lipschitz maps are dense in $W^{1,p}(M,\mathbf{H}^n)$ for all $1\le p

Motivation & Objective

• To determine the conditions under which Lipschitz mappings are dense in Sobolev spaces with Heisenberg group targets.
• To investigate the topological obstructions to Lipschitz approximation in $W^{1,p}(M, \mathbb{H}^n)$ via the geometry of horizontal embeddings.
• To introduce and study Lipschitz homotopy groups for sub-Riemannian spaces, particularly $\mathbb{H}^n$.
• To construct explicit horizontal bi-Lipschitz embeddings of spheres into $\mathbb{H}^n$ using complex and symplectic geometry.

Proposed method

• Construct smooth horizontal embeddings of $\mathbb{S}^n$ into $\mathbb{H}^n$ using complex hyperbolic geometry and symplectic structures.
• Use the existence of such embeddings to demonstrate that certain Sobolev maps cannot be approximated by Lipschitz maps.
• Apply degree theory and the Fundamental Theorem of Calculus along curves to analyze the image of approximating sequences.
• Employ Fubini’s theorem to restrict Sobolev norms to circles and control the $L^\infty$-norm of error terms.
• Use Kirszbraun-type extension theorems (Lang–Schroeder) to show that maps avoiding the $y$-axis in $\mathbb{H}^1$ extend Lipschitzly to the disk.
• Analyze the curvature and completeness of Riemannian components in $\mathbb{H}^n$ to justify extension theorems.

Experimental results

Research questions

• RQ1 When are Lipschitz maps dense in $W^{1,p}(M, \mathbb{H}^n)$ for a compact Riemannian manifold $M$?
• RQ2 What topological or geometric obstructions prevent Lipschitz approximation in Sobolev spaces with Heisenberg targets?
• RQ3 How do horizontal embeddings of spheres into $\mathbb{H}^n$ relate to the nontriviality of Lipschitz homotopy groups?
• RQ4 Can every Lipschitz map from $\mathbb{S}^1$ to $\mathbb{H}^1$ that avoids the $y$-axis be extended to a Lipschitz map on $\mathbb{B}^2$?
• RQ5 What role does the sub-Riemannian structure of $\mathbb{H}^n$ play in the density of Lipschitz maps?

Key findings

• Lipschitz maps are dense in $W^{1,p}(M, \mathbb{H}^n)$ if $\dim M \leq n$ and $1 \leq p < \infty$.
• Lipschitz maps are not dense in $W^{1,p}(M, \mathbb{H}^n)$ if $\dim M \geq n+1$ and $n \leq p < n+1$.
• The nondensity result is equivalent to the nontriviality of the $n$th Lipschitz homotopy group of $\mathbb{H}^n$.
• Explicit horizontal bi-Lipschitz embeddings of $\mathbb{S}^n$ into $\mathbb{H}^n$ were constructed using complex hyperbolic and symplectic geometry.
• Maps from $\mathbb{S}^1$ to $\mathbb{H}^1$ that avoid the $y$-axis admit Lipschitz extensions to $\mathbb{B}^2$, as shown via curvature and completeness arguments.
• The proof relies on a contradiction argument: if a Sobolev map were approximable by Lipschitz maps, its image would intersect the $y$-axis in positive measure, violating a key geometric constraint.