[Paper Review] An implicit function theorem for Lipschitz mappings into metric spaces

Piotr Hajłasz, Scott Zimmerman|arXiv (Cornell University)|Sep 18, 2018

Geometric Analysis and Curvature Flows15 references4 citations

TL;DR

This paper establishes a metric implicit function theorem for Lipschitz mappings from Euclidean space into arbitrary metric spaces, replacing the classical rank condition with a positive upper n-density of the pullback of Hausdorff content. The key result shows that under this condition, locally, the mapping becomes equivalent to a projection onto the first n coordinates via a bi-Lipschitz change of variables and a Lipschitz quotient map to R^n, generalizing the classical IFT to metric space targets.

ABSTRACT

We prove a version of the implicit function theorem for Lipschitz mappings $f:\mathbb{R}^{n+m}\supset A o X$ into arbitrary metric spaces. As long as the pull-back of the Hausdorff content $\mathcal{H}_{\infty}^n$ by $f$ has positive upper $n$-density on a set of positive Lebesgue measure, then, there is a local diffeomorphism $G$ in $\mathbb{R}^{n+m}$ and a Lipschitz map $π:X o \mathbb{R}^n$ such that $π\circ f\circ G^{-1}$, when restricted to a certain subset of $A$ of positive measure, is a the orthogonal projection of $\mathbb{R}^{n+m}$ onto the first $n$-coordinates. This may be seen as a qualitative version of a similar result of Azzam and Schul. The main tool in our proof is the metric change of variables introduced in a paper of Hajlasz and Malekzadeh.

Motivation & Objective

• To extend the classical implicit function theorem to mappings into arbitrary metric spaces, where the derivative rank condition is replaced by a geometric density condition.
• To establish a structural characterization of Lipschitz mappings $ f: \mathbb{R}^{n+m} \to X $ when the pullback of $ \mathcal{H}_\infty^n $ has positive upper n-density on a set of positive measure.
• To show that such mappings locally factor through a bi-Lipschitz diffeomorphism and a Lipschitz projection onto $ \mathbb{R}^n $, preserving fiber structure and local bi-Lipschitz behavior.
• To demonstrate that the condition $ \Theta^{*n}(f,x) > 0 $ on a positive measure set is strictly stronger than $ \mathcal{H}^n(f(A)) > 0 $, and cannot be replaced by the latter for global bi-Lipschitz structure.

Proposed method

• The main tool is the metric change of variables technique from prior work, adapted to analyze the behavior of Hausdorff content under Lipschitz mappings.
• The proof relies on the upper n-density $ \Theta^{*n}(f,x) $, defined as the limsup of $ \mathcal{H}_\infty^n(f(B(x,r)\cap A)) / (\omega_n r^n) $, which acts as a generalized Jacobian.
• A key step uses the fact that $ \Theta^{*n}(f,x) > 0 $ implies positive n-dimensional content in small balls, enabling the construction of a bi-Lipschitz diffeomorphism $ G $ on a set $ K \subset A $ of positive measure.
• The construction ensures $ \pi \circ f \circ G^{-1} $ restricts to the orthogonal projection onto the first $ n $ coordinates on $ G(K) $, with $ \pi: X \to \mathbb{R}^n $ being $ \sqrt{n} $-Lipschitz.
• The proof establishes a quantitative lower bound: $ \|x_1 - x_2\|_\infty \leq d(F(x_1,y), F(x_2,y)) $ for $ F = f \circ G^{-1} $, ensuring injectivity on fibers.
• Counterexamples are constructed to show that the $ \Theta^{*n}(f,x) > 0 $ condition cannot be weakened to $ \mathcal{H}^n(f(A)) > 0 $, as global bi-Lipschitz structure fails in such cases.

Experimental results

Research questions

• RQ1 Can the classical implicit function theorem be generalized to mappings with values in arbitrary metric spaces?
• RQ2 What geometric condition on a Lipschitz map $ f: \mathbb{R}^{n+m} \to X $ ensures that it locally factors through a projection onto $ \mathbb{R}^n $ via a bi-Lipschitz change of variables?
• RQ3 How does the upper n-density $ \Theta^{*n}(f,x) $ serve as a substitute for the Jacobian determinant in the absence of differentiability?
• RQ4 Is the condition $ \Theta^{*n}(f,x) > 0 $ on a positive measure set sufficient to guarantee local bi-Lipschitz structure of fibers and projection behavior?
• RQ5 Can the $ \Theta^{*n}(f,x) > 0 $ condition be weakened to $ \mathcal{H}^n(f(A)) > 0 $ without losing the ability to construct a global bi-Lipschitz factorization?

Key findings

• If $ \Theta^{*n}(f,x) > 0 $ on a set of positive Lebesgue measure, then $ \mathcal{H}^n(f(A)) > 0 $, ensuring the image has positive n-dimensional content.
• There exists a set $ K \subset A $ of positive measure, a bi-Lipschitz $ C^1 $-diffeomorphism $ G $, and a $ \sqrt{n} $-Lipschitz map $ \pi: X \to \mathbb{R}^n $ such that $ \pi \circ f \circ G^{-1} $ acts as orthogonal projection onto the first $ n $ coordinates on $ G(K) $.
• The preimage of any point under $ f \circ G^{-1} $ is contained in a fiber $ \{x\} \times \mathbb{R}^m $, preserving the fiber structure of the projection.
• The restriction of $ f \circ G^{-1} $ to any $ \mathbb{R}^n \times \{y\} $ is bi-Lipschitz, with a quantitative lower bound: $ \|x_1 - x_2\|_\infty \leq d(F(x_1,y), F(x_2,y)) $.
• The condition $ \Theta^{*n}(f,x) > 0 $ is strictly stronger than $ \mathcal{H}^n(f(A)) > 0 $; counterexamples show that the latter does not guarantee a global bi-Lipschitz factorization.
• For $ \mathcal{H}^n $-a.e. $ x \in X $, the preimage $ f^{-1}(x) $ is countably $ \mathcal{H}^m $-rectifiable, a consequence of the main theorem.