[Paper Review] $C^0$-coerciveness of Moser's problem and smoothing area preserving homeomorphisms

Yong‐Geun Oh|ArXiv.org|Jan 9, 2006

Geometric and Algebraic TopologyMathematics9 references19 citations

TL;DR

This paper establishes $C^0$-coerciveness of Moser's problem for volume forms via weak topology of measures, proving that any area-preserving homeomorphism on a 2-dimensional surface can be $C^0$-approximated by area-preserving diffeomorphisms. The key result confirms that the group of symplectic homeomorphisms coincides with the group of area-preserving homeomorphisms in dimension two, resolving a conjecture in symplectic topology.

ABSTRACT

In this paper, we establish the $C^0$-coerciveness of Moser's problem of mapping one smooth volume form to another in terms of the weak topology of measures associated to the volume forms. The proof relies on our analysis of Dacorogna-Moser's solution to Moser's problem of mapping one volume form to the other with the same total mass. As an application, we give a proof of smoothing result of area preserving homeomorphisms and its parametric version in two dimension, (or more generally in any dimension in which the smoothing theorem of homeomorphisms is possible, e.g., in dimension 3 but not necessarily in dimension 4). This in turn results in coincidence of the area-preserving homeomorphism group and the symplectic homeomorphism group in two dimension.

Motivation & Objective

• To establish $C^0$-coerciveness of Moser’s problem in terms of the weak topology of measures associated with volume forms.
• To prove that any area-preserving homeomorphism on a compact 2-dimensional surface can be $C^0$-approximated by area-preserving diffeomorphisms.
• To resolve a conjecture in symplectic topology by showing that the group of symplectic homeomorphisms coincides with the group of area-preserving homeomorphisms in dimension two.
• To extend the smoothing result to parametric isotopies of area-preserving homeomorphisms in two dimensions.

Proposed method

• Utilizes Dacorogna-Moser’s solution to Moser’s problem as a foundation for constructing volume-preserving diffeomorphisms.
• Applies a coercive reformulation of the Dacorogna-Moser construction to control the $C^0$-norm of the resulting diffeomorphisms.
• Employs triangular maps and a linearization scheme on the cube to reduce the problem to local estimates.
• Uses the open mapping theorem and measure-theoretic estimates to bound the difference in volume forms under perturbations.
• Analyzes higher-order terms in the Dacorogna-Moser construction to control the $C^0$-norm of the solution.
• Establishes uniform bounds on the measure distortion using $\zeta$-dependent perturbations and $L^\infty$-norms of the vector field components.

Experimental results

Research questions

• RQ1 Can Moser’s problem be formulated in a $C^0$-coercive way with respect to the weak topology of measures?
• RQ2 Is every area-preserving homeomorphism on a 2-dimensional surface $C^0$-approximable by area-preserving diffeomorphisms?
• RQ3 Does the group of symplectic homeomorphisms coincide with the group of area-preserving homeomorphisms in dimension two?
• RQ4 Can the smoothing result be extended to isotopies of area-preserving homeomorphisms in two dimensions?
• RQ5 Does the measure-preserving property enable $C^0$-approximation of homeomorphisms by diffeomorphisms in higher dimensions?

Key findings

• The $C^0$-coerciveness of Moser’s problem is established via the weak topology of measures, ensuring that small $C^0$-perturbations of volume forms correspond to small $C^0$-perturbations of the diffeomorphism solving the problem.
• Any area-preserving homeomorphism on a compact 2-dimensional surface can be $C^0$-approximated by area-preserving diffeomorphisms, proving Theorem I′.
• The group of symplectic homeomorphisms $\mathrm{Sympeo}(\Sigma,\omega)$ coincides with the group of area-preserving homeomorphisms $\mathrm{Homeo}^\Omega(\Sigma)$ in dimension two.
• The parametric version of the smoothing result holds: any isotopy of area-preserving homeomorphisms can be $C^0$-approximated by a smooth isotopy of area-preserving diffeomorphisms.
• The proof extends to any compact manifold where diffeomorphic approximation of homeomorphisms is possible, such as in dimension 3, but not necessarily in dimension 4.
• A uniform bound on the $C^0$-norm of the Dacorogna-Moser solution is derived, with error terms controlled by $|\zeta \cdot u||u|$ and $\varepsilon_1(\zeta)$, confirming coerciveness.