Volume Preserving Maps

• Volume preserving maps are diffeomorphisms that keep the volume measure exactly constant, ensuring the Jacobian determinant remains 1 almost everywhere.
• They are constructed using methods like Moser's flow and energy minimization to accurately map complex domains in geometry and dynamics.
• These maps are essential in simulating divergence-free systems, modeling invariant structures, and advancing numerical integration algorithms.

A volume-preserving map is a diffeomorphism or continuous mapping between domains—typically manifolds, polyhedra, or subsets of Euclidean space—that exactly preserves the underlying volume measure (Lebesgue or more general densities). Formally, for a map $f: M \to N$ between $n$-dimensional manifolds $M,N$ with volume forms $\mu, \nu$, $f$ is volume-preserving if $f^*\nu = \mu$, or, in coordinates, $\det(Df(x)) \equiv 1$ almost everywhere. Such maps play an essential role across differential geometry, dynamical systems, elasticity, computational geometry, and the numerical integration of divergence-free vector fields.

1. Fundamental Properties and Definitions

Volume-preserving maps are defined by invariance of the volume form under pullback: for $f:M\to N$ and smooth volume forms $\omega_M$, $\omega_N$, $f^*\omega_N = \omega_M$ (or, commonly in Euclidean coordinates, $\det Df = 1$) (Tan et al., 1 Feb 2024, Policastro, 2017). For discrete, piecewise-linear maps (such as those on meshes), the condition specializes to the equality of pre-image and post-image simplex volumes up to prescribed density scaling.

For a local diffeomorphism $f: U \subset \mathbb{R}^n \to \mathbb{R}^n$, $f$ is volume-preserving if and only if $\det Df(x) = 1$ almost everywhere. In the context of measure theory, volume preservation may also be characterized by pushforward: $f_{\#}(\mu_M) = \nu_N$. For piecewise-simplicial/affine maps on meshes, volume preservation can be enforced tetrahedron-wise: $|f(\tau)| = |\,\tau\,|$ for all tetrahedra $\tau$ (Tan et al., 1 Feb 2024, Huang et al., 2022).

2. Analytical Constructions and Algorithmic Realization

Several distinct approaches exist for constructing or approximating volume-preserving maps, both in smooth and computational contexts:

2.1 Moser's Method and Correction Flows

Moser's flow constructs a volume-preserving diffeomorphism between two domains by first finding any orientation-preserving diffeomorphism and then correcting residual Jacobian distortions via integration of a suitable time-dependent vector field $v$, solving $\operatorname{div} v = 1 - \det Df_0$ with boundary conditions, and integrating (Sandhu et al., 2012). This approach is a backbone for algorithms mapping 3-manifolds onto canonical domains such as the unit cube or ball with $\det \nabla\varphi(x) = 1$ everywhere.

2.2 Variational and Energy Minimization Approaches

Energy minimization frameworks, such as volumetric stretch energy minimization (VSEM) or isovolumetric energy minimization (IEM), penalize deviations from uniform local volume scaling. The volumetric stretch energy is defined as

$E_V(f) = \sum_{\tau} \frac{|f(\tau)|^2}{|\tau|}$

with exact preservation achieved iff $|f(\tau)| = |\tau|$ on all elements (Tan et al., 1 Feb 2024, Huang et al., 2022, Liu et al., 27 Jul 2024). The IEM further incorporates global scaling into the objective. Algorithms alternate between boundary (spherical) and interior (ball) subproblems to numerically optimize $E_V$ under constraints (e.g., mapping boundary meshes to the sphere) (Tan et al., 1 Feb 2024, Liu et al., 27 Jul 2024).

2.3 Explicit Analytic Constructions: Polyhedra, Spheres, and Balls

Explicit, bijective, and volume-preserving maps have been constructed between polyhedral and spherical domains. As in Holhoș–Roșca (Holhoş et al., 2015), area-preserving maps are extended to volume-preserving via radial homotheties and shell-wise application of angular corrections, allowing the construction of uniform, refinable grids on the ball or recovery of established pixelizations like HEALPix.

Bijective analytic maps between a 3D ball and regular octahedron, together with their inverses, enable the construction of multiresolution orthonormal bases and uniform cells for $L^2(\mathbb{B}^3)$ (Holhos et al., 2019).

3. Dynamical Systems and Volume-Preserving Maps

Volume-preserving maps are the discrete-time analogs of divergence-free flows and are of central interest in multi-degree-of-freedom Hamiltonian and non-Hamiltonian dynamics.

3.1 Invariant Structures, Resonances, and Transport

In near-integrable settings, phase space is foliated by invariant tori which act as absolute transport barriers (Meiss, 2011, Fox et al., 2012, Dullin et al., 2010). Perturbations induce resonances, and the destruction of tori is marked by a breakdown cascade analogous to 2D symplectic systems, including a Greene-residue criterion in 3D (Fox et al., 2012). Volume-preserving twist condensation is subtler than the symplectic case and necessitates analysis of resonant crossings and nondegeneracy (Dullin et al., 2010).

3.2 Long-time Diffusion and Drift

In volume-preserving—but not symplectic—maps, Nekhoroshev-type exponential stability and confinement of actions can fail, with polynomial drift possible along resonant channels, revealed via averaging theory and numerics (Guillery et al., 2017). Accelerator modes and associated anomalous diffusion, characterized by algebraically-decaying trapping times and superdiffusive statistics, are generic to 3D volume-preserving maps with periodic directions (Meiss et al., 2018).

3.3 Stickiness and Anomalous Transport

Power-law trapping phenomena, well-studied in 2D maps, extend robustly to higher-dimensional volume-preserving contexts, exemplified by the mixed phase structure and partial barriers in the Arnold–Beltrami–Childress map (Das et al., 2019). Frequency-space analysis identifies resonance chambers responsible for distinct trapping time scales, and decomposition of Poincaré recurrences reveals the mechanisms underpinning stickiness and slow decay.

4. Volume-Preserving Maps in Geometry and Numerics

4.1 Generating Forms and Numerical Integrators

Volume-preserving integrators are crucial for simulating divergence-free dynamics. Exact volume-preserving maps in $\mathbb{R}^3$ can be generated using closed and exact one-forms, classified up to variable permutation and adjunction into five inequivalent classes (three admitting Hamiltonian/Lagrangian interpretations, two genuinely new) (Verdier et al., 2014). These underpin discrete-time schemes that inherit volume-preservation and first-order consistency for general divergence-free vector fields.

4.2 Registration, Atlas Construction, and Applications

Volumetric parameterizations with local and global volume conservation are foundational for solid registration and template construction in imaging and physical simulation. The projected gradient and FISTA-accelerated algorithms yield robust, scalable, and provably convergent methods for computing such maps, serving medical imaging (e.g., MRI brain atlas registration), shape deformation, and tracking incompressible deformation in materials (Huang et al., 2022, Liu et al., 27 Jul 2024).

5. Theoretical Results and Approximation Theory

5.1 Approximation by Volume-Preserving Maps

Mapping an arbitrary $W^{1,\infty}$ map to its nearest volume-preserving counterpart can be quantified in terms of the $L^p$ deviation of Jacobian determinant from one (Policastro, 2017). Quantitative Brenier-type decompositions and matrix-nearness results facilitate the construction of such approximations with explicit error control, with applications in incompressible elasticity and computational mechanics.

5.2 Structure Theorems and Rational Mappings

In algebraic geometry, rational endomorphisms of the projective plane that preserve a specific volume form (e.g., $\frac{dx}{x}\wedge\frac{dy}{y}$) have strong rigidity properties, conjecturally only permitting birational transformations, with necessary and sufficient criteria linked to second K-group invariance (Belousov, 2016).

5.3 Realization as Poincaré Maps

Every volume-preserving diffeomorphism isotopic to the identity on a compact manifold can be realized as the Poincaré map of a divergence-free flow on its suspension manifold (Treschev, 2019). The proof builds on nonautonomous suspension, isotopy, and Cartan's lemma, with implications for the correspondence between discrete-time and continuous-time dynamical systems in ergodic theory and geometry.

6. Tables: Landmark Algorithms for Volume-Preserving Maps

Algorithm/Construction	Principle	Key Reference
Moser's Flow Correction	PDE Integration	(Sandhu et al., 2012)
Volumetric Stretch Energy (VSEM)	Energy Minimization	(Tan et al., 1 Feb 2024, Huang et al., 2022)
Isovolumetric Energy Minimization	Preconditioned CG	(Liu et al., 27 Jul 2024)
Explicit Polyhedron–Ball Maps	Analytic Formulae	(Holhoş et al., 2015, Holhos et al., 2019)
Generating One-Form Integrators	Variational Splitting	(Verdier et al., 2014)

7. Connections, Limitations, and Open Problems

While concrete algorithmic and analytic frameworks exist for the construction and approximation of volume-preserving maps, several areas remain open:

• Generalization to high-genus and singular domains in computational algorithms
• Structural classification of all rational or algebraic volume-preserving mappings in $\mathbb{P}^n$
• Full characterization of invariant structures and diffusion processes in high-dimensional volume-preserving dynamical systems
• Theoretical guarantees of injectivity (global bijectivity) in mesh-based implementations and relaxation frameworks (Tan et al., 1 Feb 2024, Liu et al., 27 Jul 2024)

The geometric, analytic, and numerical study of volume-preserving maps continues to yield deep insights in both pure mathematics and computational science, with ongoing research advancing theory, algorithms, and application domains.