Characterization of foliations via disintegration maps

Abstract

In this paper, we present a novel approach for analyzing the relationship between the supports of conditional measures and their geometric arrangement in Wasserstein space via the disintegration map. Our method establishes criteria to determine when such conditional measures arise from a metric measure foliation. Additionally, we provide a example demonstrating how this framework can be applied to study perturbations of disintegration-induced foliations.

Figures (4)

• Figure 1: Disintegration of $\text{Leb}^2$ with respect to $\text{Leb}^1$: the red line $\{x\} \times [0,1]$ carries $\mu_x = \text{Leb}^1$.
• Figure 2: Illustration for the case in which the conditional measures do not have full support: $d(\pi^{-1}(y), \pi^{-1}(y')) = W_p(\mu_{y'}, \mu_{y"})$ for every $y', y" \in Y$, but $\{ \pi^{-1}(y) \}$ (the ellipses) are not a metric foliation.
• Figure 3: Normalized arc length in function of $\theta$ for different $\lambda$.
• Figure 4: Energy as a function of the eccentricity of the leaves.

Theorems & Definitions (12)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 4.1
• Definition 4.2
• Theorem A
• Remark 4.3
• proof
• Example 4.4