Trace and Extension Theorems for Homogeneous Sobolev and Besov Spaces for Unbounded Uniform Domains in Metric Measure Spaces

Gibara, Ryan; Shanmugalingam, Nageswari

doi:10.1134/S0081543823050061

Given a map f:X → M from a topological space X to a metric space M, a decorated Reeb space consists of the Reeb space, together with an attribution function whose values recover geometric information lost during the construction of the Reeb space. For example, when M = ℝ is the real line, the Reeb space is the well-known Reeb graph, and the attributions may consist of persistence diagrams summarizing the level set topology of f. In this paper, we introduce decorated Reeb spaces in various flavors and prove that our constructions are Gromov-Hausdorff stable. We also provide results on approximating decorated Reeb spaces from finite samples and leverage these to develop a computational framework for applying these constructions to point cloud data.

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