We study metric measure spaces that have quantitative topological control, as well as a weak form of differentiable structure. In particular, let X be a pointwise doubling metric measure space. Let U be a Borel subset on which the blowups of X are topological planes. We show that U can admit at most 2 independent Alberti representations. Furthermore, if U admits 2 Alberti representations, then the restriction of the measure to U is 2-rectifiable. This is a partial answer to the case n=2 of a question of the second author and Schioppa.
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This content will become publicly available on November 22, 2024
Identifying 1-rectifiable measures in Carnot groups
We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of the first author and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group, we develop tests that identify the part of the measure that is carried by rectifiable curves and the part of the measure that is singular to rectifiable curves. Our main result is entwined with an extension of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in R^2 (P. Jones, 1990), in R^n (K. Okikolu, 1992), or in an arbitrary Carnot group (the second author) in terms of local geometric least squares data called Jones' beta numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in R^n that charges a rectifiable curve in an arbitrary complete, doubling, locally quasiconvex metric space.
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- Award ID(s):
- 2154047
- NSF-PAR ID:
- 10508417
- Publisher / Repository:
- De Gruyter Open Access
- Date Published:
- Journal Name:
- Analysis and Geometry in Metric Spaces
- Volume:
- 11
- Issue:
- 1
- ISSN:
- 2299-3274
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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