Examples of Ricci limit spaces with non-integer Hausdorff dimension
Jiayin Pan, Guofang Wei
TL;DR
This paper constructs the first examples of collapsing Ricci limit spaces where the singular set's Hausdorff dimension surpasses that of the regular set, with the overall dimension possibly being non-integer, answering a key open question.
Contribution
It provides the first known examples of Ricci limit spaces with non-integer Hausdorff dimension and a singular set larger than the regular set, addressing a question by Cheeger-Colding.
Findings
Singular set can have larger Hausdorff dimension than the regular set.
Hausdorff dimension of Ricci limit spaces can be non-integer.
Examples demonstrate collapsing Ricci limit spaces with these properties.
Abstract
We give the first examples of collapsing Ricci limit spaces on which the Hausdorff dimension of the singular set exceeds that of the regular set; moreover, the Hausdorff dimension of these spaces can be non-integers. This answers a question of Cheeger-Colding about collapsing Ricci limit spaces.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Mathematical Dynamics and Fractals · Topological and Geometric Data Analysis