Here you can find a list of open problems in the study of the geometry and topology of RCD-spaces discussed among the participants of the BIRS CMO Workshop Metric measure Spaces with Symmetry and Lower
For more information visit the report.
Given a Lie group action by measure preserving isometries, prove a suitable version of the Slice Theorem for RCD-spaces.
Yau’s conjecture: A Ricci curvature lower bound implies an L1-universal bound on scalar curvature. The Alexandrov case was settled by Petrunin in the affirmative.
The boundary conjecture: The boundary of an Alexandrov space is also an Alexandrov space with the same lower curvature bound.
Relaxed conjecture: The boundary of an Alexandrov space is an RCD-space with the appropriate curvature bound. This problem is also interesting for Ricci limit spaces, trying to use localization techniques.
Even more relaxed conjecture: Any totally geodesic subspace of an RCD-space with the induced intrinsic metric and an “enlarged” measure is again an RCD-space with the same curvature bound.
Address the “Fundamental gap” problem in the RCD-setting.
Collapsed vs. Non-collapsed: Given a metric measure space (X, d, m-Hausdorff measure) which satsfies the RCD(K, N ) condition for some K and N > m, is it true that the space also satisfies the RCD(K, m) condition? This is true in the smooth case by writing the Bakry-Emery estimate.
Given a principal fiber bundle over an RCD-space, give sufficient conditions on the fibers or total space to lift the RCD-condition to the total space.
Stratification of RCD-spaces vs. Stratification of Alexandrov spaces: Stratification of limit spaces is nicer, by Cheeger-Jiang-Naber.
Topology of 3-dimensional RCD-spaces: Mondino conjectures that they are homeomorphic to orbifolds, possibly with boundary.
Obtain the analogues of Finsler geometry results on CD-spaces; For example, show that there exists at least one tangent cone which is linearly isomorphic to the Euclidean n-space equipped with a norm.
Find a condition P for a metric measure space (X, d, m), which is weaker than the property “X is infinitesimal Hilbertian”, to ensure that if a metric measure space X satisfies the CD-condition and P , then the space is essentially non-branching.