Every RCD space with curvature bounded above is a topological manifold with boundary whose interior is the regular set, a smooth geodesically convex manifold.
New differential operator and non-collapsed $RCD$ spaces
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abstract
We show characterizations of non-collapsed compact $RCD(K, N)$ spaces, which in particular confirm a conjecture of De Philippis-Gigli on the implication from the weakly non-collapsed condition to the non-collapsed one in the compact case. The key idea is to give the explicit formula of the Laplacian associated to the pull-back Riemannian metric by embedding in $L^2$ via the heat kernel. This seems the first application of geometric flow to the study of RCD spaces.
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math.DG 1years
2019 1verdicts
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On the structure of RCD spaces with upper curvature bounds
Every RCD space with curvature bounded above is a topological manifold with boundary whose interior is the regular set, a smooth geodesically convex manifold.